arXiv:cond-mat/0603341v1 [cond-mat.str-el] 13 Mar 2006 Functional Renormalization-Group Analysis of Luttinger Liquids with Impurities Sabine Andergassen Dissertation accepted by the University of Stuttgart1 for the degree of Doctor of Natural Sciences Examiner: Prof. Dr. Walter Metzner Co-examiner: Prof. Dr. Ulrich Weiss Max-Planck-Institut f¨r Festk¨rperforschung u o Stuttgart, Germany 2006 1 Version without the German summary; full version available online at http://elib.uni-stuttgart.de/opus/volltexte/2006/2534/ Abstract In one-dimensional quantum wires the interplay of electron correlations and impurities strongly influences the low-energy physics. The diversity of energy scales and the competition of correlations in interacting Fermi systems can be treated very efficiently with the functional renormalization group (fRG), describing the gradual evolution from a microscopic model Hamiltonian to the effective low-energy action as a function of a continuously decreasing energy cutoff. The fRG provides the universal low-energy asymptotics as well as nonuniversal properties, and in particular an answer to the important question at what scale the ultimate asymptotics sets in. The lowest order truncation of the fRG hierarchy of flow equations consid1 ered previously for spinless fermions is generalized to spin- 2 systems and extended including renormalization of the two-particle interaction, in addition to renormalization of the impurity potential. The underlying approximations are devised for weak interactions and arbitrary impurity strengths. A comparison with numerical density-matrix renormalization results for systems with up to 1000 sites shows that the fRG is remarkably accurate even for intermediate interaction strengths. We investigate the influence of impurities on spectral and transport properties of fermionic lattice models with short-range interactions. The results capture relevant energy scales and crossover phenomena, in addition to the universal low-energy asymptotics. For weak and intermediate impurity strengths the asymptotic behavior is approached only at rather low energy scales, accessible only for very large systems. For spin- 1 systems two-particle backscattering leads to striking effects, 2 which are not captured if the bulk system is approximated by its low-energy fixed point, the Luttinger model. In particular, the expected decrease of spectral weight near the impurity and of the conductance at low energy scales is often preceded by a pronounced increase, and the asymptotic power laws are modified by logarithmic corrections. 2 Acknowledgments Several people contributed to the realization of the present thesis, and I would like to express my gratitude for their support during the last years. First of all, I would like to thank my supervisor Prof. Walter Metzner, who taught me to appreciate the beauty of simple ideas. I would like to thank him for his heedful supervision, for guiding my first steps in scientific research, for always having time for discussions, the numerous valuable suggestions, for his constant encouragement and support, and for the opportunity to attend several international workshops and conferences. It has been a great pleasure to work in his theory group, and I wish to thank for the opportunity to join his collaboration with Prof. Kurt Sch¨nhammer and Prof. Volker Meden in G¨ttingen, and Prof. Uli Schollw¨ck in o o o Aachen. It has been a particularly enriching experience and fruitful collabaration. I would like to thank Prof. Kurt Sch¨nhammer and Prof. Volker Meden for their supo port, the numerous discussions and encouragement, the stimulating correspondence, the careful proofreading of the manuscript, and the warm hospitality. To Prof. Uli Schollw¨ck I would like to express my thanks for his precious suggestions. o I would like to thank Prof. Ulrich Weiss for his willingness to co-report on the thesis. I wish to thank Prof. Manfred Salmhofer for the excellent lectures on the functional renormalization-group technique, and Prof. Carsten Honerkamp for his support and suggestions of further extensions and applications. A special thank goes to my officemate Dr. Tilman Enss, for the numerous discussions, his helpfulness and patient assistence in computer issues. It was a great pleasure to share these years. I would also like to thank him for a critical reading of the manuscript. Let me also commemorate Xavier Barnab´-Th´riault from G¨ttingen, with e e o whom I started, together with Dr. Tilman Enss, to implement a small shared library, and who unfortunately died in a tragic accident on August 15, 2004. He conveyed the curiosity and enthusiasm in approaching open problems. 3 I wish to thank Dr. Daniel Rohe for always finding time for my questions, his encouragement and a critical proofreading, Julius Reiss for his assistence and his sociocultural activities at the institute, Roland Gersch for a careful reading of the manuscript, Dr. habil. Karsten Held and Dr. habil. Dirk Manske for their help and encouragement, as well as all other members of the Department Metzner, for the numerous discussions and interesting conversations, their support and the friendly atmosphere. I would also like to thank our secretary Mrs. Ingrid Knapp for her help in all organizational matters, and the Computer Department of the Max Planck Institute for their support. I owe particular thanks to Prof. Carlo Di Castro, Prof. Claudio Castellani, and Dr. Massimo Capone in Rome for advices and valuable discussions. 4 Contents 1 Introduction 7 2 Impurities in Luttinger liquids 2.1 Luttinger liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Impurity effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Experimental realization . . . . . . . . . . . . . . . . . . . . . . . . . 11 11 14 17 3 Functional RG technique: a short overview 3.1 Introduction . . . . . . . . . . . . . . . . . 3.2 Generating functional . . . . . . . . . . . . 3.3 RG differential flow equation for Γ . . . . 3.4 Expansion in the fields and exact hierarchy 3.5 Comparison to other RG schemes . . . . . . . . . . 19 19 21 23 25 28 . . . . . . . . . . . . . . . . 30 30 31 32 33 33 34 39 42 48 49 50 53 58 58 59 60 4 Functional RG for Luttinger liquids 4.1 Microscopic models . . . . . . . . . . . 4.1.1 Spinless fermions . . . . . . . . 1 4.1.2 Spin- 2 fermions . . . . . . . . . 4.2 Cutoff and flow equations . . . . . . . 4.2.1 Cutoff . . . . . . . . . . . . . . 4.2.2 Truncation schemes . . . . . . . 4.2.3 Spinless fermions . . . . . . . . 1 4.2.4 Spin- 2 fermions . . . . . . . . . 4.2.5 Extension to finite temperature 4.3 Calculation of Kρ . . . . . . . . . . . . 4.3.1 Spinless fermions . . . . . . . . 1 4.3.2 Spin- 2 fermions . . . . . . . . . 4.4 Observables . . . . . . . . . . . . . . . 4.4.1 Single-particle excitations . . . 4.4.2 Density profile . . . . . . . . . . 4.4.3 Conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . of flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 Solution of fRG equations and results 5.1 Spinless fermions . . . . . . . . . . 5.1.1 Effective impurity potential 5.1.2 Local density of states . . . 5.1.3 Friedel oscillations . . . . . 5.1.4 Scaling of the conductance . 5.2 Spin- 1 fermions . . . . . . . . . . . 2 5.2.1 Single-particle excitations . 5.2.2 Density profile . . . . . . . . 5.2.3 Conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 62 62 65 73 77 81 81 88 90 102 A Evaluation of vertex flow for spin- 1 fermions 104 2 A.1 Functional RG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 A.2 One-loop g-ology calculation . . . . . . . . . . . . . . . . . . . . . . . 108 B Bethe-ansatz calculation of Kρ for the Hubbard model 111 Bibliography 114 6 1 Introduction In one dimension metallic electron systems are strongly affected by interactions. Differently from the conventional Fermi-liquid behavior, the generic low-energy physics is described by the Luttinger-liquid phenomenology [Giamarchi 2004]. For various correlation functions Luttinger-liquid theory predicts anomalous power laws; for spin-rotation invariant systems the exponents can be expressed in terms of a single interaction-dependent parameter Kρ . An important aspect concerns the peculiar effects due to the interplay of impurities and interactions. For Luttinger liquids with repulsive interactions already a single static impurity has a strong effect [Luther and Peschel 1974; Mattis 1974; Apel and Rice 1982; Giamarchi and Schulz 1988]. At low energy scales even a weak impurity effectively “cuts” the system into two parts with open boundary conditions at the end points, and physical observables are controlled by the open chain fixed point [Kane and Fisher 1992a,c]. In particular, the impurity potential becomes dressed by long-range oscillations leading to a characteristic power-law suppression of the local density of states near the impurity and the conductance through the impurity down to zero in the low-energy limit. The asymptotic behavior is universal in the sense that the exponents depend only on the properties of the bulk system via Kρ , while they do not depend on the impurity strength or shape. These power laws are generally modified by logarithmic corrections in the presence of two-particle backscattering. The asymptotic low-energy properties of Luttinger liquids with a single impurity are rather well understood. Universal power laws and scaling functions have been obtained by bosonization, conformal field theory and exact solutions for the lowenergy asymptotics in special integrable cases [Giamarchi 2004]. Numerical methods as exact diagonalization and the density-matrix renormalization group (DMRG) confirm the field-theoretical predictions and the validity of the underlying assumptions for microscopic fermionic systems with Luttinger-liquid behavior. The limited system size accessible to numerical solutions is however a serious constraint for a systematic analysis beyond the perturbatively accessible weak and strong-impurity regimes. The important question arises at what scale the ultimate asymptotics sets 7 1 Introduction in and asymptotic power laws are actually valid. That scale can indeed be surprisingly low, and the properties above it very different from the asymptotic behavior. Recently a functional renormalization group (fRG) method has been introduced for a direct treatment of microscopic models of interacting fermions, which does not only capture correctly the universal low-energy asymptotics, but allows to compute observables on all energy scales, providing thus also nonuniversal properties, and a possible key to the understanding of the behavior at intermediate scales accessible in experiments. Some of the nonuniversal properties can be computed numerically by the DMRG, but this method is limited to lattice systems with about 1000 sites, and only a restricted set of observables can be evaluated with affordable computational effort. The fRG provides a powerful computational tool to study interacting Fermi systems, especially low-dimensional systems with competing instabilities and entangled infrared singularities. Starting point is an exact hierarchy of differential flow equations for the Green or vertex functions of the system, describing the gradual evolution from the microscopic model Hamiltonian to the effective action as a function of a continuously decreasing energy cutoff introduced in the free propagator [Salmhofer 1998]. Approximations are then constructed by truncating the hierarchy and parametrizing the vertex functions with a manageable set of variables or functions. The fRG captures the expected universal power laws at low energy, as well as relevant energy scales and nonuniversal crossover phenomena at intermediate scales, as for the temperature dependence of the conductance through a double barrier [Enss et al. 2005; Meden et al. 2005]. The direct application to microscopic models allows for a flexible modeling of different geometries, as mesoscopic rings threaded by a magnetic flux [Meden and Schollw¨ck 2003a,b] and Y junctions o [Barnab´-Th´riault et al. 2005a,b]. e e In previous applications to spinless Luttinger liquids with impurities [Meden et al. 2002a,b] the fRG hierarchy of flow equations was truncated at first order, where the renormalized vertex is approximated by the bare interaction. Despite the simplicity of this scheme the effects of a single static impurity are captured qualitatively, and for spinless fermions in the weak coupling limit also quantitatively. It turned out that the asymptotic behavior typically holds only at very low energy scales and for very large systems, except for very strong bare impurities. In the present work we further develop and extend the fRG approach for Lut1 tinger liquids with impurities to spin- 2 fermions and include two-particle vertex 8 renormalization, in addition to the renormalization of the impurity potential. The underlying approximations are devised for weak interactions and arbitrary impurity strength. A comparison with exact numerical DMRG results for systems with up to 1000 sites shows however that the fRG with the inclusion of vertex renormalization is remarkably accurate even for intermediate interaction strengths. For spinless fermions this extension improves considerably the quantitative accuracy of 1 the results in particular at intermediate interaction strengths, whereas for spin- 2 systems vertex renormalization is necessary to take into account that backscattering of particles with opposite spins at opposite Fermi points scales to zero in the low-energy limit. Explicit flow equations are derived for various lattice fermion models supplemented by different types of impurity potentials. We present results for spectral properties of single-particle excitations, the oscillations in the density profile induced by impurities or boundaries and the linear conductance for chains with up to 106 lattice sites. Two-particle backscattering leads to peculiar effects, which are not captured if the bulk system is approximated by its low-energy fixed point, the Luttinger model. In particular, the expected decrease of spectral weight near the impurity and of the conductance at low energy scales is often preceded by a pronounced increase, and the asymptotic power laws are modified by logarithmic corrections. The outline of the thesis is as follows. • In Chapter 2 we give a short overview on general aspects of Luttinger liquids with impurities. • The fRG formalism is developed in Chapter 3. We briefly review the fRG for interacting Fermi systems, and derive the hierarchy of differential flow equations for the one-particle irreducible (1PI) vertex functions. • In Chapter 4 we describe the implementation of the fRG technique for various one-dimensional microscopic lattice models with impurities, providing details on the parametrization of the two-particle vertex and different truncation schemes. Parts of this chapter are published in S. Andergassen, T. Enss, V. Meden, W. Metzner, U. Schollw¨ck, and K. Sch¨no o hammer, Functional renormalization group for Luttinger liquids with impurities, Phys. Rev. B 70, 075102 (2004), cond-mat/0403517; 9 1 Introduction S. Andergassen, T. Enss, V. Meden, W. Metzner, U. Schollw¨ck, and K. Sch¨no o hammer, Renormalization group analysis of the one-dimensional extended Hubbard model with a single impurity, cond-mat/0509021. • In Chapter 5 we present results for spectral properties of single-particle excitations near an impurity or boundary, the density profile, and transport properties in the presence of a single and a double barrier. In the first part we focus on spinless fermions; the modifications due to the spin degree of freedom are addressed in the second part. Parts of this chapter are presented in the above publications, and for the conductance in V. Meden, S. Andergassen, W. Metzner, U. Schollw¨ck, and K. Sch¨nhammer, o o Scaling of the conductance in a quantum wire, Europhys. Lett. 64, 769 (2003), cond-mat/0303460; V. Meden, T. Enss, S. Andergassen, W. Metzner, and K. Sch¨nhammer, o Correlation effects on resonant tunneling in one-dimensional quantum wires, Phys. Rev. B 71, 041302(R) (2005), cond-mat/0403655; T. Enss, V. Meden, S. Andergassen, X. Barnab´-Th´riault, W. Metzner, and e e K. Sch¨nhammer, Impurity and correlation effects on transport in one-dimeno sional quantum wires, Phys. Rev. B 71, 155401 (2005), cond-mat/0411310; S. Andergassen, T. Enss, and V. Meden, Kondo physics in transport through a quantum dot with Luttinger liquid leads, cond-mat/0509576. • We conclude in Chapter 6 with a summary and an outlook on further applications and extensions of the present work. 10 2 Impurities in Luttinger liquids The exactly soluble Luttinger model provides a generic scenario for one-dimensional Fermi systems with repulsive interactions, denoted as “Luttinger liquid”. The lowenergy physics is completely determined by a few interaction-dependent characteristic parameters describing the power-law exponents of the correlation functions. Already a single static impurity leads to peculiar modifications of the electronic properties of Luttinger liquids. Even for a weak impurity potential, physical observables behave as if the system is split into two parts in the low-energy limit. The local density of states near the impurity and the conductance through the impurity vanish as power laws. 2.1 Luttinger liquids In one-dimensional interacting Fermi systems Fermi-liquid theory is not valid. The breakdown of Fermi-liquid theory is indicated already in second order perturbation theory, where the reduction of the quasi-particle weight at the Fermi surface due to interactions diverges logarithmically. These divergencies can be treated by a weak-coupling renormalization-group method applied to an effective low-energy theory known as g-ology model [S´lyom 1979]. Depending on the values of the o bare couplings, the renormalized couplings flow either to strong coupling, and hence out of the perturbatively controlled regime, or to a fixed-point Hamiltonian, the exactly soluble Luttinger model [Tomonaga 1950; Luttinger 1963; Mattis and Lieb 1965]. The term “Luttinger liquid” has been introduced for the latter systems, in analogy with the mapping of low-energy states of interacting electron systems onto the Fermi gas in higher dimensions for Fermi liquids [Haldane 1981b]. The normal 11 2 Impurities in Luttinger liquids gapless metallic phase is characterized by i) a continuous momentum distribution with a power-law singularity at the Fermi surface, described by a nonuniversal exponent α; ii) a single-particle density of states which vanishes as |ω|α near the Fermi energy, implying the absence of fermionic quasi-particles; iii) finite charge and spindensity responses for long wavelengths and the existence of collective bosonic charge and spin-density modes; iv) power-law singularities in various correlation functions with interaction-dependent exponents; v) separation of spin and charge degrees of freedom. There are several good reviews on one-dimensional Fermi systems, recent reviews are presented in Refs. [Voit 1995; Giamarchi 2004]. In the following we will summarize the most important results. Theoretical work on interacting fermions in one dimension has progressed along different lines. Besides the perturbative investigation of the weak-coupling limit [S´lyom 1979], Luttinger-liquid theory is usually formulated using the bosonization o technique [Mattis and Lieb 1965; Haldane 1980, 1981a,b; Luther and Peschel 1974; Mattis 1974]. A different approach is based on the Bethe-ansatz method for special integrable models [Giamarchi 2004]. The computation of correlation functions is however very difficult from the complicated expressions for the eigenfunctions. As proposed in a seminal work by Haldane [Haldane 1981b], the low-energy physics of the Luttinger model is generic for interacting fermions in one dimension with repulsive interactions. In the language of the renormalization group the Luttinger model Hamiltonian is the fixed-point Hamiltonian for a large class of onedimensional fermions with repulsive interactions. The Luttinger model can be solved exactly at any interaction; it is characterized by a linear dispersion relation, and the electron-electron interaction is limited to forward scattering only [Giamarchi 2004]. Umklapp and backscattering processes, as well as additional terms for more general models arising from band curvature are irrelevant and vanish in the low-energy limit [Haldane 1981b]. As in the Landau Fermi liquid a few parameters completely determine the low-energy physics. The charge degrees of freedom of Luttinger liquids are described by a sound velocity vρ and the dimensionless parameter Kρ , and the spin degrees of freedom are characterized by a spin-wave velocity vσ and Kσ . All correlation functions are uniquely parametrized by Kν and the velocities of the collective modes vν , with ν = ρ, σ, in the low-energy limit; the corresponding exponents are determined by Kν . For noninteracting particles Kρ = Kσ = 1. In the absence of a magnetic field, the ground state is spin-rotationally invariant and Kσ = 1, while Kρ < 1 (> 1) for repulsive (attractive) forces. 12 2.1 Luttinger liquids For Kσ = 1 the momentum distribution function exhibits a power-law singular−1 ity at the Fermi level with exponent α = (Kρ + Kρ − 2)/4 for any nonvanishing interaction [Giamarchi 2004]. For α < 1 the momentum distribution function near kF obeys a power law |n(k) − n(kF )| ∼ |k − kF |α . (2.1) The spectral function has the form N(ω) ∼ |ω|α (2.2) in the low-energy limit. Landau quasi-particle excitations are absent in the Luttinger liquid. The power laws hold also for nonsoluble generalizations of the model with a nonlinear dispersion [Haldane 1981b]. In the Luttinger model the charge and spin density modes are exact undamped eigenstates, and any excited state of the model is a superposition of these elementary excitations. This becomes particularly explicit in the bosonized form of the Luttinger model [Mattis and Lieb 1965]. The Luttinger model Hamiltonian conserves charge and the z component of spin separately on each Fermi point. Charge and spin excitations are completely independent, as the respective terms in the Hamiltonian commute. This phenomenon is called “spin-charge separation” [Giamarchi 2004], charge and spin propagate with different velocities. Concerning the leading low-energy long-wavelength response functions there is no difference between Fermi and Luttinger liquids [Giamarchi 2004]. Thermodynamic properties as the compressibility and the susceptibility do not differ from the Fermi-liquid description and the modification due to the interaction leads to renormalized coefficients depending on Kν and vν . Differences between Fermi and Luttinger-liquid behavior arise only from the enhanced phase space for forward scattering in one dimension. Marked differences appear in the single-particle propagator, which determines the momentum distribution function and the spectral density for single-particle excitations. In a Fermi liquid residual interactions modify the propagator only on a subleading level, leading for example to a small quasi-particle decay rate, while in a Luttinger liquid forward scattering affects the leading low-energy behavior. Another distinctive feature of Luttinger liquids is the singular behavior of density correlations with momenta near 2kF [Giamarchi 2004]. Conservation laws play a crucial role in one-dimensional Fermi systems [Metzner et al. 1998]. In addition to the usual charge and spin conservation the discrete structure 13 2 Impurities in Luttinger liquids of the Fermi surface in one dimension leads to an additional conservation law: separate charge conservation in low-energy scattering processes for particles near the left and right Fermi points, respectively. Separate spin conservation is spoiled by the backscattering process generally present in models of spin- 1 fermions. In most 2 cases of interest, in particular for the models considered in the present work, the backscattering amplitude scales to zero at low energies, and the separate spin conservation is restored asymptotically. The velocities associated with the corresponding conserved currents provide a complete parametrization of the low-energy physics [Haldane 1981b; Metzner and Di Castro 1993]. 2.2 Impurity effects An important aspect of Luttinger-liquid behavior concerns the peculiar modification of the electronic properties in the presence of impurities. For Luttinger liquids with repulsive interactions (Kρ < 1) already a single static impurity has a strong effect at low energy scales, even if the impurity potential is relatively weak [Luther and Peschel 1974; Mattis 1974; Apel and Rice 1982; Giamarchi and Schulz 1988; Kane and Fisher 1992a,c; Furusaki and Nagaosa 1993a,b; Yue et al. 1994]. In general the interplay of disorder and interactions is still a challenging issue, although the properties of noninteracting disordered electronic systems are rather well understood. In one-dimensional noninteracting systems disorder leads to localization of all electrons; the localization length characterizing the exponential decay of the wave function is of the same order as the mean free path [Giamarchi 2004]. On the other hand interactions strongly affect the properties of the pure system, leading to Luttinger-liquid behavior. Thus in one dimension a particularly strong mutual influence of disorder and interactions is expected. Relevant parameters in the description of disorder are the strength of the individual impurity V and the impurity density nimp . The variation of these two parameters leads to different physical effects. In the limiting case of very weak individual impurities with a dense distribution the effect of a single impurity is negligible and collective effects dominate; the corresponding relevant length scale is ∼ 1/nimp. As a consequence of the central-limit theorem, for continuous systems the disorder can be described by a Gaussian distribution in the limit nimp → ∞ and V → 0 for constant nimp V 2 measuring the disorder strength [Giamarchi 2004]. The main results 14 2.2 Impurity effects for Gaussian disorder from a perturbative treatment can be summarized as follows [Giamarchi 2004]. Interactions are effectively renormalized by disorder, which is reversely affected by interactions. Repulsive interactions generally enhance localization whereas attractive ones reduce this effect. For spinless fermions attractive interactions enhance superconducting fluctuations, leading to an effective screening of the disorder. For spin- 1 fermions a competing effect arises. The tendency 2 towards a uniform charge distribution inhibits the coupling to disorder, leading to an increase in the localization length for strong interactions in the pure Hubbard model. For the extended Hubbard model with a local as well as nearest-neighbor interaction this effect is reduced. The opposite limit examined in the present work corresponds to strong and dilute impurities. In this case collective effects do not play any role and the problem essentially reduces to a single isolated impurity. An interesting unsolved problem concerns the combination of single impurity and collective effects at intermediate scales: depending on whether collective effects become important before the individual impurities renormalize to high barriers, a different characteristic behavior is expected. In the following we consider the case of a single or double impurity, where the effects of coherent scattering from many impurities are absent. The asymptotic low-energy properties of Luttinger liquids with a single impurity have been investigated by mapping the problem onto an effective field theory, where terms which are expected to be irrelevant in the low-energy limit are neglected. For attractive interactions the impurity is irrelevant in the renormalization-group sense and scales to zero at low energies. For repulsive electron systems with Kρ < 1 the essential properties from the perturbative bosonic renormalization-group calculation and the boundary conformal field-theory analysis can be summarized as follows. The backscattering amplitude generated by a weak impurity is a relevant perturbation which grows as Λ(Kρ −1)/z , for a decreasing energy scale Λ, where z is the number of spin components. This behavior can be traced back to the power-law singularity of the 2kF density response function in a Luttinger liquid. On the other hand, the tunneling amplitude through a weak link between two otherwise separate wires is irrelevant and scales to zero as ΛαB , with the boundary exponent 1 −1 αB = (Kρ − 1) z (2.3) depending only on the interaction strength and band filling, but not on the impurity parameters. At low energy scales any impurity thus effectively “cuts” the system into 15 2 Impurities in Luttinger liquids two parts with open boundary conditions at the end points, and physical observables are controlled by the open chain fixed point. In particular, the local density of states near the impurity is suppressed as ρ(ω) ∼ |ω|αB (2.4) for |ω| → 0. Long-range Friedel oscillations in the density profile induced by boundaries or impurities decay with a power law at long distances [Egger and Grabert 1995] as n(x) ∼ x−Kρ (2.5) for spinless fermions, where x measures the distance from the impurity or boundary. 1 For spin- 2 fermions Kρ is replaced by (Kρ + 1)/2 in Eq. (2.5). The conductance through an infinite Luttinger liquid with a single impurity vanishes at low temperatures as G(T ) ∼ T 2αB . (2.6) The conductance through a single impurity of variable strength can be collapsed onto a single curve by a one-parameter scaling ansatz. For resonant scattering at double barriers the distance between the two barriers and the detuning from resonance introduce additional scales and a more complex behavior is observed. The Lorentzian resonance line shape for noninteracting electrons is modified by the interaction, and for appropriate parameters the conductance exhibits distinctive power-law scaling as a function of temperature [Kane and Fisher 1992b; Furusaki and Nagaosa 1993a; Furusaki 1998; Nazarov and Glazman 2003; Polyakov and Gornyi 2003; Yue et al. 1994]. Note that the above power laws are strictly valid only in the absence of two1 particle backscattering. For spin- 2 fermions they are in general modified by logarithmic corrections. The asymptotic behavior is universal in the sense that the exponents depend only on the properties of the bulk system, via Kρ , while they do not depend on the impurity strength or shape, except in special cases such as resonant scattering at double barriers, which require fine-tuning of parameters. The asymptotic low-energy properties of Luttinger liquids with a single impurity are rather well understood. Universal power laws and scaling functions have been obtained by bosonization, conformal field theory, and exact solutions for the low-energy 16 2.3 Experimental realization asymptotics in special integrable cases [Giamarchi 2004]. Numerical results from exact diagonalization and DMRG applied to the lattice model of spinless fermions with nearest-neighbor interaction confirmed the field theoretical scenario and the validity of the underlying assumptions [Eggert and Affleck 1992; Meden et al. 1998]. These methods are however limited to lattice systems with about 1000 sites and do not allow for a systematic analysis of the crossover between the weak and strong-impurity limit. Moreover, only a restricted set of observables can be evaluated with affordable computational effort. In this context the fRG provides a complementary technique for microscopic models of interacting fermions with impurities, which does not only capture correctly the universal low-energy asymptotics, but allows one to compute observables on all energy scales, providing thus also nonuniversal properties, and in particular an answer to the important question at what scale the ultimate asymptotics sets in. That scale can indeed be surprisingly low, and the properties above it very different from the asymptotic behavior. 2.3 Experimental realization The progress in the fabrication of artificial low-dimensional structures led to advanced experimental verification of the theoretical predictions. We present a short list of the most promising systems and of the employed experimental techniques. For a detailed discussion and references to the most recent publications and review articles on the subject we refer to Ref. [Sch¨nhammer 2004]. o Strictly one-dimensional systems are a theoretical idealization, the coupling to an experimental probe as well as the coupling between several Luttinger liquids is not completely understood [Giamarchi 2004]. The coupling between the chains in a strongly anisotropic three-dimensional compound leads to the development of long-range order at very low temperatures in the phase for which the algebraic decay of the corresponding correlation function of the single-chain Luttinger liquid is the slowest. In appropriate temperature and energy regimes Luttinger-liquid behavior can be expected in several systems with a predominantly one-dimensional character, as highly anisotropic quasi one-dimensional conductors, organic conductors like the Bechgaard salts, as well as inorganic materials, artificial quantum wires in semiconductor heterostructures or on surface substrates, carbon nanotubes, and fractional quantum Hall fluids [Sch¨nhammer 2004]. In particular, single-wall caro 17 2 Impurities in Luttinger liquids bon nanotubes are expected to show Luttinger-liquid behavior with Kρ ∼ 0.2 − 0.3 down to very low temperatures, despite the presence of two low-energy channels [Egger and Gogolin 1997; Kane et al. 1997]. Experimental techniques used to verify Luttinger-liquid behavior involve mainly high resolution photoemission and transport measurements, in addition to optical properties [Sch¨nhammer 2004]. A careful analysis of experimental data indicating o power-law behavior and signatures of spin-charge separation reveals partly inconsistent interpretations. The discussion on the modification of the quantized value e2 /h for noninteracting electrons in a single channel by the interaction to Kρ (e2 /h) indicates a sensitive dependence on the schematization of the contacts, a challenging theoretical as well as experimental problem [Sch¨nhammer 2004]. Experimental o results for cleaved-edge overgrowth quantum wires and carbon nanotubes indicate power laws of the conductance consistent with Luttinger-liquid behavior. In the last few years, ultracold gases in optical lattices have opened up an entirely new area of physics, where strong correlations can be studied with unprecedented flexibility and control of the parameters. Further work is necessary for clear experimental evidence of Luttinger-liquid behavior. 18 3 Functional RG technique: a short overview We review the functional renormalization-group approach for interacting Fermi systems in the 1PI version. Introducing an infrared cutoff Λ in the free propagator and differentiating the effective action with respect to Λ, an exact hierarchy of differential flow equations for the 1PI vertex functions is derived, describing the gradual evolution from the microscopic model Hamiltonian to the effective action as a function of the continuously decreasing energy cutoff. We briefly discuss the relation to alternative formulations of the fRG approach. 3.1 Introduction The renormalization-group is a powerful method in the study of low-dimensional Fermi systems, providing in particular a systematic and unbiased method to study competing instabilities and entangled infrared singularities at weak coupling. Early renormalization-group approaches for one-dimensional systems, combined with exact solutions of fixed-point models, have been a major source of physical insight [S´lyom o 1979; Giamarchi 2004]. From the renormalization-group point of view, the existence of the Luttinger liquid requires the cancellation of contributions to the flow of the two-particle vertex to all orders [Metzner et al. 1998]. This is a one-dimensional phenomenon, in higher dimensions the interactions in general diverge and the flow in the fermionic variables breaks down indicating a possible opening of a gap in the fermionic excitation spectrum. Wilson’s renormalization-group approach [Wilson 1971; Wilson and Kogut 1974] of successive integration of degrees of freedom with different energy scales determines the evolution of the bare action of the system, given by the microscopic 19 3 Functional RG technique: a short overview Hamiltonian, to the final effective action, from which all physical quantities can be extracted. The hierarchy of coupled differential flow equations for the Green or vertex functions describing the full functional evolution of the effective action has been first implemented for bosonic field theories in the context of critical phenomena [Wegner and Houghton 1973; Polchinski 1984; Wetterich 1993]. The intuition of the relevance of fRG methods for interacting Fermi systems followed in the 1990s [Benfatto and Gallavotti 1990; Feldman and Trubowitz 1990; Shankar 1991, 1994], together with important rigorous work [Salmhofer 1999]. The infinite hierarchy of flow equations can be solved exactly only in special cases, for instance the Luttinger model [Sch¨ tz et al. 2004]. Truncations however preserve the successive handling of u energy scales and the consequent treatment of infrared singularities, characteristic of a renormalization-group treatment. There are several variants of the fRG flow equations. The flow equations for the connected amputated Green functions correspond to the Polchinski scheme, first derived in Ref. [Polchinski 1984; Keller et al. 1992]. The expansion of the connected amputated Green functions in 1PI vertex functions led to the respective flow equations [Wegner and Houghton 1973; Weinberg 1976], subsequently derived from the Legendre transform of the generating functional [Wetterich 1993; Morris 1994; Salmhofer and Honerkamp 2001]. The Wick-ordered scheme is obtained from the Polchinski scheme by expanding the generating functional of the connected amputated Green functions in Wick-ordered polynomials [Wieczerkowski 1988; Salmhofer 1998, 1999]. Important applications of the fRG in condensed-matter physics include the two-dimensional Hubbard model using the Polchinski scheme [Zanchi and Schulz 1998, 2000], the Wick-ordered scheme [Halboth and Metzner 2000] and also the 1PI scheme [Honerkamp et al. 2001]. In the context of classical disordered systems a fRG approach is necessary to overcome the problem of dimensional reduction [Wiese 2003]. One-dimensional impurity problems and Luttinger-liquid physics are most conveniently investigated in the 1PI scheme, as self-energy contributions are included to all orders. In the following the hierarchy of differential flow equations for the 1PI vertex functions is derived, which is obtained by differentiating the corresponding generating functional with respect to an infrared cutoff introduced in the free propagator [Salmhofer 1998]. 20 3.2 Generating functional 3.2 Generating functional We consider a system of interacting fermions with single-particle propagator of the noninteracting system G0 . The properties of the system are determined by the action ¯ ¯ ¯ S[ψ, ψ] = (ψ, G−1 ψ) − V [ψ, ψ] , 0 (3.1) ¯ where ψ and ψ are Grassmann variables associated with creation and annihilation ¯ operators, and V [ψ, ψ] is an arbitrary many-body interaction. Here we introduced ¯ ¯ the short-hand notation (ψ, G−1 ψ) = K,K ′ ψK [G−1 ]K,K ′ ψK ′ , where K contains the 0 0 Matsubara frequency in addition to the single-particle quantum numbers and K stands for summation over the discrete indices and integrals over the continuous ones. All connected Green functions are obtained from the generating functional [Negele and Orland 1987] defined by η e−G[η,¯] = = 1 Z0 ¯ ¯ η ¯ dψdψ eS[ψ,ψ] e−(ψ,η)−(¯,ψ) ¯ ¯ η ¯ dµQ [ψ, ψ] e−V [ψ,ψ] e−(ψ,η)−(¯,ψ) , (3.2) with Grassmann source terms η and η . The normalized Gaussian measure with ¯ −1 covariance Q = G0 1 ¯ ¯ ¯ dψdψ e(ψ,Qψ) dµQ [ψ, ψ] = Z0 (3.3) includes the exponential of the quadratic part of the action and the noninteracting ¯ partition function Z0 , such that dµQ [ψ, ψ] = 1. The generating functional for the connected Green functions is related to the partition function of the physical system with action (3.1) by G[η, η] = −ln Z[η, η] . ¯ ¯ (3.4) ¯ In the noninteracting case V [ψ, ψ] = 0, and the Gaussian integral ¯ η η ¯ dµQ [ψ, ψ] e−(ψ,η)−(¯,ψ) = e−(¯,G0 η) (3.5) implies that G[η, η ] = (¯, G0 η). ¯ η 21 3 Functional RG technique: a short overview The connected m-particle Green functions are given by the derivatives of the generating functional G[η, η ] with respect to the source terms at η = η = 0 ¯ ¯ ′ ′ ¯ ′ ′ ¯ Gm (K1 , . . . , Km ; K1 , . . . , Km ) = (−1)m ψK1 . . . ψKm ψKm . . . ψK1 = c ∂m ∂m G[η, η] ¯ ′ ′ ¯ ∂ηK1 . . . ∂ηKm ∂ ηKm . . . ∂ ηK1 ¯ , η=¯=0 η (3.6) where . . . c is the connected average of the product of Grassmann variables between the brackets. The connected amputated Green functions are generated by the effective interaction V[χ, χ] defined by ¯ ¯ e−V[χ,χ] = ¯ ¯ ¯ dµQ [ψ, ψ] e−V [ψ+χ,ψ+χ] . (3.7) The substitution χ = G0 η and χ = GT η , where GT is the transposed propagator, ¯ 0 ¯ 0 relates V[χ, χ] to the functional G[η, η] by ¯ ¯ V[χ, χ] = G[η, η] − (¯, G0 η) . ¯ ¯ η (3.8) The functional derivatives of V[χ, χ] generate connected Green functions divided by ¯ ′ ′ G0 (K1 ) . . . G0 (Km ) G0 (K1 ) . . . G0 (Km ), that is, propagators amputated from external legs in the corresponding Feynman diagrams. The term (¯, G0 η) cancels the η ¯ noninteracting part of G[η, η] such that V[χ, χ] = 0 for V [ψ, ψ] = 0. Hence, the non¯ ¯ interacting propagator is subtracted from the one-particle Green function generated by V[χ, χ]. ¯ ¯ The generating functional Γ[φ, φ] for the 1PI vertex functions γm is derived from the Legendre transform of G[η, η] by ¯ ¯ ¯ ¯ Γ[φ, φ] + (φ, Qφ) = G[η, η ] + (φ, η) − (¯, φ) , ¯ η (3.9) with ∂G ∂η ¯ η= ∂Γ ¯ + Qφ ∂φ ¯ ∂G φ= ∂η η= ¯ ∂Γ ¯ − QT φ ∂φ φ= 22 (3.10) 3.3 RG differential flow equation for Γ and δ2G δ2Γ = ¯+Q δη δ η ¯ δφ δ φ −1 . (3.11) ¯ For the special case without interaction G[η, η ] = (¯, G0 η) leads to Γ[φ, φ] = 0. ¯ η The choice of the appropriate generating functional for a convenient formulation of a renormalization-group approach depends on the physical problem under investigation. For a detailed description of the different schemes we refer to Ref. [Enss 2005]; here we will concentrate on the 1PI version of the fRG. 3.3 RG differential flow equation for Γ In this section we briefly review the general renormalization-group setup, introduced as a transformation that leaves the generating functional for the correlation functions invariant, and concentrate subsequently on the derivation of a continuous renormalization-group equation for the 1PI functions, following the derivation in the context of interacting Fermi systems in Ref. [Salmhofer and Honerkamp 2001]. The addition principle for Gaussian fields implies that for the decomposition G0 = G< + G> the corresponding Gaussian measure factorizes as 0 0 ¯ e−W[Φ,Φ] = = ¯ ¯ ¯ dµQ [Ψ, Ψ]e−Y[Ψ+Φ,Ψ+Φ] ¯ dµQ< [Ψ< , Ψ< ] ¯ ¯ ¯ ¯ dµQ> [Ψ> , Ψ> ]e−Y[Ψ< +Ψ> +Φ,Ψ< +Ψ> +Φ] , (3.12) with Ψ = Ψ< + Ψ> . The generating functional W corresponds to W = V for the particular choice Y = V . This leads to the semigroup law of the renormalization group ¯ e−W[Φ,Φ] = ′ ¯′ ¯ ¯ dµQ< [Ψ′ , Ψ′]e−W> [Ψ +Φ,Ψ +Φ] , (3.13) where in W> = W(Q> , Y) the fields with propagator Q> have been integrated out. The semigroup law implies that the system (Q, Y) under analysis is exactly equivalent to the system (Q< , W(Q> , Y)). In the present case Q> is a covariance with infrared cutoff Λ, and Q< has support only for fields with energies smaller than Λ. 23 3 Functional RG technique: a short overview Set up in this way, the renormalization group is simply a symmetry of the generating functional W(Q, Y). In differential form W(Q, Y) is independent of Λ, that is, ∂ W(Q, Y) = 0 . ∂Λ (3.14) Inserting the right-hand side of Eq. (3.13) leads to the flow equation describing the gradual evolution from Y to the effective functional W as a function of the continuously decreasing energy cutoff Λ. W> is an infinite power series in the fields; the quadratic and quartic terms correspond to the self-energy and the effective interaction, higher order terms are however always present and the convergence of the infinite series is a nontrivial problem [Salmhofer and Honerkamp 2001]. ¯ In the following the differential equation for the generating functional ΓΛ [φ, φ] of the 1PI functions, starting point for the hierarchy of differential flow equations, is derived. Introducing an infrared cutoff at an energy scale Λ > 0 in the bare propagator leads to a Λ-dependent generating functional for the connected Green functions defined by e−G Λ [η,¯] η = ¯ ¯ η ¯ dµQΛ [ψ, ψ] e−V [ψ,ψ] e−(ψ,η)−(¯,ψ) . (3.15) The original functional is recovered in the limit Λ → 0. Similarly the functional ¯ ΓΛ [φ, φ] generating the 1PI vertex functions is constructed with GΛ replacing G0 in 0 Λ Eqs. (3.9 - 3.11). Differentiating the above Eq. (3.15) for G [η, η ] with respect to Λ ¯ yields − ∂G Λ [η, η] −G Λ [η,¯] ¯ Λ η η ˙ e = − Tr (GΛ QΛ ) e−G [η,¯] 0 ∂Λ + ¯ ¯ η ¯ ¯ ˙ dµQΛ [ψ, ψ] (ψ, QΛ ψ) e−V0 [ψ,ψ] e−(ψ,η)−(¯,ψ) Λ η ˙ = − Tr (GΛ QΛ ) + ∆QΛ e−G [η,¯] , ˙ 0 (3.16) where the first term comes from the derivative of the normalization factor (3.3), and Tr denotes the sum over all space-time indices. The functional Laplace operator ∆Q is defined as ∆Q = 24 δ δ ,Q δη δ η ¯ = K δ δ QK . δηK δ ηK ¯ (3.17) 3.4 Expansion in the fields and exact hierarchy of flow equations The flow of G Λ [η, η] is then ¯ 2 Λ ∂G Λ [η, η] ¯ ¯ Λ ˙Λ ˙ Λ δ G [η, η ] + = Tr (G0 Q ) − Tr Q ∂Λ δη δ η ¯ δG Λ [η, η] ˙ Λ δG Λ [η, η] ¯ ¯ ,Q δη δη ¯ . (3.18) ¯ Using the Legendre transform (3.9) the derivative of ΓΛ [φ, φ] reads ¯ ∂G Λ [η, η ] ¯ ∂ΓΛ [φ, φ] ¯ ˙ = − (φ, QΛ φ) ∂Λ ∂Λ δG Λ [η, η] ˙ Λ δG Λ [η, η ] ¯ ¯ ,Q δη δη ¯ = 2 Λ ¯ ˙ δ G [η, η ] − GΛ − Tr QΛ 0 δη δ η ¯ ˙ = − Tr QΛ ¯ δ 2 ΓΛ [φ, φ] Λ ¯ +Q δφ δ φ ¯ ˙ − (φ, QΛ φ) −1 − GΛ 0 , (3.19) leading to the exact renormalization-group equation ∂(GΛ )−1 ∂ Λ ¯ 0 − Tr Γ [φ, φ] = Tr GΛ 0 ∂Λ ∂Λ ¯ δ 2 ΓΛ [φ, φ] Λ −1 ¯ + (G0 ) δφ δ φ −1 ∂(GΛ )−1 0 ∂Λ . (3.20) With the initial condition ¯ ¯ ΓΛ0 [φ, φ] = V [φ, φ] (3.21) Eq. (3.20) determines the flow of ΓΛ uniquely for all Λ < Λ0 . 3.4 Expansion in the fields and exact hierarchy of flow equations Λ The renormalization-group equations for the 1PI m-particle vertex functions γm are ¯ derived by expanding ΓΛ [φ, φ] in Eq. (3.20) as a power series in the fields. The Λ ¯ coefficients in the expansion of ΓΛ [φ, φ] determine γm by ¯ Γ [φ, φ] = Λ ∞ 1 (m!)2 m=0 m ¯ ′ φK j φK j . Λ ′ ′ γm (K1 , . . . , Km ; K1 , . . . , Km ) ′ ′ K1 ...Km K1 ...Km j=1 (3.22) 25 3 Functional RG technique: a short overview Due to the antisymmetry properties of the Grassmann variables only antisymmetric vertex functions contribute. Similarly the second derivative on the right-hand side of Eq. (3.20) can be expanded. Separating the φ-independent part corresponding to the self-energy yields ¯ ¯ δ 2 ΓΛ [φ, φ] δ 2 ΓΛ [φ, φ] = ¯ ¯ δφ δ φ δφ δ φ ¯ φ=φ=0 ¯ ¯ ˜ ˜ + ΓΛ [φ, φ] = −ΣΛ + ΓΛ [φ, φ] , (3.23) ¯ ˜ where the remaining functional ΓΛ [φ, φ] is defined by Eq. (3.22) with indices m and j starting from 2. The second term on the right-hand side of the flow equation (3.20) then reads ¯ δ 2 ΓΛ [φ, φ] Λ −1 ¯ + (G0 ) δφ δ φ −1 ˜ = ΓΛ + (GΛ )−1 =G Λ ∞ −1 ˜ (−1)l ΓΛ GΛ l , (3.24) l=0 with the full propagator GΛ defined via the Dyson equation (GΛ )−1 = (GΛ )−1 − ΣΛ . 0 Introducing the single-scale propagator S Λ as S Λ = GΛ ∂(GΛ )−1 Λ 0 G ∂Λ (3.25) the differential equation (3.20) for ΓΛ is ∂(GΛ )−1 ∂ Λ 0 + Γ = Tr GΛ 0 ∂Λ ∂Λ = Tr GΛ − GΛ 0 ∞ (−1)l+1 Tr GΛ l=0 ∂(GΛ )−1 0 + ∂Λ ∞ ∂ ˜ [GΛ ]−1 GΛ ΓΛ ∂Λ 0 ˜ ˜ (−1)l Tr S Λ ΓΛ GΛ ΓΛ l l . (3.26) l=0 The first term corresponds to a vacuum energy not entering the correlation functions, ˜ while the second one contains one-loop diagrams with (l + 1) vertices ΓΛ connected ˜ by one single-scale propagator S Λ and l full propagators GΛ . The term linear in ΓΛ generates self-energy corrections. Λ Inserting the components γm on the left-hand side and the components γm on ˜Λ the right-hand side of the flow equation (3.26) for ΓΛ we obtain a system of equations 26 3.4 Expansion in the fields and exact hierarchy of flow equations Λ for γm . In a graphical representation the equations for m ≤ 3 are SΛ = (3.27) SΛ SΛ + = (3.28) GΛ SΛ SΛ = + GΛ GΛ SΛ + GΛ (3.29) The initial conditions for the vertex functions at Λ = ∞ are given by the bare interactions of the system. In particular, the flow of the two-particle vertex starts from the antisymmetrized bare two-particle interaction while m-particle vertices of higher order vanish at Λ = ∞, in the absence of bare m-body interactions with m > 2. Note that the right-hand side of the equation for γm contains γm+1 . The infinite system of differential equations contains only one-loop terms in every equation, as the differential formulation of Eq. (3.20) contains only a single trace, and for the 1PI scheme no tree terms appear. The infinite hierarchy produces the full Green functions, generating graphs with an arbitrary number of loops; truncations amount to a partial inclusion of higher order contibutions generated during the flow, where the internal lines contain only modes above the cutoff scale Λ. Consequences of symmetries are discussed in Ref. [Salmhofer and Honerkamp 2001]. 27 3 Functional RG technique: a short overview 3.5 Comparison to other RG schemes Infrared divergencies arising in the context of perturbative expansions or in proximity of phase transitions can alternatively be regularized by temperature, a weak coupling strength or a finite system size. In the fRG approach the cutoff scale is introduced only in the quadratic part of the bare action, and the regularization is implemented with respect to energy scales. The temperature and interaction flows are derived in Refs. [Honerkamp and Salmhofer 2001; Honerkamp et al. 2004] respectively, a pedagogic introduction is given in Ref. [Enss 2005]. The renormalizationgroup equations describe the flow of the correlation functions as the cutoff scale is lowered. The choice of the basis set for the correlation functions determines a particular scheme. In addition to the 1PI scheme described previously, the various generating functionals introduced in Sec. 3.2 correspond to different schemes. Starting point for the Polchinski scheme is the effective interaction, generating functional of the connected amputated Green functions. The flow equation for V Λ [χ, χ] is derived by ¯ Λ replacing Q by Q in Eq. (3.7) and taking the derivative with respect to Λ. An expansion in powers of χ and χ of the functional V Λ [χ, χ] in the renormalization¯ ¯ group equation leads to Polchinski’s flow equations for amputated connected Green functions [Polchinski 1984; Keller et al. 1992], with a similar structure as for the connected Green functions. The connected amputated Green functions are the expansion coefficients of the generating functional V Λ [χ, χ] in terms of monomials of ¯ the source fields χ and χ. Alternatively, one can also expand V Λ [χ, χ] with re¯ ¯ spect to Wick-ordered polynomials, leading to the Wick-ordered Green functions as expansion coefficients [Wieczerkowski 1988; Salmhofer 1998, 1999]. The flow equations are characterized by a bilinear structure in the vertices on the right-hand side connected by bare Λ-dependent propagators. The Wick ordering also implies that except for the differentiated propagator the internal lines are supported below scale Λ instead of above it. Thus, for a momentum cutoff only momenta close to the Fermi surface contribute at low cutoff scale Λ. This justifies a parametrization of the coupling functions by projecting onto the Fermi surface [Halboth and Metzner 2000]. Self-energy corrections are however most conveniently taken into account in the 1PI formalism with full propagators on the internal lines. In an exact treatment all schemes are equivalent, differences arise with truncations of the infinite hierarchy of flow equations. While the full hierarchy of flow 28 3.5 Comparison to other RG schemes equations leads to the correct solution to all orders in perturbation theory independently of the scheme, in the computation of the lowest orders a particular scheme might be more suitable than others, depending on the considered physical problem and properties. An important point for the choice concerns the possibility of an efficient parametrization of the effective interactions by a manageable number of variables. Continuous symmetries in the bare action lead to conservation laws and Ward identities relating Green and response functions, as a consequence of the Noether theorem. These are generally not preserved for the truncated flow equations, in contrast to the solution of the infinite flow-equation hierarchy, as shown in detail in Ref. [Enss 2005]. For a gauge-invariant construction however, as for the temperatureflow scheme, the Ward identities between Green and response functions are satisfied exactly despite truncations. The related property of self-consistency is satisfied by construction in conserving approximations [Baym and Kadanoff 1961], but generally violated in truncated fRG flows. However, in the one-dimensional lattice models for Luttinger liquids, the truncated fRG is nevertheless surprisingly successful and selfconsistency does not appear to play an important role. 29 4 Functional RG for Luttinger liquids We apply the fRG in the one-particle irreducible version to one-dimensional Fermi systems with impurities. The lowest order truncation of the fRG hierarchy of flow equations, where the two-particle vertex is approximated by the bare interaction, considered previously for spinless fermions, is extended including two-particle vertex renormalization, and generalized to spin- 1 systems. For spinless fermions the 2 quantitative accuracy of the results improves considerably, whereas for spin- 1 sys2 tems vertex renormalization is necessary to take into account that backscattering of particles with opposite spins at opposite Fermi points scales to zero in the low-energy limit. The underlying approximations are devised for weak interactions and arbitrary impurity strengths. Details on the computation of the relevant observables from the solution of the flow equations are presented. 4.1 Microscopic models We consider various lattice fermion systems with spinless and spin- 1 fermions sup2 plemented by different types of impurity potentials. The Hamiltonian has the form H = H0 + HI + Himp (4.1) where H0 is the kinetic energy, HI a short-range interaction, and Himp a static local or nonlocal impurity potential. 1 We distinguish between spinless and spin- 2 fermions. 30 4.1 Microscopic models 4.1.1 Spinless fermions For the spinless fermion model c† cj + c† cj+1 j+1 j H0 = −t (4.2) j describes nearest-neighbor hopping processes with an amplitude t and HI = U nj nj+1 (4.3) j is a nearest-neighbor interaction of strength U, as shown in Fig. 4.1. We use standard second quantization notation, where c† and cj are creation and annihilation operators j on site j respectively, and nj = c† cj is the local density operator. The impurity is j represented by V j ′ j c† ′ cj , j Himp = (4.4) j,j ′ where Vj ′j is a static potential. For “site impurities” Vj ′j = Vj δjj ′ (4.5) this potential is local. For the special case of a single site impurity Vj = V δjj0 (4.6) the potential acts only on one site j0 . We also consider “hopping impurities” described by the nonlocal potential Vj ′j = Vjj ′ = −tj,j+1 δj ′,j+1 . (4.7) For a single hopping impurity tj,j+1 = (t′ − t) δjj0 (4.8) the hopping amplitude t is replaced by t′ on the bond linking the sites j0 and j0 + 1. In the following we will set the bulk hopping amplitude t equal to one, that is, all energies are expressed in units of t. The clean spinless fermion model H0 +HI is exactly soluble via the Bethe ansatz [Yang and Yang 1966]. The system is a Luttinger liquid for all particle densities n 31 4 Functional RG for Luttinger liquids V U t j0 1 L Figure 4.1: Spinless fermion model with nearest-neighbor hopping amplitude t, nearestneighbor interaction U , and a local site potential V on site j0 . and any interaction strength, except at half filling for |U| > 2. For U > 2 a charge density wave with wave vector π forms; for U < −2 the system undergoes phase separation. The Luttinger-liquid parameter Kρ , which determines all the critical exponents of the liquid, can be computed exactly from the Bethe ansatz solution [Haldane 1980]. At half filling Kρ is related to U by the simple explicit formula −1 Kρ = 2 U arccos − π 2 (4.9) for |U| ≤ 2. 1 4.1.2 Spin- 2 fermions 1 For spin- 2 fermions, the kinetic energy is given by H0 = −t † c† j+1,σ cjσ + cjσ cj+1,σ , (4.10) j,σ where c† and cjσ are creation and annihilation operators for fermions with spin jσ projection σ on site j. The interaction term of the extended Hubbard model contains a local interaction U and a nearest-neighbor interaction U ′ nj↑ nj↓ + U ′ HI = U nj nj+1 , (4.11) j j with njσ = c† cjσ and nj = nj↑ + nj↓ , as shown in Fig. 4.2. For the pure Hubbard jσ model only the local interaction U is finite, while U ′ = 0. The impurity term Vj ′ j c†′ σ cjσ j Himp = j,j ′ 32 σ (4.12) 4.2 Cutoff and flow equations U’ U 1 L t Figure 4.2: Extended Hubbard model with nearest-neighbor hopping amplitude t, local interaction U and nearest-neighbor interaction U ′ . differs from the spinless case only by the spin sum. In the absence of impurities, the Hubbard model can be solved exactly using the Bethe-ansatz [Lieb and Wu 1968], while the extended Hubbard model is not integrable. The Hubbard model is a Luttinger liquid for arbitrary repulsive interactions at all particle densities except half filling, where the system becomes a Mott insulator [Voit 1995; Giamarchi 2004]. The phase diagram of the extended Hubbard model is more complex. Away from half filling it is a Luttinger liquid at least for sufficiently weak repulsive interactions [Voit 1995]. For the Hubbard model the Luttinger-liquid parameter Kρ can be computed exactly from the Bethe ansatz solution [Frahm and Korepin 1990; Kawakami and Yang 1990; Schulz 1990]. 4.2 Cutoff and flow equations 4.2.1 Cutoff The cutoff introduced in Sec. 3.3 can be imposed in many different ways. The only requirement is that the infrared singularities must be regularized such that the flow equations allow for a regular perturbation expansion in powers of the renormalized two-particle vertex. Since translation invariance is spoiled by the impurity, a Matsubara frequency cutoff is the most efficient choice, while a momentum cutoff is less suitable. At T = 0 the cutoff is sharp [Andergassen et al. 2004], the extension to T > 0 will be addressed subsequently. The cutoff is imposed by excluding modes with frequencies below scale Λ from the functional integral representation of the system, or equivalently, by introducing a regularized bare propagator GΛ (iω) = Θ(|ω| − Λ) G0(iω) . 0 (4.13) 33 4 Functional RG for Luttinger liquids Here G0 is the bare propagator of the pure system, involving neither interactions nor impurities. Instead of the sharp cutoff imposed by the step function Θ one may also choose a smooth cutoff function, but the sharp cutoff has the advantage that it reduces the number of integration variables on the right-hand side of the flow equations. Note that we will frequently write expressions which are well defined only if the sharp cutoff is viewed as a limit of increasingly sharp smooth cutoff functions. The suppression of frequencies below scale Λ affects all Green and vertex functions of the interacting system, which become thus functions of Λ. The original system is recovered in the limit Λ → 0. 4.2.2 Truncation schemes The truncation of the fRG hierarchy of differential flow equations for the one-particle irreducible m-particle vertex functions and their parametrization with a manageable set of variables or functions leads to different approximation schemes. In the lowest order truncation of the fRG hierarchy (cf. Sec. 3.4) the renormalized two-particle vertex is approximated by the bare interaction [Meden et al. 2002a,b]. This truncation scheme, denoted by Scheme I, includes only the first equation in the hierarchy for the one-particle vertex function ΓΛ = −ΣΛ , where the 1 self-energy ΣΛ is related to the interacting propagator by the usual Dyson equation GΛ = (GΛ )−1 − ΣΛ 0 −1 . (4.14) Here and below GΛ , ΣΛ etc. are operators, which do not refer to any particular single-particle basis, unless we write matrix indices explicitly. The right-hand side of the flow equation for ΣΛ (3.27) involves the two-particle vertex ΓΛ and the singlescale propagator S Λ introduced in Eq. (3.25) which has support only on a single frequency scale |ω| = Λ. The flow equation for the self-energy reads 1 ∂ Λ ′ Σ (1 , 1) = − ∂Λ β + eiω2 0 S Λ (2, 2′) ΓΛ (1′ , 2′ ; 1, 2) , (4.15) 2,2′ where β is the inverse temperature. The numbers 1, 2, etc. are a shorthand for Matsubara frequencies and labels for single-particle states such as site and spin ′ ′ indices. Note that ω1 = ω1 and ω2 = ω2 due to Matsubara frequency conservation. The exponential factor in the above equation is irrelevant at any finite Λ, but is necessary to define the initial conditions of the flow at Λ = Λ0 → ∞. 34 4.2 Cutoff and flow equations For a sharp frequency cutoff the frequency sum on the right-hand side of the flow equation can be carried out analytically in the zero temperature limit, where the Matsubara sum becomes an integral. At this point one has to deal with products of delta functions δ(|ω| − Λ) and expressions involving step functions Θ(|ω| − Λ). These at first sight ambiguous expressions are well defined and unique if the sharp cutoff is implemented as a limit of increasingly sharp broadened cutoff functions Θǫ , with the broadening parameter ǫ tending to zero. The expressions can then be conveniently evaluated by using the following relation [Morris 1994], valid for arbitrary continuous functions f : 1 δǫ (x − Λ) f [Θǫ(x − Λ)] → δ(x − Λ) f (t) dt , (4.16) 0 where δǫ = Θ′ǫ . Note that the functional form of Θǫ for finite ǫ does not affect the result in the limit ǫ → 0. In Scheme I ΓΛ in Eq. (4.15) is replaced by the antisymmetrized bare two-particle interaction ΓΛ′0 ′ ;1,2 = I1′ ,2′ ;1,2 , where the lower indices 1, 2, etc. label single-particle 1 ,2 states (not frequencies). Since I1′ ,2′ ;1,2 is frequency independent, no frequency dependence of the self-energy is generated in the flow. Carrying out the frequency integration in the flow equation for the self-energy (4.15) one obtains 1 ∂ Λ Σ1′ ,1 = − ∂Λ 2π ω=±Λ + ˜ eiω0 GΛ ′ (iω) I1′ ,2′ ;1,2 , 2,2 (4.17) 2,2′ where ˜ GΛ (iω) = G−1 (iω) − ΣΛ 0 −1 . (4.18) ˜ Note that GΛ has no jump at |ω| = Λ, in contrast to GΛ . The flow is determined uniquely by the differential flow equation and the initial condition at Λ = ∞. The self-energy at Λ = ∞ is given by the bare impurity (site or hopping) potential V . In a numerical solution the flow starts at some large finite initial cutoff Λ0 . Here one has to take into account that, due to the slow decay of the right-hand side of the flow equation for ΣΛ at large Λ, the integration of the flow from Λ = ∞ to Λ = Λ0 yields a contribution which does not vanish in the ˜ limit Λ0 → ∞, but rather tends to a finite constant. Since GΛ ′ (iω) → δ2,2′ /(iω) for 2,2 |ω| = Λ → ∞, this constant is determined as − 1 lim 2π Λ0 →∞ Λ0 + eiω0 dΛ ∞ ω=±Λ 2,2′ δ2,2′ 1 I1′ ,2′ ;1,2 = iω 2 I1′ ,2;1,2 . (4.19) 2 35 4 Functional RG for Luttinger liquids Including the bare impurity potential V1,1′ , the initial conditions for the self-energy at Λ = Λ0 → ∞ is 1 I1′ ,2;1,2 . (4.20) ΣΛ0 ′ = V1,1′ + 1,1 2 2 + For the flow at Λ < Λ0 the factor eiω0 in Eq. (4.17) can be discarded. A further development of the fRG approach for impurities in Luttinger liquids includes the two-particle vertex renormalization, denoted by Scheme II and used in the following if not specified otherwise. For spinless fermions this extension does not matter qualitatively, but the quantitative accuracy of the results improves considerably, in particular at intermediate interaction strengths. By contrast, for spin- 1 2 systems vertex renormalization is necessary to take into account that backscattering of particles with opposite spins at opposite Fermi points scales to zero in the lowenergy limit. The right-hand side of the flow equation for the two-particle vertex ΓΛ (3.28) involves ΓΛ itself, but also the three-particle vertex ΓΛ . Neglecting the 3 contribution of the three-particle vertex to the flow of the two-particle vertex, the coupled system of flow equations for the two-particle vertex ΓΛ and the self-energy ΣΛ is closed. In terms of an expansion in the bare coupling function, this truncation is exact up to second order. However, the fRG provides more than just a secondorder calculation: the evolution of the interaction and the self-energy is continually fed back into the fRG differential equation. This effectively sums up contributions from arbitrarily high orders and thus produces a scale-dependent resummation of perturbation theory. We note that it does not correspond to an expansion to a fixed loop order: the flow equations appear to be one loop, but they also take into account two-loop effects by iteration. The relevant question is whether higher orders significantly change the flow, they certainly do so if the coupling functions get too large. The contribution of ΓΛ to ΓΛ is small as long as ΓΛ is sufficiently small, 3 Λ because Γ3 is initially (at Λ0 ) zero and is generated only from terms of third order in ΓΛ . A comparison of the fRG results to exact DMRG results and exact scaling properties shows that the truncation error is often surprisingly small, even for rather large interactions [Meden et al. 2002a,b]. The explicit form of the truncated flow equation for the two-particle vertex reads 1 ∂ Λ ′ ′ Γ (1 , 2 ; 1, 2) = ∂Λ β GΛ (3, 3′ ) S Λ (4, 4′ ) ΓΛ (1′ , 2′ ; 3, 4) ΓΛ(3′ , 4′ ; 1, 2) 3,3′ 4,4′ − ΓΛ (1′ , 4′ ; 1, 3) ΓΛ(3′ , 2′ ; 4, 2) − (3 ↔ 4, 3′ ↔ 4′ ) 36 4.2 Cutoff and flow equations + ΓΛ (2′ , 4′ ; 1, 3) ΓΛ(3′ , 1′ ; 4, 2) + (3 ↔ 4, 3′ ↔ 4′ ) . (4.21) Diagrammatically, the individual contributions for the particle-particle and particlehole channels written explicitly are 2 2′ 1 1′ ∂ Λ Γ = ∂Λ 1 1′ 1 1′ − − 2 2′ 1 2′ + 2 2′ 1 2′ + 2 1′ 2 1′ (4.22) Instead of solving the frequency integrated flow equation in full generality, we implement the following approximation: the frequency-dependent flow of the renormalized two-particle vertex ΓΛ is replaced by its value at vanishing (external) frequencies, such that ΓΛ remains frequency independent. As a consequence, also the self-energy remains frequency independent. Since the bare interaction is frequency independent, neglecting the frequency dependence leads to errors only at second order (in the interaction strength) for the self-energy, and at third order for the vertex function at zero frequency. In addition to the quantitative errors we miss qualitative properties related to the frequency dependence of the self-energy, such as the suppression of the one-particle spectral weight in the bulk of a pure Luttinger liquid. On the other hand, a comparison with exact numerical results and asymptotic analytical results shows that the impurity effects are not qualitatively affected by the frequency dependence of Σ, at least for weak interactions. The frequency-integrated flow equation for the two-particle vertex, evaluated at 37 4 Functional RG for Luttinger liquids vanishing external frequencies, has the form 1 ∂ Λ Γ1′ ,2′ ;1,2 = ∂Λ 2π ω=±Λ 3,3′ 4,4′ 1 ˜Λ ˜ 4,4 G3,3′ (iω) GΛ ′ (−iω) ΓΛ′ ,2′ ;3,4 ΓΛ′ ,4′ ;1,2 1 3 2 ˜ ˜ + GΛ ′ (iω) GΛ ′ (iω) −ΓΛ′ ,4′ ;1,3 ΓΛ′ ,2′ ;4,2 + ΓΛ′ ,4′ ;1,3 ΓΛ′ ,1′ ;4,2 3,3 4,4 1 3 2 3 , (4.23) with the initial condition ΓΛ′0 ′ ;1,2 = I1′ ,2′ ;1,2 . 1 ,2 A crucial point is to devise an efficient parametrization of the vertex by a manageable number of variables. For a finite lattice system with L sites the flow of the two-particle vertex ΓΛ′ ,2′ ;1,2 involves O(L3 ) independent flowing variables, if transla1 tion invariance is assumed, and O(L4 ) variables, if the influence of the impurity on the flow of the two-particle vertex is taken into account. For a treatment of large systems it is therefore necessary to reduce the number of variables by a suitable approximate parametrization of the vertex. In the low-energy limit (small Λ) the flow is dominated by a very small number of variables, the others being irrelevant according to standard renormalization-group arguments [Voit 1995]. In particular, the frequency dependence of the vertex, discarded already above, is irrelevant for the flow of ΓΛ at small Λ. For larger Λ one can use perturbation theory as a guide for a simple but efficient parametrization of ΓΛ . We neglect the influence of the impurity on the flow of the two-particle vertex, such that ΓΛ remains translation invariant. While this is sufficient for capturing the effects of isolated impurities in otherwise pure systems, it is known that impurity contributions to vertex renormalization become important in macroscopically disordered systems [Giamarchi 2004]. We also neglect the feedback of the bulk self-energy into the flow of ΓΛ , which yields higher order contributions in the renormalized interaction. The two-particle vertex is parametrized approximately by a renormalized static short-range interaction, which allows us to capture various features: the lowenergy flow of the vertex at kF in the pure system is obtained correctly to second order in the renormalized couplings; the nonuniversal contributions at finite energy scales are correct to second order in the bare interaction; the algorithm for the flow of the self-energy remains as fast as in the absence of vertex renormalization, such that one can easily deal with up to 107 lattice sites! For a more concrete treatment of the vertex renormalization, we now focus on a specific model. 38 4.2 Cutoff and flow equations 4.2.3 Spinless fermions For spinless fermions the two-particle vertex and the self-energy are fully characterized by either site or momentum variables. In the low-energy limit, the flow of the vertex is dominated by contributions with momenta close to the Fermi points, such that the right-hand side of the flow equation is determined by momentum compo′ ′ with k1 , k2 , k1 , k2 = ±kF . Due to the antisymmetry of nents of the vertex ΓΛ′ ′ k1 ,k2 ;k1 ,k2 the vertex, there is only one such component which is nonzero: g Λ = ΓΛF ,−kF ;kF ,−kF . k (4.24) In the low-energy limit the momentum dependence of the vertex away from ±kF is irrelevant. There are therefore many possible choices for the functional form of , which all lead to the correct low-energy asymptotics. For a model with a ΓΛ′ ′ k1 ,k2 ;k1 ,k2 bare nearest-neighbor interaction U, a natural and efficient choice is to parametrize the flowing vertex simply by a renormalized nearest-neighbor interaction U Λ , which leads to a real space vertex of the form ΓΛ′ ,j ′ ;j j 1 2 1 ,j2 = UjΛ ,j2 (δj 1 ′ 1 ,j1 δj ′ 2 ,j2 − δj ′ 1 ,j2 δj ′ 2 ,j1 ), (4.25) with UjΛ ,j2 = U Λ (δj1 ,j2−1 +δj1 ,j2 +1 ) . This yields the following structure in momentum 1 space: ΓΛ′ ,k′ ;k k 1 2 1 ,k2 ′ ′ = 2U Λ [cos(k1 − k1 ) − cos(k2 − k1 )] δ (2π) ′ ′ k1 +k2 ,k1 +k2 , (4.26) where the Kronecker δ implements momentum conservation (modulo 2π). The flowing coupling constant U Λ is linked to the value of the vertex at the Fermi points by the relation g Λ = 2U Λ [1 − cos(2kF )] . (4.27) The flow equation for g Λ becomes ∂g Λ 1 = ∂Λ 2π ω=±Λ dp ( P P + P H + P H′ ) , 2π (4.28) with the particle-particle and particle-hole contributions 1 P P = G0 (iω) G0 (−iω) ΓΛF ,−kF ;p,−p ΓΛ −p k p,−p;kF ,−kF 2 p 39 4 Functional RG for Luttinger liquids P H = − [G0 (iω)]2 ΓΛF ,p;kF ,p ΓΛ F ;p,−kF p k p,−k P H ′ =G0 F (iω) G0 F (iω) ΓΛ F ,p+kF ;kF ,p−kF ΓΛ F ,kF ;p+kF ,−kF , p−k p+k −k p−k (4.29) where ΓΛ on the right-hand side of the flow equation is given by Eq. (4.26). Using Eq. (4.27) to replace ∂Λ g Λ by ∂Λ U Λ on the left-hand side of Eq. (4.28), one obtains a flow equation for U Λ of the simple form ∂Λ U Λ = h(Λ) (U Λ )2 . (4.30) The function h(Λ) depends only on the cutoff Λ and the Fermi momentum kF . An explicit formula for h(Λ) can be obtained by carrying out the momentum integral in Eq. (4.28) using the residue theorem. For finite systems the momentum integral should be replaced by a discrete momentum sum; however, this leads to sizable corrections only for very small systems. Inserting the momentum structure of ΓΛ (4.26) into the flow equation (4.28) and replacing g Λ by U Λ on the left-hand side yields (U Λ )2 ∂U Λ = ∂Λ 2π sin2 kF 2π ω=±Λ 0 dp f (p, ω) , 2π (4.31) where f (p, ω) = 2 sin2 kF sin2 p (cos kF − cos p)2 [cos(2kF ) − cos p]2 . − + 0 0 0 0 0 (iω − ξp )(−iω − ξ−p ) (iω − ξp )2 (iω − ξp−kF )(iω − ξp+kF ) (4.32) 0 Here ξk = −2 cos k − µ0 , with µ0 = −2 cos kF , is the bare dispersion relation relative to the bare Fermi level. Since f (p, ω) can be written as a rational function of cos p and sin p, the p -integral can be carried out analytically using the substitution z = eip and the residue theorem. After a lengthy but straightforward calculation one obtains the following result for the coefficient h(Λ) in (4.30): h(Λ) = − × 40 i 1 − Re (µ0 + iΛ) 2π 2 1− 4 (µ0 + iΛ)2 3iµ4 − 10µ3 Λ − 12iµ2 (Λ2 + 1) + 6Λ3 µ0 + 18Λµ0 + 6iΛ2 + iΛ4 0 0 0 π(2µ0 + iΛ)(4 − µ2 + Λ2 − 2iΛµ0 )2 0 . (4.33) 4.2 Cutoff and flow equations The flow equation (4.30) can be integrated to UΛ = U , 1 − U H(Λ) (4.34) where H(Λ) is the primitive function of h(Λ) with H(Λ) → 0 for Λ → ∞. Integrating h(Λ) one obtains H(Λ) = − Λ 1 (4 − µ2 )Λ2 − 2iµ0 (2 − µ2 )Λ + µ4 − 6µ2 + 8 0 0 0 0 + Re 2 2 − 2iµ Λ + 4 − µ2 2π π 2 (4 − µ0 ) Λ 0 0 4 + µ2 + iµ0 Λ 0 µ4 0 tanh−1 + 2 3/2 2(4 − µ0 ) − Λ − iµ0 iµ0 sinh−1 2 2 (4 + µ2 + iµ0 Λ)2 + 4(Λ − 2iµ0 )2 0 , (4.35) where sinh−1 and tanh−1 denote the main branch of the inverse of the complex functions sinh and tanh respectively. At half filling, corresponding to kF = π/2, the function h(Λ) is particularly simple h(Λ) = − 1 2π 1−Λ 6 + Λ2 (4 + Λ2 )3/2 (4.36) such that U Λ reduces to U UΛ = 1+ Λ− 2+Λ2 √ 4+Λ2 . (4.37) U/(2π) In Fig. 4.3 we show results for the renormalized nearest-neighbor interaction U Λ as obtained from the flow equation at various densities n, for a bare interaction U = 1. While the renormalization does not follow any simple rule at intermediate scales Λ, all curves saturate at a finite value U ∗ in the limit Λ → 0, corresponding to a finite g ∗ , as expected for a Luttinger-liquid fixed point [Voit 1995]. Parametrizing ΓΛ by a renormalized nearest-neighbor interaction has the enormous advantage that the self-energy, as determined by the flow equation (4.17), is a tridiagonal matrix in real space, that is, only the matrix elements ΣΛ and ΣΛ j,j j,j±1 are Λ nonzero. Inserting Γ from Eq. (4.25) into (4.17), one obtains the following simple coupled flow equations for the diagonal and off-diagonal matrix elements: UΛ ∂ Λ Σj,j = − ∂Λ 2π ˜ GΛ j+r,j+r (iω) ω=±Λ r=±1 41 4 Functional RG for Luttinger liquids 1.2 n = 1/2 1.1 n = 3/8 UΛ 1 0.9 n = 1/4 0.8 0.7 10−3 n = 1/8 10−2 10−1 100 Λ 101 102 103 Figure 4.3: Flow of the renormalized nearest-neighbor interaction U Λ for the spinless fermion model, for U = 1 and various densities n. ∂ Λ Σ = ∂Λ j,j±1 UΛ 2π ˜ GΛ (iω) . j,j±1 (4.38) ω=±Λ Note that the self-energy enters also the right-hand side of these equations, via ˜ GΛ = (G−1 − ΣΛ )−1 . Since ΣΛ and G−1 are both tridiagonal in real space, the 0 0 matrix inversion required to compute the diagonal and first off-diagonal elements ˜ of GΛ from ΣΛ can be carried out very efficiently. An algorithm for the numerical solution of the flow equation for ΣΛ scaling linearly with the system size is described in Ref. [Andergassen et al. 2004; Enss 2005]. Very large systems with up to 107 sites can be treated without extensive numerical effort. 1 4.2.4 Spin- 2 fermions We now describe the parametrization of the spatial (or momentum) dependences of the two-particle vertex ΓΛ for spin- 1 fermions [Andergassen et al. 2005b], employing 2 a natural extension of our previous parametrization for the spinless case in Sec. 4.2.3. We consider spin-rotation invariant lattice systems with local and nearest-neighbor interactions. This includes the extended Hubbard model. For a spin-rotation invariant system the spin structure of the two-particle vertex 42 4.2 Cutoff and flow equations can be decomposed in a singlet and a triplet part: + ΓΛ Tσ′ ,σ′ ;σ t ΓΛ = ΓΛ Sσ′ ,σ′ ;σ s 1 2 1 ,σ2 , 1 1 ,σ2 2 (4.39) with Sσ′ ,σ′ ;σ 1 2 1 ,σ2 Tσ′ ,σ′ ;σ 1 2 1 ,σ2 1 2 1 = 2 = δσ δσ − δσ ′ 1 σ2 δσ δσ δσ + δσ ′ 1 σ2 δσ ′ 1 σ1 ′ 1 σ1 ′ 2 σ2 ′ 2 σ2 ′ 2 σ1 ′ 2 σ1 . (4.40) Since the total vertex is antisymmetric in the incoming and outgoing particles, the singlet part ΓΛ has to be symmetric and the triplet part ΓΛ antisymmetric. s t Proceeding in analogy to the case of spinless fermions in Sec. 4.2.3, we first list momentum components of the vertex with all momenta at ±kF . For the triplet vertex the antisymmetry allows once again only one such component Λ gt = ΓΛ kF ,−kF ;kF ,−kF . t| (4.41) For the singlet vertex there are several distinct components at ±kF . Since we will neglect the influence of the impurity on the vertex renormalization, the renormalized vertex remains translation invariant. Hence the momentum components are ′ ′ restricted by momentum conservation: k1 + k2 = k1 + k2 , modulo integer multiples of 2π. The remaining independent (not related by obvious symmetries) components are Λ gs2 = ΓΛ kF ,−kF ;kF ,−kF s| Λ gs4 = ΓΛ kF ,kF ;kF ,kF s| (4.42) and in the case of half filling, for which kF = π/2, also Λ gs3 = ΓΛ π/2,π/2;−π/2,−π/2 . s| (4.43) The labels 2, 3, 4 are chosen in analogy to the conventional g-ology notation for onedimensional Fermi systems [S´lyom 1979]. In order to parametrize the vertex in a o Λ uniform way in all cases, we will include the umklapp component gs3 not only at half filling, but at any density. The effect on the other components is negligible for the range of interactions and fillings considered. Extending our treatment of the spinless case in Sec. 4.2.3, we now parametrize the vertex by renormalized local and nearest-neighbor interactions in real space. 43 4 Functional RG for Luttinger liquids For the triplet part, there is no local component, and only one nearest-neighbor component compatible with the antisymmetry, namely Λ Ut′ = ΓΛ j,j+1;j,j+1 , t| (4.44) which has the same form as the nearest-neighbor interaction in the spinless case. Note that ΓΛ j,j+1;j,j+1 does not depend on j and is equal to ΓΛ j,j−1;j,j−1. For the t| t| symmetric singlet part, there is one local component Λ Us = ΓΛ j,j;j,j s| (4.45) and three different components involving nearest neighbors: Λ ′ Us = ΓΛ j,j+1;j,j+1 s| PsΛ = ΓΛ j+1,j+1;j,j s| WsΛ = ΓΛ j+1,j;j,j . s| (4.46) For the Hubbard model, the bare vertex is purely local and the initial condition for Λ the vertex is given by Us 0 = 2U, while all the other components vanish. For the ′ extended Hubbard model, Us Λ0 = Ut′ Λ0 = U ′ are nonzero. The triplet vertex is parametrized by only one renormalized real space coupling, which leads to a momentum representation of the form ΓΛ k′ ,k′ ;k t| 1 2 Λ 1 ,k2 ′ ′ = 2Ut′ [cos(k1 − k1 ) − cos(k2 − k1 )] δ (2π) ′ ′ k1 +k2 ,k1 +k2 , (4.47) where the Kronecker δ implements momentum conservation (modulo 2π). The flowing coupling Ut′ Λ is thus linked in a one-to-one correspondence to the Fermi momenΛ tum coupling gt by Λ Λ gt = 2Ut′ [1 − cos(2kF )] (4.48) as in the spinless case in Sec. 4.2.3. In the singlet channel we have found four real space couplings, that is, one more than necessary to match the three singlet Λ Λ Λ couplings in momentum space, gs2 , gs3 , gs4 . We discard the interaction WsΛ , because it does not appear in the bare Hubbard model, where it is generated only at third order in U, while the pair hopping PsΛ appears already in second-order perturbation theory. Fourier transforming the remaining interactions yields the singlet vertex in k-space ΓΛ k′ ,k′ ;k s| 1 44 2 Λ 1 ,k2 Λ ′ ′ ′ = Us + 2Us [cos(k1 − k1 ) + cos(k2 − k1 )] 4.2 Cutoff and flow equations +PsΛ cos(k1 + k2 ) δ (2π) ′ ′ k1 +k2 ,k1 +k2 (4.49) Λ from which we obtain a linear relation between the momentum space couplings gs2 , Λ Λ Λ ′ gs3 , gs4 and the renormalized interaction parameters Us , Us Λ , PsΛ : Λ Λ Λ ′ gs2 = Us + 2Us [1 + cos(2kF )] + 2PsΛ Λ Λ Λ ′ gs3 = Us − 4Us − 2PsΛ Λ Λ Λ ′ gs4 = Us + 4Us + 2PsΛ cos(2kF ) . (4.50) The determinant of this linear system is positive for all kF , except for kF = 0 and π. Hence the equations can be inverted for all densities except the trivial cases of an empty or completely filled band. We can now set up the flow equations for the four independent couplings Ut′ Λ , Λ ′ Us , Us Λ , and PsΛ which parametrize the vertex. Consider the case T = 0 first. Inserting the spin structure (4.39) into the general flow equation for the two-particle vertex (4.23), and using the momentum representation for a translation invariant vertex, the flow equation for the singlet and triplet vertices ΓΛ , for a = s, t, can be a written as 1 ∂ Λ Γa| k1 ,k2 ;k1 ,k2 = − ′ ′ ∂Λ 2π ω=±Λ b,b′ =s,t dp ( P P + P H + P H′ ) , 2π (4.51) with the particle-particle and particle-hole contributions PP P P = Ca,bb′ G0 (iω) G01 +k2 −p (−iω) ΓΛ k1 ,k2 ;p,k1+k2 −p ΓΛ′ | p,k1 +k2 −p;k1,k2 p k b b| ′ ′ PH P H = Ca,bb′ G0 (iω) G0 1−k1 (iω) ΓΛ k1 ,p+k1 −k1 ;k1 ,p ΓΛ′ | p,k2;p+k1−k1 ,k2 ′ ′ ′ ′ p p+k b| ′ b ′ PH P H ′ = Ca,bb′ G0 (iω) G0 1−k2 (iω) ΓΛ k2 ,p+k1−k2 ;k1 ,p ΓΛ′ | p,k1;p+k1 −k2 ,k2 . ′ ′ ′ ′ p p+k b| ′ b (4.52) The coefficients Ca,bb′ are obtained from the spin sums as PP Cs,ss = 1 PP PP PP Cs,st = Cs,ts = Cs,tt = 0 PP Ct,tt = 1 PP PP PP Ct,ss = Ct,st = Ct,ts = 0 PH Cs,ss = −1/4 PH PH PH Cs,st = Cs,ts = Cs,tt = 3/4 PH Ct,tt = 5/4 PH PH PH Ct,ss = Ct,st = Ct,ts = 1/4 45 4 Functional RG for Luttinger liquids ′ PH PH Cs,bb′ = − Cs,bb′ ′ PH PH Ct,bb′ = Ct,bb′ . (4.53) Note that we have neglected the self-energy feedback in the flow of ΓΛ , such that only bare propagators G0 enter. On the right-hand side of the flow equation we insert the parametrization (4.47) for ΓΛ and (4.49) for ΓΛ . The flow of the triplet t s ′ ′ vertex ΓΛ ′ ′ is evaluated only for (k1 , k2 , k1 , k2 ) = (kF , −kF , kF , −kF ) as in t| k1 ,k2 ;k1 ,k2 Λ Eq. (4.41), which yields the flow of gt , while the flow of the singlet vertex ΓΛ ′ ′ (k1 , k2 , k1, k2 ) ′ ′ s| k1 ,k2 ;k1 ,k2 Λ Λ of gs2 , gs3 , is computed for the three choices of which yield the flow Λ gs4 corresponding to Eqs. (4.42) and (4.43). Using the linear equations (4.48) and (4.50) to replace the couplings g Λ by the renormalized real space interactions on the left-hand side of the flow equations, we obtain a complete set of flow equations for Λ ′ the four renormalized interactions Ut′ Λ , Us , Us Λ , and PsΛ of the form Λ ∂Λ Uα = Λ Λ hα′ α′′ (Λ) Uα′ Uα′′ , α (4.54) α′ ,α′′ where α = 1, 2, 3, 4 labels the four different interactions. The functions hα′ α′′ (Λ) can α be computed analytically by carrying out the momentum integrals in Eq. (4.51) via the residue theorem, details are reported in App. A.1. The flow equations can then be solved numerically very easily. For finite systems the momentum integral should be replaced by a discrete momentum sum; however, this leads only to negligible corrections for the physical observables presented in Sec. 4.4. After computing the flow of the real space interactions, one can also calculate the flow of the momentum space couplings g Λ by using the linear relation between the two. In the low-energy limit (small Λ) one recovers the one-loop flow of the g-ology model, the general effective low-energy model for one-dimensional fermions [S´lyom 1979], for details see Sec. A.2. In addition, our vertex renormalization o captures also all nonuniversal second-order contributions to the vertex at ±kF from higher energy scales. In Fig. 4.4 we show results for the renormalized real space interactions together with the corresponding momentum space couplings, as obtained by integrating the flow equations for the Hubbard model at quarter filling and T = 0. Note that the couplings converge to finite fixed-point values in the limit Λ → 0, but the Λ Λ convergence is very slow, except for the momentum space couplings gs3 and gs4 . This can be traced back to the familiar behavior of the so-called backscattering 1 Λ Λ Λ coupling g1⊥ = 2 (gs2 − gt ), that is, the amplitude for the exchange of two particles 46 4.2 Cutoff and flow equations with opposite spin at opposite Fermi points. Backscattering is known to vanish logarithmically in the low-energy limit for spin-rotation invariant spin- 1 Luttinger 2 liquids [Voit 1995]. We emphasize that this logarithmic behavior is not promoted to a power law by higher order terms beyond our approximation. By contrast, the linear combination of couplings which determines the Luttinger-liquid parameter Kρ converges very quickly to a finite fixed-point value. 2.5 2 1.5 Us ′ Us Ps Ut′ 1 0.5 0 -0.5 -1 2 1.5 1 gs2 gs3 gs4 gt 0.5 0 10−10 10−8 10−6 10−4 10−2 100 102 104 Λ Figure 4.4: Vertex flow for the Hubbard model at quarter filling (n = 1/2) and U = 1; upper panel: flow of the renormalized real space interactions, lower panel: flow of the momentum space couplings. 47 4 Functional RG for Luttinger liquids Due to the above parametrization of the vertex by real space interactions which do not extend beyond nearest neighbors on the lattice, the self-energy generated by the flow equations is frequency independent and tridiagonal in real space. Inserting the spin and real space structure of ΓΛ into the general flow equation for the selfenergy (4.17), one obtains 1 ∂ Λ Σj,j = − ∂Λ 4π ∂ Λ 1 Σj,j±1 = − ∂Λ 4π Λ Λ Λ ˜ ′ Us GΛ (iω) + (Us + 3Ut′ ) j,j r=±1 ω=±Λ ω=±Λ ˜ GΛ j+r,j+r (iω) Λ ˜ ′Λ ˜ (Us − 3Ut′ ) GΛ (iω) + PsΛ GΛ (iω) j,j±1 j±1,j , (4.55) ˜ where GΛ = (G−1 − ΣΛ )−1 . 0 Due to the slow decay of G at large frequencies, the integration of the flow equation for Σ from Λ = ∞ to Λ = Λ0 yields a contribution which remains finite even in the limit Λ0 → ∞, as described in Sec. 4.2.2. For the extended Hubbard model this contribution is given by ΣΛ0 = U/2 + 2U ′ for j = 2, . . . , L − 1 and j,j ΣΛ0 = ΣΛ0 = U/2 + U ′ . The numerical integration of the flow is started at a 1,1 L,L sufficiently large Λ0 with ΣΛ0 as initial condition. 4.2.5 Extension to finite temperature At finite temperatures the Matsubara frequencies ωn are discrete. The sum over ωn of a function f can be written as an integral over a continuous variable ω by introducing the distribution function P with a normalization |ω−ωn |≤ πT dω P (ω) = 1 for all n, dω P (ω) f (ωn) = f (ωn ) = ωn ωn dω P (ω)f (ωn, ω ) , (4.56) |ω−ωn |≤ πT where ωn, ω denotes the discrete Matsubara frequency closest to ω. Introducing a sharp frequency cutoff in the continuous variable ω, the extension of the flow equations to finite temperatures is fairly simple, as pointed out by T. Enss [Andergassen et al. 2005b]. The general form of the flow equation for the generating functional for the 1PI vertex functions Υ at T = 0, ∂ Λ Υ = ∂Λ 48 dω δ(|ω| − Λ) F Θ(|ω| − Λ), ΥΛ (ω) , (4.57) 4.3 Calculation of Kρ is modified to ∂ Λ Υ =T ∂Λ dω P (ω) δ(|ω| − Λ) F Θ(|ω| − Λ), ΥΛ (ωn, ω ) (4.58) at T > 0. Applying the lemma (4.16) the integral over ω can be carried out analytically 1 ∂ Λ Υ =T ∂Λ dω P (ω) δ(|ω| − Λ) 0 dt F t, ΥΛ (ωn, ω ) P (ω) F ΥΛ (ωn, ω ) , =T (4.59) ω=±Λ 1 where F (·) = 0 dt F (t, ·). The contribution to the flow on the interval ωn, Λ − πT ≤ Λ < ωn, Λ + πT is described by an autonomous differential equation, as the only explicit Λ dependence appears in P . As a consequence the result is independent of the particular choice of the distribution function P ; for simplicity we choose the constant P (ω) = 1 . 2πT (4.60) This leads to the final form of the flow equation ∂ Λ 1 Υ = ∂Λ 2π F ΥΛ (ω) (4.61) ω=±ωn, Λ for Υ. Hence, in the flow equations for the self-energy and the two-particle vertex, Eqs. (4.17) and (4.23) respectively, the expression ω = ±Λ at T = 0 is replaced by ω = ±ωn, Λ at finite temperature, the functional dependence on ω remains the same. Note that for P (ω) = δ(ω − ωn, ω ) the flow equation cannot be simplified by (4.16) and a smooth frequency cutoff has to be chosen [Enss et al. 2005; Enss 2005]. 4.3 Calculation of Kρ The Luttinger-liquid parameter Kρ , which determines the critical exponents of Luttinger liquids, can be computed from the fixed-point couplings as obtained from the fRG. A relation between the fixed-point couplings and Kρ can be established via the exact solution of the fixed-point Hamiltonian of Luttinger liquids, the Luttinger model. A comparison of the fRG result for Kρ with the exact Bethe-ansatz result 49 4 Functional RG for Luttinger liquids for the bulk model (without impurity) serves also as a check for the accuracy of our vertex renormalization. Since the above simplified flow equations yield not only the correct low-energy asymptotics to second order in the renormalized interaction, but contain also all nonuniversal second-order corrections at ±kF from higher energy scales, the resulting Kρ is obtained correctly to second order in the interaction. 4.3.1 Spinless fermions For spinless fermions, Kρ is determined by the Luttinger model parameters g and vF as Kρ = 1 − g/(2πvF ) , 1 + g/(2πvF ) (4.62) where g is the interaction between left and right movers and vF the effective Fermi velocity of the model, that is, the slope of the (linear) dispersion relation, with a possible shift due to interactions between particles moving in the same direction (g4 -coupling) already included [Voit 1995]. We therefore need to extract g and vF from the fRG flow in the limit Λ → 0. In order to obtain Kρ correctly to order U 2 , it is sufficient to obtain vF correctly to linear order in U. The Luttinger model interaction g and the fixed-point coupling g ∗ = ΓΛ→0 F ;kF ,−kF kF ,−k from the fRG are not simply identical, in contrast to what one might naively expect. To find the true relation between g and g ∗ , one has to take into account that the forward scattering limit of the dynamical two-particle vertex is generally not unique (in the absence of cutoffs), and depends on whether momentum or frequency transfers tend to zero first. This ambiguity is well-known in Fermi-liquid theory, where it leads to the distinction between quasi-particle interactions and scattering amplitudes [Negele and Orland 1987], but is equally present in Luttinger liquids, for the same reason in all cases: the ambiguity of the small momentum, small frequency limit of particle-hole propagators contributing to the vertex function. In the dynamical limit, where the momentum transfer q vanishes first, the singular particle-hole propagators do not contribute. In Fermi liquids this limit yields the quasi-particle interaction. In the opposite static limit the frequency transfer ν vanishes first and particle-hole propagators yield a finite contribution. In the presence of an infrared cutoff Λ > 0 the forward scattering limit of the vertex function is unique, since the ambiguity in the particle-hole propagator is due to the infrared pole of the singleparticle propagator. Hence ΓΛF ,−kF ;kF ,−kF is well defined. However, ΓΛF ,−kF ;kF ,−kF k k 50 4.3 Calculation of Kρ and also its limit for Λ → 0 depend on the choice for the cutoff function. For a momentum cutoff, which excludes states with excitation energies below Λ around the Fermi points, particle-hole excitations with small momentum transfers q are impossible. Hence particle-hole propagators with infinitesimal q do not contribute to the vertex at any Λ > 0, such that ΓΛF ,−kF ;kF ,−kF converges to the dynamical k forward scattering limit, which is simply given by the bare coupling constant g in the Luttinger model. For a frequency cutoff the particle-hole propagators with vanishing momentum and frequency transfer yield a finite contribution at Λ > 0, which tends to the static limit for Λ → 0. This can be seen directly by integrating dp [G0 (iω)]2 over Λ from infinity to zero. Hence the vertex ΓΛF ,−kF ;kF ,−kF p k ω=±Λ obtained from our frequency cutoff fRG tends to the static forward scattering limit. For the Luttinger model the static forward scattering limit of the vertex can be obtained from the dynamical effective interaction between left and right movers D(q, iν), which is defined as the sum of particle-hole chains D(q, iν) = g + g Π0 (q, iν) g Π0 (q, iν) g + · · · = − + g 1− g 2 Π0 (q, iν) Π0 (q, iν) − + , (4.63) where Π0 (q, iν) = ± ± 1 q 2π iν ∓ vF q (4.64) is the bare particle-hole bubble for right (+) and left (−) movers. Note that only odd powers of g contribute to the effective interaction between left and right movers. This effective interaction appears naturally in the exact solution of the Luttinger model via Ward identities [Dzyaloshinskii and Larkin 1973; Metzner and Di Castro 1993]. For the static limit one obtains lim D(q, 0) = q→0 g 1 − [g/(2πvF )]2 (4.65) which we identify with our fixed-point coupling g ∗ as obtained from the fRG with frequency cutoff. Inverting this relation between g and g ∗ we obtain g= 2πvF g∗ −πvF + (πvF )2 + (g ∗ )2 . (4.66) For spinless fermions the difference between g and g ∗ appears only at third order in the coupling, but for models with spin the distinction becomes important already at second order. 51 4 Functional RG for Luttinger liquids 1.6 0.05 ∆Kρ 1.4 0 Kρ 1.2 -0.05 -1 1 0 1 0.5 U 1 1.5 2 0.8 0.6 -1 -0.5 0 2 Figure 4.5: Luttinger-liquid parameter Kρ as a function of U at various densities (as in Fig. 4.3) for the spinless fermion model; the inset shows the difference between the fRG result and the exact Bethe ansatz result for Kρ . The Fermi velocity vF can be computed from the (frequency-independent) selfenergy in momentum space as 0 vF = vF + ∂k Σk |kF , (4.67) 0 where vF = ∂k ǫk |kF is the bare Fermi velocity. The self-energy is computed from the flow equation (4.38), which can be rewritten in momentum space as UΛ ∂ Λ Σ =− ∂Λ k π ω=±Λ dp 1 − cos(k − p) , 2π iω − ξp − ΣΛ p (4.68) where ξp = ǫp − µ. The chemical potential µ has to be fixed by the final condition ξkF + ΣkF = 0, where kF = πn depends only on the density, not the interaction. From the tridiagonal structure of Σ in real space, but also from the above expression it follows that ΣΛ has the form ΣΛ = aΛ + bΛ cos k. The functional flow equation k k for ΣΛ yields a coupled set of ordinary flow equations for the coefficients aΛ and k Λ b , with initial conditions aΛ0 = U and bΛ0 = 0. The momentum integrals can be evaluated analytically via the residue theorem, such that the remaining set of two coupled differential equations (with U Λ as input) can be easily solved numerically. The result for vF is correct at least to first order in U, but not necessarily to second 52 4.3 Calculation of Kρ order, since our simplified parametrization captures the two-particle vertex correctly to second order only at the Fermi points. Inserting g and vF into the Luttinger model formula (4.62) we can now compute Kρ as a function of U and density for the microscopic spinless fermion model. In Fig. 4.5 we show results for Kρ (U) for various fixed densities. A comparison with exact results from the Bethe ansatz solution of the spinless fermion model [Haldane 1980] in the inset shows that the fRG results are correct to second order in U and the vertex renormalization scheme described above is very accurate. 1 4.3.2 Spin- 2 fermions The Luttinger-liquid parameter Kρ for spin- 1 fermions is given by 2 Kρ = 1 + (gρ4 − gρ2 )/(πvF ) . 1 + (gρ4 + gρ2 )/(πvF ) (4.69) The coupling constants gρ2 and gρ4 parametrize forward scattering interactions in the charge channel (that is, spin symmetrized) between opposite and equal Fermi points respectively. They are related to the bare singlet and triplet vertices of the Luttinger model by 1 γs| kF ,−kF ;kF ,−kF + 3γt| kF ,−kF ;kF ,−kF 4 1 = γs| kF ,kF ;kF ,kF . 4 gρ2 = gρ4 (4.70) These bare vertices are identical to the dynamical forward scattering limits of the full vertex ΓΛ . On the other hand, the vertex ΓΛ obtained from the fRG with a frequency cutoff yields the static forward scattering limit for Λ → 0 (cf. Sec. 4.3.1). For the Luttinger model, the static forward scattering limit for the vertex can be computed from the effective interactions Dρ2 (q, iν) and Dρ4 (q, iν), which are defined as the sum over all particle-hole chains with the bare interactions gρ2 and gρ4 [S´lyom 1979]. o The summation becomes a simple geometric series if one introduces symmetric and antisymmetric combinations gρ± = gρ4 ±gρ2 and Dρ± (q, iν) = Dρ4 (q, iν)±Dρ2 (q, iν). The static limit of the effective interaction Dρ± (q, iν) yields the relation ∗ gρ± = gρ± 1 − gρ± /(πvF ) (4.71) 53 4 Functional RG for Luttinger liquids between the Luttinger model couplings gρ± and the fixed-point couplings ∗ gρ± = 1 ∗ ∗ ∗ gs4 ± gs2 + 3gt 4 (4.72) from the fRG with frequency cutoff. Inverting (4.71) one obtains Kρ = ∗ 1 − gρ+ /(πvF ) . ∗ 1 − gρ− /(πvF ) (4.73) The Fermi velocity vF can be computed from the self-energy for the translationinvariant pure system as in the spinless case, using the momentum representation of the flow equations (4.55). The results for Kρ from the above procedure are correct to second order in the bare interaction for the Hubbard model and also for the extended Hubbard model. Λ Λ While the flowing couplings gs2 and gt converge only logarithmically to their fixedΛ Λ point values for Λ → 0, the linear combination gs2 + 3gt which enters Kρ converges much faster. 1 0.9 Kρ 0.8 0.7 fRG BA 0.6 0.5 0 0.2 0.4 0.6 0.8 1 n Figure 4.6: Luttinger-liquid parameter Kρ for the Hubbard model as a function of electron density; results from the fRG are compared to exact results from the Bethe ansatz; the upper curves are for U = 1 and the lower ones for U = 2. In Fig. 4.6 we show results for Kρ for the Hubbard model as obtained from the fRG and, for comparison, from the exact Bethe ansatz solution [Frahm and Korepin 54 4.3 Calculation of Kρ 1990; Kawakami and Yang 1990; Schulz 1990]. Details on the solution of the corresponding integral equations are reported in App. B. The truncated fRG yields accurate results at weak coupling except for low densities and close to half filling. In the latter case this failure is expected since umklapp scattering interactions renormalize toward strong coupling, even if the bare coupling is weak. At low densities already the bare dimensionless coupling U/vF is large for fixed finite U, simply because vF is proportional to n for small n, such that neglected higher order terms become important. Note, for comparison, that for spinless fermions with a fixed nearest-neighbor interaction the bare dimensionless coupling at the Fermi level vanishes in the low-density limit. 1 fRG DMRG g-ology 0.9 U = 0.5 Kρ 0.8 U =1 0.7 0.6 0.5 0 0.5 1 1.5 U′ 2 2.5 3 Figure 4.7: Luttinger-liquid parameter Kρ for the quarter-filled extended Hubbard model as a function of U ′ for U = 0.5 and 1; results from the fRG are compared to DMRG data and to results from a one-loop g-ology calculation. For the extended Hubbard model Fig. 4.7 shows a comparison of fRG results for Kρ to DMRG data [Ejima et al. 2005]. The fRG results are exact to second order in the interaction and are thus very accurate for weak U and U ′ . Results from a standard one-loop g-ology calculation as described in Sec. A.2 deviate quite strongly already for U ′ > 0.5. In the g-ology approach interaction processes are classified into backward scattering (g1⊥ ), forward scattering involving electrons from opposite Fermi points (g2⊥ ), from the same Fermi points (g4⊥ ), and umklapp scattering (g3⊥ ). All further momentum dependences of the vertex are discarded. This is justified by 55 4 Functional RG for Luttinger liquids the irrelevance of these momentum dependences in the low-energy limit, but leads to deviations from the exact flow at finite scales, and therefore to less accurate results for the fixed-point couplings. Λ The flow of gi⊥ , i = 1, ..., 4, is plotted in Fig. 4.8, in the upper panel for the quarter-filled Hubbard model with bare interaction U = 1, and for the extended √ Hubbard model with U ′ = U/ 2 in the lower. The fRG result is compared to the result from a one-loop g-ology calculation. The backscattering coupling g1⊥ vanishes logarithmically in both cases, as expected for the Luttinger-liquid fixed point [Voit 1995]. For the pure Hubbard model the good agreement with g-ology results stems from the purely local interaction in real space, since in that case pronounced momentum dependences of the vertex develop only in the low-energy regime where the g-ology parametrization is a good approximation. By contrast, for the extended Hubbard model momentum dependences of the vertex which are not captured by the g-ology classification (except for small Λ) are obviously more important. A generalization of the g-ology parametrization of the vertex to higher dimensions, which amounts to neglecting the momentum dependence normal to the Fermi surface, is frequently used in one-loop fRG calculations in two dimensions [Halboth and Metzner 2000; Zanchi and Schulz 2000; Honerkamp et al. 2001; Kampf and Katanin 2003]. The above comparison indicates that this parametrization works well for the pure Hubbard model, but could be improved for models with nonlocal interactions. The parametrization of the vertex by an effective short-range interaction used here could be easily extended to higher dimensions, where it will probably yield more accurate results, too. The relevance of an improved parametrization of the vertex beyond the conventional g-ology classification has also been demonstrated in a recent fRG analysis of the phase diagram of the half-filled extended Hubbard model [Tam et al. 2005]. The inclusion of the momentum dependence due to the nearest-neighbor interaction component in real space on the right-hand side of the flow equation for the two-particle vertex modifies the flow of the couplings at intermediate scales, before reaching the regime where a g-ology description at weak coupling applies. A small repulsive initial backscattering amplitude may renormalize to an effective attractive one. For negative g1⊥ the renormalization group scales to strong coupling, indicating an instability of the model towards a different ground state characterized by a gap in the spin excitation spectrum [Voit 1995]. In Fig. 4.9 the phase boundary for the Luttinger liquid and spin gap is shown as a function of n and U ′ , as obtained 56 4.3 Calculation of Kρ g4⊥ 1 g3⊥ 0.5 g2⊥ fRG g-ology g1⊥ 0 3 g4⊥ 2 g2⊥ 1 g1⊥ 0 g3⊥ -1 −10 10 10−5 100 Λ 105 1010 Figure 4.8: Flow of vertex on the Fermi points (in g-ology notation) at quarter filling and U = 1; upper panel: Hubbard model, lower panel: extended Hubbard model with √ U ′ = U/ 2; the fRG flow is compared to the one-loop g-ology flow; note that in the upper panel fRG and g-ology results almost coincide. from the fRG together with the result from a one-loop g-ology calculation. The fRG results confirm the spin gap phase at low densities found with numerical Quantum Monte Carlo methods [Clay et al. 1999], where the spin gap develops with increasing U ′ from U ′ = 0. At low densities and close to half filling the truncated fRG results are not meaningful, since renormalization towards strong coupling occurs in 57 4 Functional RG for Luttinger liquids these limits and neglected higher order terms become important, see also Fig. 4.6. Close to half filling the spin gap opens with increasing U ′ from U ′ = U/2, as for the one-loop g-ology calculation. A full functional implementation of the momentum dependence of the two-particle vertex would allow a more detailed analysis of the phase diagram. 3 U =1 U′ 2 g1⊥ > 0 1 0 0 0.2 0.4 0.6 0.8 1 n Figure 4.9: Phase boundary between the Luttinger liquid (g1⊥ > 0) and spin gap phase (g1⊥ < 0) for the extended Hubbard model as a function of n and U ′ for U = 1; results from the fRG (solid lines) are compared to results from a one-loop g-ology calculation (dashed line). 4.4 Observables In the next section we will present results for spectral properties of single-particle excitations near an impurity or boundary, the density profile and the linear conductance. Here we describe how the relevant observables are computed from the solution of the flow equations. 4.4.1 Single-particle excitations Integrating the flow equation for the self-energy ΣΛ down to Λ = 0 yields the physical (cutoff-free) self-energy Σ and the single-particle propagator G = (G−1 −Σ)−1 . From 0 58 4.4 Observables the latter spectral properties of single-particle excitations can be extracted. We focus on local spectral properties, which are described by the local spectral function 1 ρj (ω) = − Im Gjj (ω + i0+ ) , π (4.74) where Gjj (ω + i0+ ) is the local propagator, analytically continued to the real frequency axis from above. In our approximation the self-energy is frequency independent and can therefore be viewed as an effective single-particle potential. The propagator G is thus the Green function of an effective single-particle Hamiltonian. In real space representation this Hamiltonian is given by the tridiagonal matrix heff = h0 + Σ, where the matrix elements of h0 are the hopping amplitudes in H0 , Eq. (4.2). For a lattice with L sites this matrix has L (including possible multiplicities) eigenvalues ǫλ and an orthonormal set of corresponding eigenvectors ψλ . For the spectral function ρj (ω) one thus obtains a sum of δ peaks ρj (ω) = λ wλj δ(ω − ξλ) , (4.75) where ξλ = ǫλ − µ, and the spectral weight wλj is the squared modulus of the amplitude of ψλ on site j. For large L the level spacing between neighboring eigenvalues is usually of order L−1 , except for one or a few levels outside the band edges which correspond to bound states. Due to even-odd effects etc. the spectral weight wλj generally varies quickly from one eigenvalue to the next one. A smooth function of ω which suppresses these usually irrelevant finite-size details can be obtained by averaging over neighboring eigenvalues. In addition, dividing the spectral weight wλj by the level spacing between eigenvalues yields the local density of states, which we denote by Dj (ω). 4.4.2 Density profile Boundaries or impurities induce a density profile with long-range Friedel oscillations, which are expected to decay with a power law at long distances [Egger and Grabert 1995]. The expectation value of the local density nj could be computed from the local one-particle propagator Gjj , if G was known exactly. However, the approximate flow equations for Σ can be expected to describe the asymptotic behavior of G correctly only at long distances between creation and annihilation operator in time and/or 59 4 Functional RG for Luttinger liquids space, while in the local density operator time and space variables coincide. In the standard renormalization-group terminology nj is a composite operator, which has to be renormalized separately [Zinn-Justin 2002]. The flow equation for nΛ can be derived by computing the shift of the grand j Λ canonical potential Ω generated by a small field φj coupled to the local density. Alternatively to the numerical differentiation one may carry out the φj derivative analytically in the flow equations, which yields a flow equation for the density in terms of the density response vertex. The general structure at T = 0 is described in Refs. [Andergassen et al. 2004; Enss 2005]; the final form of the flow equation for the density is 1 ∂ Λ nj = − ∂Λ 2π + Λ ˜ tr eiω0 GΛ (iω) Rj (iω) , (4.76) ω=±Λ with the density response vertex given by 1 ∂ Λ Rj;1′ ,1 = − ∂Λ 2π Λ ˜ 2,3 ˜3 GΛ (iω) Rj;3,3′ GΛ′ ,2′ (iω) ΓΛ′ ,2′ ;1,2 . 1 ω=±Λ 2,2′ (4.77) 3,3′ Note that within the approximate treatment of the two-particle vertex described in Sec. 4.2.2, the density-response vertex is frequency independent. For ΓΛ parametrized by local and nearest-neighbor interactions in real space, Λ Λ Λ the matrix Rj is tridiagonal, that is, only the components Rj;3,3 and Rj;3,3±1 are nonzero. The initial condition for the density is nΛ0 = 1 for any filling, due to j 2 the slow convergence of the flow equation (4.76) at large frequencies, which yields a finite contribution to the integrated flow from Λ = ∞ to Λ0 for arbitrarily large finite Λ0 , as in the case of the self-energy discussed in more detail in Sec. 4.2.2. The Λ0 initial condition for the response vertex is Rj;l,l′ = δjl δll′ . To avoid the interference of Friedel oscillations emerging from the impurity or one boundary with those coming from the (other) boundaries of our systems we suppress the influence of the latter by coupling the finite chain to semi-infinite noninteracting leads, with a smooth decay of the interaction at the contacts, as described in detail in the next section. 4.4.3 Conductance For the calculation of the conductance G a finite interacting chain (with sites 1, . . . , L) is coupled to noninteracting leads at both ends, corresponding to an ex- 60 4.4 Observables perimental setup where the Luttinger-liquid wires are connected to (higher dimensional) Fermi-liquid leads. Using a projection method the system with leads can be reduced to an effective L-site problem [Taylor 1972]. The leads are modeled by a one-dimensional tight-binding lattice with nearest-neighbor hopping amplitude t. The influence of the leads on the interacting chain can be taken into account by an additional dynamical boundary potential Vjlead (iωn ) = iωn + µ0 2 1− 1− 4 (iωn + µ0 )2 (δ1,j + δL,j ) (4.78) in the bare propagator G0 of the interacting chain [Enss et al. 2005]. The parameter µ0 is the chemical potential, which is related to the density n in the leads by µ0 = −2 cos kF . Uncontrolled conductance drops due to scattering at the contacts between leads and the interacting part of the chain can be avoided by switching off the interaction potential smoothly near the contacts. As long as the switching on of the interaction is smooth enough and the bulk part of the wire is large compared to the switching region, the results are independent of the microscopic details of this procedure. In addition, interaction-induced bulk shifts of the density have to be compensated by a suitable bulk potential [Enss et al. 2005]. In linear response the conductance is computed via the Kubo formula from the current-current correlation function [Mahan 2000]. Within our approximation scheme the self-energy has no imaginary part, which implies that there are no vertex corrections [Oguri 2001]. In this approximation the Kubo formula reduces to the Landauer-B¨ ttiker formula [Datta 1995], relating the transmission probability diu rectly to the linear conductance G(T, L) of noninteracting fermions. For a detailed derivation we refer to Refs. [Enss et al. 2005; Enss 2005]; the final expression for the conductance is given by G(T, L) = −z e2 h 2−µ0 −2−µ0 |t(ε, T, L)|2 f ′ (ε) dε , (4.79) with |t(ε, T, L)|2 = [4 − (µ0 + ε)2 ]|G1,L (ε, T )|2, and f the Fermi function; z is the number of spin components. 61 5 Solution of fRG equations and results In this section we present results for the local density of states near boundaries and impurities, the density profile, and the linear conductance for the spinless fermion model and the extended Hubbard model, as obtained from the solution of the fRG flow equations. A comparison with exact DMRG data is made for those observables and system sizes for which such data could be obtained. The asymptotic low-energy behavior for weak and intermediate impurity strengths is approached only at rather low 1 energy scales, accessible only for very large systems. For spin- 2 systems two-particle backscattering leads to striking effects, which are not captured if the bulk system is approximated by its low-energy fixed point, the Luttinger model. In particular, the expected decrease of spectral weight near the impurity and of the conductance at low energy scales is often preceded by a pronounced increase, and the asymptotic power laws are modified by logarithmic corrections. 5.1 Spinless fermions 5.1.1 Effective impurity potential The typical shape of the self-energy representing the effective impurity potential can be seen in Fig. 5.1, where we plot the diagonal elements Σj,j and the off-diagonal elements Σj,j+1 near a site impurity of strength V = 1.5 added to the spinless fermion model with interaction strength U = 1 at quarter filling. Recall that the selfenergy is tridiagonal in real space and frequency independent within our treatment. The diagonal elements can be interpreted as a local effective potential, the offdiagonal elements as a nonlocal effective potential which renormalizes the hopping 62 5.1 Spinless fermions amplitudes. At long distances from the impurity both Σj,j and Σj,j+1 tend to a constant. The former describes just a bulk shift of the chemical potential, the latter a bulk renormalization of the hopping amplitude toward larger values. The oscillations around the bulk shifts are generated by the impurity. The wave number of the oscillations is 2kF = π/2, where kF is the Fermi wave vector of the bulk system at quarter filling. Fig. 5.2 shows ΣΛ ,j0 for a site impurity of strength V = 1.5 as a j0 function of Λ for different system sizes L. For finite L the flow is effectively cut off on a scale ∼ 1/L, a sequence of L provides an extrapolation to the thermodynamic limit. The renormalized potential at the impurity site remains finite in the limit L → ∞, while the Fourier transform ΣΛ ′ for momenta with k − k ′ = ± 2kF k,k diverges. Similarly for a hopping impurity the effective amplitude does not scale to zero in the limit L → ∞, and the weak-link behavior is associated to the long-range oscillations in real space. The straight line in a log-log plot of the difference between the asymptotic value Σ00 ,j0 (L = ∞) and Σ00 ,j0 (L) shown in the inset indicates a j j power law dependence. 0.5 0.4 Σj,j 0.3 -0.12 -0.16 Σj,j+1 -0.2 -75 -50 -25 j − j0 0 25 Figure 5.1: Self-energy near a site impurity of strength V = 1.5 for the spinless fermion model at quarter filling and interaction strength U = 1; the impurity is situated at the center of a chain with L = 1025 sites. The amplitude of the oscillations generated by an impurity in Σ decays slower than the inverse distance from the impurity at intermediate length scales, but approaches a decay proportional to 1/|j − j0 | for |j − j0 | → ∞. This can be seen most 63 5 Solution of fRG equations and results L = 25 L = 26 L = 27 L = 28 L = 29 1.7 ΣΛ0 ,j0 j 10−2 10−3 1.6 10−4 10−5 10−6 10−7 102 1.5 −4 10 10−3 103 10−2 104 105 10−1 Λ 100 101 102 Figure 5.2: ΣΛ ,j0 as a function of Λ for a site impurity of strength V = 1.5 for spinless j0 fermions at half filling and U = 1; the impurity is situated at the center of a chain of length L; the inset shows the difference from the asymptotic value Σ00,j0 (∞) − Σ00 ,j0 (L) as j j a function of L. clearly by plotting an effective exponent βj for the decay of the oscillations, defined as the negative logarithmic derivative of the oscillation amplitude with respect to the distance |j − j0 |. In Fig. 5.3 we show the effective exponent resulting from the oscillations of Σj,j as a function of the distance from a site impurity, for U = 1 and half filling. The impurity is situated at the center of a long chain with L = 218 + 1 sites. To avoid interferences with oscillations from the boundaries we have attached semi-infinite noninteracting leads to the ends of the interacting chain, as described in Sec. 4.4.3. Only for relatively large impurity strengths the asymptotic regime corresponding to βj = 1 is reached before finite-size effects set in. For small V one can see that βj increases from values below one, but the asymptotic long-distance behavior is cut off by the boundaries of the interacting region. For very small V (for example V = 0.01 in Fig. 5.3) we observe a plateau in βj for intermediate distances from the impurity site. In this regime βj is close to Kρ which can be understood by analytically solving the flow equations for small V [Meden et al. 2002a,b]. 64 5.1 Spinless fermions exponent 1 0.9 0.8 0.7 101 102 103 j − j0 104 105 Figure 5.3: Effective exponent for the decay of oscillations of Σj,j as a function of the distance from a site impurity of strengths V = 0.01, 0.1, 0.3, 1, 10 (from bottom to top), for the spinless fermion model at half filling and interaction strength U = 1; the impurity is situated at the center of a chain with L = 218 + 1 sites. 5.1.2 Local density of states The long-range 2kF -oscillations of the self-energy lead to a marked suppression of the spectral weight for single-particle excitations at the Fermi level, that is, at ω = 0. In Fig. 5.4 we show the local density of states Dj (ω) on the site next to a site impurity of strength V = 1.5 for the spinless fermion model at half filling. The result for the interacting system at U = 1 is compared to the noninteracting case. Even-odd effects have been eliminated by averaging over neighboring eigenvalues (cf. Sec. 4.4.1). δ peaks outside the band edges corresponding to bound states are not plotted. The interaction leads to a global broadening of the band, which is due to an enhancement of the bulk hopping amplitude, and also to a strong suppression of Dj (ω) at low frequencies which is not present in the noninteracting system. For a finite system (here L = 1025) the spectral weight at the Fermi level remains finite, but tends to zero with increasing system size. In Fig. 5.5 we show results for the density of states choosing the same parameters as in Fig. 5.4, but now for densities away from half filling: n = 1/4 and n = 3/4. In addition to the dip near ω = 0 a second singularity appears at a finite frequency. This effect is due to the fact that a long-range potential with a wave number 2kF does not only strongly 65 5 Solution of fRG equations and results scatter states with momenta near kF , but also those with momenta close to π − kF . Indeed the singularity is situated at ω = ǫπ−kF − µ, where ǫk is the renormalized (bulk) dispersion. In the half-filled case only one singularity is seen simply because π − kF = kF for kF = π/2. 0.4 U =0 U =1 Dj0 −1 0.3 0.2 0.1 0 -3 -2 -1 0 ω 1 2 3 Figure 5.4: Local density of states on the site next to a site impurity of strength V = 1.5 for spinless fermions at half filling and U = 1; the impurity is situated at the center of a chain with 1025 sites; the noninteracting case U = 0 is shown for comparison. Similar results are found for a hopping impurity, shown in Fig. 5.6. For an attractive interaction the density of states is strongly enhanced at the Fermi level. For comparison, typical results for the local density of states at a boundary are presented in Fig. 5.7. The spectral weight at the Fermi level is expected to vanish asymptotically as a power law |ω|αB , where αB = 1 −1 Kρ (5.1) is the boundary exponent describing the power-law suppression of the density of states at the boundary of a semi-infinite chain with repulsive interactions [Kane and Fisher 1992b]. That exponent depends only on the bulk parameters of the model, not on the impurity strength. For the spinless fermion model it can be computed exactly from the Bethe ansatz solution [Haldane 1980]. 66 5.1 Spinless fermions 0.5 n = 1/4 n = 3/4 0.4 Dj0 −1 0.3 0.2 0.1 0 -4 -3 -2 -1 0 ω 1 2 3 4 Figure 5.5: Local density of states on the site next to a site impurity as in Fig. 5.4 (same parameters), but now for densities n = 1/4 and 3/4. 1 0.8 U =1 U = −1 Dj0 0.6 0.4 0.2 0 -4 -3 -2 -1 ω 0 1 2 Figure 5.6: Local density of states on the site next to a hopping impurity t′ = 0.25 for spinless fermions at density n = 0.6 and U = 1; the impurity is situated at the center of a chain with 1024 sites; the attractive case U = −1 is shown for comparison. In the above figures the envelope of the δ peaks of weight w characterizing the spectral function of a finite system size introduced in Sec. 4.4.1 is shown. As a consequence the energy range over which a power-law suppression is observed is 67 5 Solution of fRG equations and results 1 0.8 U =1 U = −1 D1 0.6 0.4 0.2 0 -4 -3 -2 -1 ω 0 1 2 Figure 5.7: Local density of states at the boundary, for the same parameters as in Fig. 5.6. cut off by the finite size of the system at low energies. For a reliable analysis of the exponential behavior it is more convenient to consider the finite-size scaling of the spectral weight at the chemical potential. The large L dependence of the spectral weight is expected to exhibit a power law with the same exponent, and the scale where the power-law behavior sets in is given by L ∼ ω/πvF . In Fig. 5.8 the spectral weight at a boundary of a half-filled chain of length L = 106 and different interaction strengths U is shown as a function of ω. The straight line in a loglog plot corresponds to a power law. Results for the same density and interaction parameters, but now as a function of system size L are presented in Fig. 5.9. Within the extension to finite temperatures (cf. Sec. 4.2.5), the same power-law behavior is found as a function of temperature, a detailed analysis in the context of transport phenomena follows below. We now analyze the asymptotic behavior of the spectral weight at the Fermi level by defining an effective exponent α(L) as the negative logarithmic derivative of the spectral weight with respect to the system size, such that α(L) tends to a (positive) constant in case of a power law suppression. In Fig. 5.10 we show results for α(L) as obtained from the fRG for the spinless fermion model at quarter filling with up to about 106 sites, for a weak (U = 0.5) and an intermediate (U = 1.5) interaction parameter. The spectral weight has been computed either at a bound- 68 5.1 Spinless fermions 100 Lw 10−1 10−2 U = 0.5 U =1 U =2 10−3 10−4 10−6 10−5 10−4 10−3 ω 10−2 10−1 Figure 5.8: Spectral weight at a boundary as a function of ω for the spinless fermion model at half filling, L = 106 and different interaction strengths U . 100 Lw 10−1 10−2 U = 0.5 U =1 U =2 10−3 10−4 1 10 102 103 104 105 106 L Figure 5.9: Spectral weight at the Fermi level as a function of system size L, for the same parameters as in Fig. 5.8. ary, or near a hopping impurity of strength t′ = 0.5. Results obtained from the fRG without (upper panel) and with (lower panel) vertex renormalization, corresponding to Scheme I and Scheme II introduced in Sec. 4.2.2, are compared to exact numerical DMRG results (for up to 512 sites) and the exact boundary exponents 69 5 Solution of fRG equations and results 0.4 0.3 α 0.2 0.1 0 -0.1 0.4 0.3 α 0.2 0.1 0 -0.1 102 103 104 L 105 106 Figure 5.10: Logarithmic derivative of the spectral weight at the Fermi level near a boundary (solid lines) or hopping impurity (dashed lines) as a function of system size L, for spinless fermions at quarter filling and interaction strength U = 0.5 (circles) or U = 1.5 (squares); upper panel: without vertex renormalization, lower panel: with vertex renormalization; the open symbols are fRG, the filled symbols DMRG results; the horizontal lines represent the exact boundary exponents for U = 0.5 and 1.5.; in the boundary case (solid lines) the spectral weight has been taken on the first site of a homogeneous chain, in the impurity case (dashed lines) on one of the two sites next to a hopping impurity t′ = 0.5 in the center of the chain. 70 5.1 Spinless fermions αB , plotted as horizontal lines. The fRG results follow a power law for large L, with the same asymptotic exponent for the boundary and impurity case, confirming thus the expected universality. However, the asymptotic regime is reached only for fairly large systems, even for the intermediate interaction strength U = 1.5. For the fRG Scheme I developed previously for impurities in spinless Luttinger liquids the effects of a single static impurity in a spinless Luttinger liquid are fully captured qualitatively, and in the weak-coupling limit also quantitatively. Originally developed for the analysis of spectral densities of single-particle excitations, this scheme has been applied recently also to transport problems, such as persistent currents in mesoscopic rings and the conductance of interacting wires connected to noninteracting leads [Meden et al. 2003; Meden and Schollw¨ck 2003a,b]. The comparison o with the exact DMRG results and exact exponents shows that the fRG Scheme II is also quantitatively rather accurate, and that the inclusion of vertex renormalization leads to a substantial improvement at intermediate coupling strength. A quantitative estimate of the accuracy of the exponents can be obtained from a comparison to exact results from the Bethe ansatz solution of the spinless fermion model [Haldane 1980] shown in Fig. 5.11. The results obtained without vertex renormalization (Scheme I) are represented with open symbols. The fRG results are correct to first order in the interaction U. In the approximate treatment of the two-particle vertex (cf. Sec. 4.2.2) terms of order U 2 are only partially included, and an agreement to higher order can not be expected. Nevertheless the quantitative accuracy is improved considerably. The n dependence of the accuracy is related to the different importance of the vertex renormalization for different densities, as can be seen in Fig. 4.3. For half filling the vertex renormalization leads to a pronounced increase of the renormalized interaction, whereas for smaller fillings the effect is smaller, eventually leading to a decrease for n 1/3. Results for the effective exponent α in the case of a site impurity are shown in Fig. 5.12, at quarter filling and for an interaction strength U = 1. The comparison of the different curves obtained for different impurity strengths confirms once again the expected asymptotic universality, and also how the asymptotic regime shifts rapidly toward larger systems as the bare impurity strength decreases. The crossover scale depends on the bare impurity strength V as [Kane and Fisher 1992a,c; Fendley et al. 1995] V 1 ∼ Lc πvF 1 1−Kρ , (5.2) 71 5 Solution of fRG equations and results 0.06 0.04 0.02 ∆ αB 0 -0.02 -0.04 -0.06 -0.08 -0.1 -0.5 n = 1/2 n = 1/4 0 0.5 U 1 1.5 Figure 5.11: Difference between the fRG result and the exact Bethe ansatz result for the boundary exponent αB as a function of U , at densities n = 1/2 and 1/4 for the spinless fermion model as obtained from the power-law suppression of the spectral weight at the boundary; the open symbols are results without vertex renormalization, the filled symbols results with vertex renormalization. in agreement with our findings. The scale on which the impurity flows to strong coupling depends on the initial strength V and the “flow velocity” given by 1 − Kρ . The effective flowing impurity strength can be estimated by Veff ∼ V L1−Kρ , (5.3) where the crossover scale Lc in Eq. (5.2) corresponds to Veff ∼ πvF . Note that for weak impurities and intermediate system sizes the spectral weight follows a power law corresponding to the bulk behavior, and approaches the boundary exponent only at large distances. The bulk suppression of the spectral weight described by −1 the anomalous dimension α = (Kρ + Kρ − 2)/2 is not captured within the present scheme, since the self-energy is frequency independent. In the 1PI version of the fRG the frequency dependence is generated by the two-particle vertex. The anomalous dimension could be included in an improved scheme by an iterative solution of the fRG flow inserting the two-particle vertex into the flow equation for the selfenergy without neglecting its frequency dependence. This gives a two-loop diagram including the full two-particle vertex functions at scale Λ. The right-hand side of the differential equation for the self-energy is then nonlocal in Λ: the change of 72 5.1 Spinless fermions the two-particle vertex and self-energy at scale Λ involves two-particle vertices and self-energies at scales Λ ≥ Λ′ . In presence of sufficiently strong effective impurity potentials however, the boundary behavior prevails over the bulk suppression of the spectral weight, as αB ∼ U and α ∼ U 2 . 0.2 α 0.1 0 boundary V =3 V =1 V = 0.3 V = 0.1 -0.1 -0.2 102 103 104 L 105 106 Figure 5.12: Logarithmic derivative of the spectral weight at the Fermi level near a boundary (solid line) or site impurity (dashed lines) as a function of system size L, for spinless fermions at quarter filling and interaction strength U = 1; in the boundary case the spectral weight has been taken on the first site of a homogeneous chain, in the impurity case on the site next to a site impurity of strength V in the center of the chain; the horizontal line represents the exact boundary exponent for U = 1. 5.1.3 Friedel oscillations We now discuss results for the density profile nj . Boundaries and impurities induce Friedel oscillations of the local density with a wave vector 2kF . In a noninteracting system these oscillations decay proportionally to the inverse distance from the boundary or impurity. In an interacting Luttinger liquid the Friedel oscillations are expected to decay as |j − j0 |−Kρ at long distances |j − j0 |. For a very weak impurity one expects a slower decay proportional to |j − j0 |1−2Kρ at intermediate distances, and a crossover to the asymptotic power law with exponent Kρ at very long distances [Egger and Grabert 1995]. At intermediate distances the response of 73 5 Solution of fRG equations and results the density to a weak impurity can be treated in linear response theory, such that the density modulation is determined by the density-density response function at 2kF , which leads to the power-law decay with exponent 2Kρ − 1. 10−1 10−2 amplitude 10−3 10−4 10−5 10−6 10−7 U U U U = 0.1 = 0.5 = 1.0 = 1.5 10−8 1 exponent 0.9 0.8 0.7 0.6 0.5 1 10 102 103 104 105 j Figure 5.13: Amplitude (envelope) of oscillations of the density profile nj induced by a boundary as a function of the distance from the boundary, for spinless fermions with various interaction strengths U at half filling; the interacting chain with 219 + 1 sites is coupled to a semi-infinite noninteracting lead at the end opposite to the boundary; upper panel: log-log plot of the amplitude, lower panel: effective exponents for the decay, and the exact asymptotic exponents as horizontal lines. We analyze the long-distance behavior of the amplitudes more closely for the half-filled case, and compare to exact results for the asymptotic exponents. For 74 5.1 Spinless fermions incommensurate filling factors the density profile looks more complicated, but at long distances from the boundary the oscillation amplitude has a well-defined envelope which exhibits a power law as a function of j. In Fig. 5.13 we show fRG results for the amplitude of density oscillations emerging from an open boundary, for a very long spinless fermion chain with 219 + 1 sites and various interaction strengths U. The end opposite to the open boundary is smoothly connected to a noninteracting lead. In a log-log plot (upper panel of Fig. 5.13) the amplitude follows a straight line for almost all j, corresponding to a power-law dependence. Deviations from a perfect power law can be seen more neatly by plotting the effective exponent αj , defined as the negative logarithmic derivative of the amplitude with respect to j (see the lower panel of Fig. 5.13). The effective exponent is almost constant except at very short distances or when j approaches the opposite end of the interacting chain, which is not surprising. From a comparison with the exact exponent (horizontal lines in the figure) one can assess the quantitative accuracy of the fRG results. exponent 0.7 0.6 0.5 0.4 1 10 102 103 j − j0 104 105 Figure 5.14: Effective exponent for the decay of density oscillations as a function of the distance from a site impurity of strengths V = 0.01, 0.1, 0.3, 1, 3, 10 (from bottom to top); the impurity is situated at the center of a spinless fermion chain with 218 + 1 sites and interaction strength U = 1 at half filling; the interacting chain is coupled to semi-infinite noninteracting leads at both ends. Effective exponents describing the decay of Friedel oscillations generated by site impurities of various strengths are shown in Fig. 5.14, for a half-filled spinless fermion 75 5 Solution of fRG equations and results chain with 218 +1 sites and interaction U = 1. Both ends of the interacting chains are coupled to noninteracting leads to suppress oscillations coming from the boundaries. For strong impurities the results are close to the boundary result (cf. Fig. 5.13), as expected. For weaker impurities the oscillations decay more slowly, that is, with a smaller exponent, and approach the boundary behavior only asymptotically at large distances (beyond the range of our chain for V < 1). For very weak impurities (V = 0.01 in Fig. 5.14) the oscillation amplitude follows a power law corresponding to the linear response behavior with exponent 2Kρ − 1 at intermediate distances. Similar results are obtained for attractive interactions [Andergassen et al. 2004]. A comparison with the exact exponents from the Bethe ansatz solution of the spinless fermion model [Haldane 1980] is shown in Fig. 5.15, where the difference ∆Kρ is reported as a function of the interaction U at densities n = 1/2 and n = 1/4. The open symbols represent the accuracy for the weak-impurity behavior in the linear response regime and the filled ones for the asymptotic exponent for strong impurities at long distances. Deviations from the exact result are quadratic in the interaction U. 0.07 0.06 ∆ Kρ 0.05 n = 1/2 n = 1/4 0.04 0.03 0.02 0.01 0 -0.5 0 0.5 U 1 1.5 Figure 5.15: Difference between the fRG result and the exact Bethe ansatz result for the Luttinger-liquid parameter Kρ as a function of U at densities n = 1/2 and 1/4 for the spinless fermion model, as obtained from the power-law decay of Friedel oscillations generated by a strong impurity at long distances (filled symbols), and by a weak impurity at intermediate distances (open symbols). 76 5.1 Spinless fermions 5.1.4 Scaling of the conductance In this section we study the transport through an interacting wire with a single impurity connected to semi-infinite noninteracting leads. The conductance exhibits the same asymptotic scaling behavior as a function of temperature T for an infinite wire as at T = 0 as a function of L. For more than one impurity the T dependence is richer, showing nonmonotonic behavior and distinctive power-laws with different universal exponents in various regimes; an extensive analysis is reported in Ref. [Enss et al. 2005; Enss 2005]. Although this difference does not appear for a single impurity, we will mainly focus on the more physical temperature dependence of the conductance. Before analyzing the scaling behavior we present in Fig. 5.16 a comparison of fRG results to numerical DMRG data for the conductance for a short wire at T = 0, determined from the persistent current observed in the presence of a magnetic flux piercing a noninteracting ring in which the interacting wire is embedded [Meden et al. 2003; Meden and Schollw¨ck 2003a,b]. The excellent agreement o proves the reliability of the approximate fRG scheme for interactions in the range 1/2 ≤ Kρ ≤ 1. 1 G/(e2 /h) 0.8 fRG DMRG 0.6 0.4 0.2 0 0 1 2 3 4 5 U Figure 5.16: Conductance as a function of the interaction U for a homogeneous spinless fermion chain at half filling, with L = 12 sites; the interaction is turned on sharply at the contacts. Fig. 5.17 shows typical fRG results for the T dependence of the conductance 77 5 Solution of fRG equations and results through a single site impurity of strength V . The 1/T scaling observed for high temperatures (of the order of the bandwidth) results from the 1/T behavior of the derivative of the Fermi function in Eq. (4.79), together with the weak temperature dependence of |t(ε, T, L)|2 at high T . For a strong impurity V = 10, G(T ) follows a power law with exponent 2αB as indicated by the dashed line in Fig. 5.17, until saturation sets in for T ∼ πvF /L. For an intermediate impurity the slope of the data tends towards the asymptotic exponent, but is still significantly away from it when finite-size saturation sets in. This slow change of the slope is a general feature of intermediate V . For a weak impurity G(T ) approaches e2 /h. Similar behavior is found for the scaling of 1 − G(T )/(e2 /h) in the limit of a weak impurity predicted to follow T 2(Kρ −1) , which holds as long as the correction to perfect conductance stays small [Kane and Fisher 1992a,c; Fendley et al. 1995]. G/(e2 /h) 100 10−1 V = 10 V =1 V = 0.1 10−2 10−3 −4 10 10−3 10−2 10−1 100 101 T Figure 5.17: Temperature dependence of the conductance for a half-filled spinless fermion wire of length L = 104 , interaction U = 0.5 and a single site impurity of strengths V at the center of the wire; the dotted lines highlight power-law behavior. The above results are generic as long as the impurity is placed sufficiently away from the contact regions. The scale δj0 = πvF /j0 , where j0 is the impurity position, sets a lower bound for the power-law scaling with the exponents discussed above [Furusaki and Nagaosa 1996]. For T ≃ δj0 a crossover to a power-law scaling with different exponents is found. For impurity positions in the contact region the exponent αB describes the tunneling between the noninteracting and the interacting 78 5.1 Spinless fermions Luttinger liquids [Enss et al. 2005]. Restrictions on the temperature range where universal scaling behavior might be detected arise from the bandwidth from above and the finite wire length from below. For an interacting wire of L lattice sites the energy scale δL = πvF /L represents a lower bound for any temperature scaling. Depending on the impurity and interaction strength an asymptotic low-energy regime might not be reachable in experiments on finite wires. For finite temperatures systems of 104 lattice sites are considered, comparable for typical lattice constants to quantum wires in the micrometer range accessible to transport experiments. Considering the conductance as a function of temperature and impurity strength, for a fixed Kρ the renormalization-group flow from weak to strong impurity strength ˜ determines a scaling function GKρ (x) on which the data for different T and V collapse [Kane and Fisher 1992a,c; Moon et al. 1993; Fendley et al. 1995]. Using a one-parameter scaling ansatz e2 ˜ with x = [T /T0 (U, n, V )]Kρ −1 , (5.4) GKρ (x) , h the curves for G(T ) and different V can be collapsed onto the Kρ -dependent scal˜ ing function GKρ (x) for an appropriate nonuniversal scale T0 (U, n, V ). It has the ˜ ˜ limiting behavior GKρ (x) ∼ 1 − x2 for x → 0, and GKρ (x) ∼ x−2/Kρ for x → ∞; ˜ for Kρ = 1/2 and Kρ = 1/3 the functional form of GKρ was determined explicitly [Kane and Fisher 1992c; Moon et al. 1993; Fendley et al. 1995]. An example is shown in Fig. 5.18 for U = 0.5, the different colors stand for different impurity strengths V . As a consequence of the extended crossover region between weak and strong-impurity behavior, even for the fairly large system size of L = 104 sites and the large range of temperatures we can treat, it is impossible to directly demonstrate the full crossover for a single set of parameters. A power-law behavior in both limits is found, with exponents which can be expressed consistently in terms of a single approximate Luttinger-liquid parameter Kρ . Data for the same Kρ but different interaction and filling parameters collapse on the results for half filling, since the scaling function depends on U and n only via the Luttinger-liquid parameter Kρ . Considering different types of impurity potentials extending over more than one site ˜ or bond does not modify GKρ . A previous analysis of the zero temperature scaling behavior in the wire length L, replacing T in the above ansatz by πvF /L, showed that one-parameter scaling is not affected by the presence of leads if the interaction is turned on very smoothly at the contacts and no one-particle scattering terms at the contacts are considered G= 79 5 Solution of fRG equations and results 100 G/(e2 /h) 1 − x2 10−1 x−2/Kρ 10−2 10−3 10−2 10−1 100 101 x = [T /T0(U, n, V )]Kρ−1 Figure 5.18: One-parameter scaling plot of the conductance for the spinless fermion model at half filling with U = 0.5; the colors represent results obtained for different impurity strengths; the dashed lines indicate the asymptotic behavior for small and large x. [Meden et al. 2003]. In addition, fRG data are found to collapse onto the K = 1/2 local sine-Gordon scaling function known analytically [Meden et al. 2003; Enss et al. 2005]. We finally remark that one-parameter scaling represents an excellent example for the power of the fRG technique capturing complex crossover phenomena at intermediate scales. There is an interesting correspondence between the transport through impurities in a Luttinger liquid and the quantum Brownian motion in a cosine potential examined in Ref. [Weiss 1999]. The duality symmetry of the latter maps a weak impurity (small V ) exactly to a strong impurity (large V ) under the substitution Kρ → 1/Kρ. For Kρ < 1 the system becomes localized, whereas the effective barrier height vanishes for Kρ > 1. This symmetry can be extended also to finite temperatures by a frequency-dependent transformation [Weiss 1999]. The equivalence between the mobility of the Brownian particle and the conductance through an impurity in a Luttinger liquid relates the behavior for a strong impurity to the one for an appropriate weak impurity by Gstrong (T, Kρ )/(e2 /h) = 1 − Gweak (T, 1/Kρ )/(e2 /h) . (5.5) In particular, the above relation holds for expansions around V ≪ 1 and V ≫ 1. 80 1 5.2 Spin- 2 fermions The respective convergence radius defines the crossover scale between the two dual descriptions by Tcstrong (Kρ ) = Tcweak (1/Kρ). For Kρ > 1 high and low temperatures are exchanged. Similar scaling behavior is found for the nonlinear conductance, where the bias assumes the role of the temperature [Weiss 1999]. As a consequence of this symmetry and the knowledge of the analytic form for a particular value of Kρ = 1 the explicit scaling function can be derived, as well as an expression for the crossover scale [Weiss 1999]. The solution reproduces the result from the thermodynamic Bethe ansatz [Fendley et al. 1995]. 5.2 Spin- 1 fermions 2 5.2.1 Single-particle excitations For ω → 0 the spectral weights and the local density of states near a boundary or impurity are ultimately suppressed according to a power law with the boundary exponent αB = 1 1 + −1 , 2Kρ 2Kσ (5.6) with Kσ = 1 for spin-rotation invariant systems [Giamarchi 2004]. However, due to the slow logarithmic decrease of the two-particle backscattering amplitude, the fixed point value of Kσ is reached only logarithmically from above. Hence, we can expect that the asymptotic value of αB is usually reached only very slowly from below. The local density of states at the boundary of a quarter-filled Hubbard chain, computed by the fRG, is shown in Fig. 5.19 for various values of the local interaction U. Contrary to the expected asymptotic power-law suppression the spectral weight near the chemical potential is strongly enhanced. The predicted suppression occurs only at very small energies for sufficiently large systems. In the main panel of Fig. 5.19 the crossover to the asymptotic behavior cannot be observed, as the finitesize cutoff ∼ πvF /L is too large. Results for a larger system with L = 106 sites at U = 2 in the inset show the crossover to the asymptotic suppression, albeit only at very small energies. The dependence of the boundary spectral weight at the Fermi level on the system size L is plotted in Fig. 5.20. The L dependence of the spectral weight at zero energy is expected to display the same asymptotic powerlaw behavior for large L as the ω dependence discussed above. Instead of decreasing with increasing L, the spectral weight increases even for rather large systems for 81 5 Solution of fRG equations and results small and moderate values of U. For U > 2 the crossover to a suppression is visible in Fig. 5.20. For U = 0.5 only an increase is obtained up to the largest systems studied. The crossover depends sensitively on the interaction strength U, for small U it is exponentially large in πvF /U. 0.4 D1 0.3 0.4 0.2 U U U U 0.1 0 -0.4 -0.2 =0 = 0.5 =1 =2 0 0.35 -0.01 0.2 0.4 0 0.6 0.01 0.8 1 ω Figure 5.19: Local density of states at the boundary of a Hubbard chain of length L = 4096 at quarter filling and various interaction strengths U ; the inset shows results for U = 2 and L = 106 at very low ω. The above behavior of the spectral weight and the density of states near a boundary of the Hubbard chain, that is, a pronounced increase preceding the asymptotic power-law suppression, is captured qualitatively already by the Hartree-Fock approximation [Meden et al. 2000; Sch¨nhammer et al. 2000]. This is at first sight o surprising, as the Hartree-Fock theory does not capture any Luttinger-liquid features in the bulk of a translational invariant system. In particular, a self-consistent Hartree-Fock calculation leads to the unphysical result of a charge-density-wave ground state for all U > 0 [Cohen et al. 1998], since a single impurity can not modify bulk properties of the system. The initial increase of Dj (ω) near a boundary is actually obtained already within perturbation theory at first order in the interaction [Meden et al. 2000], 0 Dj (ω) = Dj (ω) 82 1+ ˜ ˜ V (0) − z V (2kF ) ˜ ln |ω/ǫF | + O(V 2 ) 2πvF (5.7) 1 5.2 Spin- 2 fermions Lw 2 1.5 U = 0.5 U =2 U =4 1 101 102 103 104 105 106 L Figure 5.20: Spectral weight at the Fermi level at the boundary of a quarter-filled Hubbard chain as a function of system size L, for various different interaction strengths. ˜ where Dj (ω)0 is the noninteracting density of states, V (q) the Fourier transform of the real space interaction, and z the number of spin components. For spinless fermions (z = 1) with repulsive interactions the coefficient in front of the logarithm is always positive such that the first-order term leads to a suppression of Dj (ω). ˜ ˜ For the Hubbard model, one has z = 2 and V (0) − 2V (2kF ) = −U is negative for repulsive U. Hence, at least for weak U the density of states increases for decreasing ω until terms beyond first oder become important. For the extended Hubbard model, ˜ ˜ V (0) − 2V (2kF ) = 2U ′ [1 − 2 cos(2kF )] − U, which can be positive or negative for U, U ′ > 0, depending on the density and the relative strength of the two interaction ˜ ˜ parameters. At quarter filling V (0) − 2V (2kF ) is negative and therefore leads to an enhanced density of states for U ′ < U/2. ˜ ˜ Using g-ology notation (cf. Sec. A.2), one can write V (0)−2V (2kF ) = g2⊥ −2g1⊥ , which reveals that substantial two-particle backscattering (g1⊥ > g2⊥ /2) is necessary to obtain an enhancement of Dj (ω) for repulsive interactions. Backscattering vanishes at the Luttinger-liquid fixed point, but only very slowly. In case of a negative ˜ ˜ V (0) −2V (2kF ) the crossover to a suppression of Dj (ω) is due to higher order terms, which are expected to become important when the first-order correction is of order 83 5 Solution of fRG equations and results one, that is, for energies below the scale 2πvF ωc = ǫF exp ˜ ˜ V (0) − 2V (2kF ) (5.8) corresponding to a system size Lc = πvF /ωc . The scale ωc is exponentially small for weak interactions. A more accurate analytical estimate of the crossover scale from enhancement to suppression has been derived for the Hubbard model within Hartree-Fock approximation in Ref. [Meden et al. 2000]. In a renormalization-group treatment ωc is somewhat enhanced by the downward renormalization of backscattering. A comparison of fRG results with DMRG data for the spectral weight at the Fermi level is shown in Fig. 5.21, for a boundary site in the upper panel, and near a hopping impurity of strength t′ = 0.5 in the lower. The agreement improves at weaker coupling, as expected, and is generally better for the impurity case, compared to the boundary case. The deviations in the boundary case are probably due to our approximate translation-invariant parametrization of the two-particle vertex. Boundaries and to a minor extent impurities spoil the translation invariance of the two-particle vertex. Although the deviations from translation invariance of the vertex become irrelevant in the low-energy or long-distance limit, and therefore do not affect the asymptotic behavior, they are nevertheless present at intermediate scales. This feedback of impurities into the vertex increases of course with the impurity strength and is thus particularly important near a boundary. The scale for the crossover from enhancement to suppression of spectral weight discussed above depends sensitively on effective interactions at intermediate scales and can therefore be shifted considerably even by relatively small errors in that regime. With the additional nearest-neighbor interaction in the extended Hubbard model it is possible to tune parameters such that the two-particle backscattering amplitude becomes negligible. In that case the asymptotic power-law suppression of spectral weight should be free from logarithmic corrections and accessible already for smaller systems and at higher energy scales. The bare backscattering interaction in the extended Hubbard model is given by g1⊥ = U + 2U ′ cos(2kF ) and therefore vanishes for U ′ = −U/[2 cos(2kF )], which is repulsive for U > 0 if n > 1/2. In a one-loop calculation a slightly different value of U ′ has to be chosen to obtain a negligible Λ renormalized g1⊥ for small finite Λ, since the flow generates backscattering terms at intermediate scales even if the bare g1⊥ vanishes. In Fig. 5.22 we show fRG and DMRG results 84 1 5.2 Spin- 2 fermions Lw 1.5 U = 0.5 U =1 1 0.7 Lw 0.6 0.5 0.4 101 102 103 L 104 105 Figure 5.21: Spectral weight at the Fermi level near a boundary ( upper panel) and a hopping impurity t′ = 0.5 ( lower panel) as a function of system size L for the Hubbard model at quarter filling and different interaction strengths U ; results from the fRG (open symbols) are compared to DMRG data (filled symbols). for the spectral weight of the extended Hubbard model at the Fermi level near a hopping impurity. In the upper panel a generic case with sizeable backscattering is shown, while the parameters leading to the curves in the lower panel have been chosen such that the two-particle backscattering amplitude is negligible at low energy. Only in the latter case a pronounced suppression of spectral weight is reached already for intermediate system size, similar to the behavior obtained pre- 85 5 Solution of fRG equations and results 0.8 0.7 Lw 0.6 0.5 U = 0.5 U =1 U =2 0.4 0.3 1 0.9 0.8 Lw 0.7 0.6 0.5 0.4 0.3 1 10 102 103 L 104 105 Figure 5.22: Spectral weight at the Fermi level near a hopping impurity t′ = 0.5 as a func- √ tion of system size L for the extended Hubbard model with U ′ = U/ 2, for various choices of U ; upper panel: n = 1/2 (leading to sizeable backscattering), lower panel: n = 3/4 (leading to small backscattering); results from the fRG (open symbols) are compared to DMRG data (filled symbols). viously for spinless fermions with nearest-neighbor interaction [Andergassen et al. 2004; Meden et al. 2002a,b]. This is also reflected in the energy dependence of the local density of states near the impurity. For parameters leading to negligible twoparticle backscattering as in Fig. 5.23 the suppression of the density of states sets in already at relatively high energies and is not preceded by any interaction-induced 86 1 5.2 Spin- 2 fermions increase; the slight increase for small system sizes is a finite-size effect present also in the noninteracting case. Note also that the fRG results are much more accurate for small backscattering, as can be seen by comparing the agreement with DMRG data in the upper and lower panel of Fig. 5.22 especially for larger U. This indicates that the influence of the impurity on the vertex flow, which we have neglected, is more important in the presence of a sizable backscattering interaction. 0.3 D1 0.2 0.1 0 -1 U U U U =0 = 0.5 =1 =2 -0.5 0 ω 0.5 1 Figure 5.23: Local density of states near a hopping impurity t′ = 0.5 in an extended √ Hubbard model with density n = 3/4 and interaction U ′ = U/ 2 (leading to a small backscattering interaction) for various choices of U ; the size of the chain is L = 4096. In the case of a negligible backscattering amplitude, the spectral weight at the Fermi level approaches a power law without logarithmic corrections for accessible system sizes if the impurity is sufficiently strong. The power law is seen most clearly by plotting the effective exponent α(L), that is, the negative logarithmic derivative of the spectral weight with respect to the system size. Fig. 5.24 shows α(L) on the site next to a site impurity of strength V for the extended Hubbard model with U = 1, U ′ = 0.65 and n = 3/4. The backscattering amplitude is very small for these parameters. The fRG results approach the expected universal V -independent power law for large L, but only very slowly for small V . For weak bare impurity potential V , the crossover to a strong effective impurity occurs only on a large length scale of order V 2/(Kρ −1) [Kane and Fisher 1992a,c]. For V = 0.1 this scale is obviously well above the largest system size reached in Fig. 5.24. The Hartree- 87 5 Solution of fRG equations and results Fock approximation also yields power laws for large L, but the exponents depend on the impurity parameters. This failure of Hartree-Fock theory was already observed earlier for spinless fermions [Meden et al. 2002a,b]. 0.2 α 0.1 0 V = 10 V =1 V = 0.1 -0.1 -0.2 102 103 104 105 L Figure 5.24: Logarithmic derivative of the spectral weight at the Fermi level on the site next to a site impurity of strength V in the center of the chain as a function of system size L, for the extended Hubbard model with interaction parameters U = 1, U ′ = 0.65 and density n = 3/4; here the filled symbols are fRG, the open symbols Hartree-Fock results. The effective exponent obtained from the fRG calculation agrees with the exact boundary exponent to linear order in the bare interaction, but not to quadratic order. A quantitative estimate of the accuracy is obtained from a comparison to exact DMRG results [Ejima et al. 2005] shown in Fig. 5.25, for the extended Hubbard √ model at n = 3/4 and with U ′ = U/ 2 leading to a negligible backscattering amplitude. To improve this, the frequency dependence of the two-particle vertex, has to be taken into account. This is also necessary to describe inelastic processes and to capture the anomalous dimension of the bulk system (see also Sec. 5.1.2). 5.2.2 Density profile Boundaries and impurities induce a density profile with long-range Friedel oscillations, which are expected to decay with a power law with exponent (Kρ + Kσ )/2 at long distances, where Kσ = 1 for spin-rotation invariant systems [Egger and Grabert 88 1 5.2 Spin- 2 fermions 0.2 DMRG fRG αB 0.15 0.1 0.05 0 0 0.2 0.4 0.6 0.8 1 U Figure 5.25: Boundary exponent αB as a function of U at densities n = 3/4, with √ U ′ = U/ 2 (small backscattering interaction) as obtained from the power-law suppression of the spectral weight at the boundary; fRG results are compared to DMRG data. 1995]. For weak impurities linear response theory predicts a decay as |j −j0 |1−Kρ −Kσ at intermediate distances. As an additional benchmark for the fRG technique, we compare in Fig. 5.26 fRG and DMRG results for the density profile nj for a quarter-filled Hubbard chain with L = 128 lattice sites and open boundaries. Friedel oscillations emerge from both boundaries and interfere in the center of the chain. The fRG results have been shifted by a small constant amount to allow for a better comparison of the oscillations. Note that the mean value of nj in the tails of the oscillations deviates from the average density by a finite-size correction of order 1/L, which is related to the asymmetry of the oscillations near the boundaries. The long-distance behavior of the density oscillations as obtained within the fRG scheme has been analyzed in detail for spinless fermions in Sec. 5.1.3. For fermions with spin, asymptotic power laws can be identified only for special parameters leading to negligible two-particle backscattering. In general, the asymptotic behavior of Friedel oscillations is realized only at very long distances, and the power laws are modified by logarithmic corrections. We finally remark the presence of 1 a 4kF -component of the Friedel oscillations for spin- 2 fermions, which decays as |j − j0 |−2Kρ . In the present weak-coupling treatment this contribution is negligible, 89 5 Solution of fRG equations and results nj 0.3 0.2 DMRG fRG 0.1 0 20 40 60 j 80 100 120 Figure 5.26: Density profile nj for the Hubbard model with 128 sites and interaction strength U = 1 at quarter filling; fRG results are compared to DMRG data. since the 2kF component dominates for Kρ > 1/3 [Egger and Grabert 1995]. 5.2.3 Conductance Single impurity For a system of spinless fermions with a single impurity it was already shown that the conductance obtained from the truncated fRG obeys the expected power laws, in particular G(T ) ∝ T 2αB at low T , and one-parameter scaling behavior [Meden et al. 2003; Enss et al. 2005]. The corresponding scaling function agrees remarkably well with an exact result for Kρ = 1/2, although the interaction required to obtain such a small Kρ is quite strong. The more complex temperature dependence of the conductance in the case of a double barrier at or near a resonance is also fully captured by the fRG [Enss et al. 2005; Meden et al. 2005]. Fig. 5.27 shows typical fRG results for the temperature dependence of the conductance for the extended Hubbard model with a single strong site impurity (V = 10). Similar results were obtained for a hopping impurity. The considered size L = 104 corresponds to interacting wires in the micrometer range, which is the typical size of quantum wires available for transport experiments. For U ′ = 0 the conductance increases as a function of decreasing T down to the lowest tem- 90 1 5.2 Spin- 2 fermions peratures in the plot. For increasing nearest-neighbor interactions U ′ a suppression of G(T ) at low T becomes visible, but in all the data obtained at quarter-filling the suppression is much less pronounced than what one expects from the asymptotic power law with exponent 2αB . By contrast, the suppression is much stronger and follows the expected power law more closely if parameters are chosen such that two-particle backscattering becomes negligible at low T , as can be seen from the conductance curve for n = 3/4 and U ′ = 0.65 in Fig. 5.27. The value of Kρ for these parameters almost coincides with the one for another parameter set in the plot, n = 1/2 and U ′ = 0.75, but the behavior of G(T ) is completely different. Note that at T ∼ πvF /L finite-size effects set in, as can be seen at the low T end of some of the curves in the figure. An enhancement of the conductance due to backscattering has been found already earlier in a renormalization-group study of impurity scattering in the g-ology model [Matveev et al. 1993; Yue et al. 1994]. G/(2e2/h) 0.05 0.04 0.03 0.02 0.01 10−4 U′ U′ U′ U′ U′ 10−3 10−2 T =0 = 0.25 = 0.5 = 0.75 =1 10−1 100 Figure 5.27: Temperature dependence of the conductance for the extended Hubbard model with L = 104 sites and a single site impurity of strength V = 10, for a Hubbard interaction U = 1 and various choices of U ′ ; the density is n = 1/2, except for the lowest curve, which has been obtained for n = 3/4 and U ′ = 0.65 (leading to a very small backscattering interaction); the dashed line is a power-law fit for the latter parameter set. Results for the conductance of the extended Hubbard model with a hopping impurity with various amplitudes t′ are shown in Fig. 5.28. The bulk parameters have been chosen such that the two-particle backscattering is practically zero at low T . From the plot of the logarithmic derivative of G(T ) in the upper panel one can 91 5 Solution of fRG equations and results ∂log [G/(2e2/h)] / ∂log T 0.3 0.2 0.1 0 -0.1 ∂log [1 − G/(2e2/h)] / ∂log T -0.2 t′ t′ t′ t′ t′ = 0.05 = 0.25 = 0.5 = 0.75 = 0.95 0.1 0 -0.1 -0.2 -0.3 10−4 10−3 10−2 T 10−1 100 Figure 5.28: Logarithmic temperature derivative of the conductance (upper panel) and of its deviation from the unitarity limit (lower panel) for the extended Hubbard model with L = 104 sites and various hopping impurities. The density is n = 3/4, interaction parameters are U = 1 and U ′ = 0.65. The dashed horizontal lines highlight power-law behavior. see that for a strong impurity (small t′ ) the conductance follows a well-defined power law G(T ) ∝ T 2αB over a large temperature range. For intermediate t′ the curves approach the asymptotic exponent at low T from below, but do not reach it before finite-size effects lead to a saturation of G(T ) for T < πvF /L. For the weakest impurity in the plot, t′ = 0.95, the conductance remains very close to the unitarity limit. However, the plot of the logarithmic derivative of 1 − G/(2e2 /h) in the lower 92 1 5.2 Spin- 2 fermions panel of Fig. 5.28 shows that 1 − G/(2e2 /h) increases as T Kρ −1 for decreasing T , as expected for a weak impurity in the perturbative regime [Kane and Fisher 1992a,c]. The effective exponents indicated by the two horizontal lines in the figure deviate from the exact values (determined from the DMRG result [Ejima et al. 2005] for Kρ ) by about 20% in the case of 2αB and only by 5% for Kρ − 1 (cf. Sec. 5.1.4). Depending on the bare impurity and interaction parameters, nonuniversal behavior dominates at intermediate energy and length scales. Moreover, in the presence of backscattering the asymptotic power laws are modified by logarithmic corrections. Double impurity We finally present results for the conductance of a wire with a double-barrier impurity [Andergassen et al. 2005a]. The setup modeling a quantum dot is shown in Fig. 5.29. Applying a gate voltage Vg on the dot sites j ∈ [jl , jr ] by Hgate = Vg nj,σ (5.9) j,σ the conductance can be tuned to resonance. LD = jr − jl + 1 is sufficiently far away from the contacts at sites 1 and L the position of the dot does not play a role. Vl Vr Vg lead lead 1 LD Figure 5.29: Quantum dot schematization. Earlier studies of tunneling through a quantum dot embedded in a spinless Luttinger liquid showed that at T = 0 and for finite L the resonances in the linear conductance G(Vg ) are characterized by an almost Lorentzian shape with unitary height for symmetric barriers and a width w vanishing as a power law with a Kρ dependent exponent in the limit L → ∞. For asymmetric barriers the resonances disappear for increasing L [Kane and Fisher 1992a,c]. At T > 0 the peak value 93 5 Solution of fRG equations and results of the conductance shows distinctive power-law behavior as a function of temperature [Enss et al. 2005; Meden et al. 2005]. Including the spin degree of freedom the physics becomes more complex due to the appearance of the Kondo effect. For noninteracting leads (L = 1) the Kondo physics was investigated theoretically for the single-impurity Anderson model [Glazman and Raikh 1988; Ng and Lee 1988]. At low temperatures and for sufficiently high tunnel barriers the Kondo effect leads to a broad plateau-like line shape of the resonance replacing the Lorentzian. On resonance the number of electrons on the dot is odd implying a local spin- 1 degree 2 of freedom responsible for the Kondo effect [Hewson 1993]. For the single-impurity Anderson model the conductance is proportional to the one-particle spectral weight of the dot at the chemical potential [Meir and Wingreen 1992]. Varying Vg within an energy range of order U the Kondo resonance of the spectral function is pinned at µ0 at height 2e2 /h explaining the broad plateau-like resonance in G(Vg ) [Hewson 1993; Gerland et al. 2000]. The problem of a single spin- 1 coupled to a Luttinger 2 liquid was investigated generalizing the Kondo model [Furusaki 2005]. The fRG approach allows for a direct computation of the electron transport through a quantum dot embedded in a Luttinger liquid in the presence of the Kondo effect. Here we address the question of the resonance line shape and the power-law scaling of G(T ) resulting from the competition between the two correlation effects. For this purpose we first consider the situation L = LD = 1 at T = 0 corresponding to the single-impurity Anderson model. Unless otherwise stated we consider symmetric dot-lead couplings. In Fig. 5.30 the conductance G as a function of gate voltage Vg for the single-impurity Anderson model is shown for different tunnel barriers t′ in the upper panel, together with the occupation of the dot in the lower. For t′ ≪ U the resonance has a plateau-like shape [Gerland et al. 2000]. In this region the occupation is close to 1 while it sharply raises/drops to 2/0 to the left/right of the plateau. Also for asymmetric barriers we reproduce the exact resonance height 4∆L ∆R (∆L + ∆R )2 (2e2 /h), where ∆ = ∆L + ∆R measures the hybridization of the dot and the (left and right) lead states [Hewson 1993; Gerland et al. 2000]. Here we are interested in the interplay of Kondo and Luttinger-liquid physics and thus focus on small t′ , that is, on tunnel barriers with small transmission. The pinning of the spectral function and the subsequent plateau-like resonance can be derived within the fRG Scheme I (cf. Sec. 4.2.2). For L = 1 the chemical potential only shifts the position of the resonance. For half filling G(Vg ) is symmetric around 0 and the flow equation for the effective onsite energy V = Vg + ΣΛ ,jD on jD 94 1 5.2 Spin- 2 fermions 1 t′ = 0.1 t′ = 0.25 t′ = 0.5 G/(2e2/h) 0.8 0.6 0.4 0.2 0 2 n 1.5 1 0.5 0 -4 -3 -2 -1 0 1 Vg Figure 5.30: Upper panel: conductance as a function of gate voltage for the Hubbard model at quarter filling, with U = 1, for L = LD = 1 and different t′ ; lower panel: average number of electrons on the dot. the dot site jD reduces to ∂ Λ U UV Λ /π Λ V = − Re GjD ,jD (iΛ) = ∂Λ π (Λ + ∆)2 + (V Λ )2 (5.10) in the limit of ∆ ≪ U. Here ∆ = 2πt′ 2 ρ is the hybridization, and ρ denotes the spectral weight at the end of the leads in the infinite band width limit [Hewson 1993]. The initial condition is V Λ0 = Vg . In this scheme the self-energy is frequency independent leading to a Lorentzian spectral function of width 2∆ and height 1/(π∆) centered around V = V Λ=0 . This implies that the spectral weight at µ0 and thus G(Vg ) is determined by V [Meir and Wingreen 1992]. The solution of the differential 95 5 Solution of fRG equations and results equation (5.10) at Λ = 0 is obtained in implicit form J0 (vg ) vJ1 (v) − δJ0 (v) = , vY1 (v) − δY0 (v) Y0 (vg ) (5.11) with v = V π/U, vg = Vg π/U, δ = ∆π/U, and Bessel functions Jn , Yn . For |Vg | < Vc this equation has a solution with a small |V |, where vc = Vc π/U is the first zero of J0 corresponding to Vc ≃ 0.77U. For U ≫ ∆ the crossover to a solution with |V | being of order U (for |Vg | > Vc ) is fairly sharp. Expanding both sides of Eq. (5.11) for small |v| and |vg | gives V = Vg exp U . π∆ (5.12) The exponential pinning of the spectral weight at µ0 = 0 for small |Vg | and the sharp crossover to a V of order U when |Vg | > Vc leads to the observed resonance line shape. For U ≫ ∆ the width of the plateau is 2Vc ≃ 1.5U, which is larger than the width U found with the numerical renormalization-group method [Gerland et al. 2000]. Our approximation furthermore slightly overestimates the sharpness of the box-shaped resonance. We expect the agreement is improved with the more accurate fRG Scheme II (cf. Sec. 4.2.2). In the complementary case of Luttinger-liquid leads and in the absence of Kondo effect, suppressed by turning off the interaction on the dot, similar results as for spinless fermions are obtained. Luttinger-liquid behavior leads to infinitely sharp resonances in the limit L → ∞, in strong contrast to the broad resonances induced by the Kondo effect [Enss et al. 2005]. To clearly observe Luttinger-liquid behavior for spin- 1 fermions at experimentally accessible scales one has to consider a situ2 ation in which the backscattering process yielding logarithmic corrections to the power-laws is small by tuning the nearest-neighbor interaction U ′ , see Sec. 5.2.1. In Fig. 5.31 the L dependence of G(Vg ) for a single-site dot computed for a small backscattering amplitude and lead length L = 104 typical for experiments is shown. At T = 0 the width of the resonance in G(Vg ) tends to zero for L → ∞. The extracted width w as a function of L reported in the lower panel with open symbols follows a power law L(Kρ −1)/2 with an fRG approximation to the Luttinger-liquid parameter that is, correct to leading order in the interaction. Off resonance G asymptotically vanishes as L−2αB [Kane and Fisher 1992a,c; Enss et al. 2005]. For Vg close to resonance and 1 − G/(2e2 /h) ≪ 1, the deviation from the unitary limit increases as L1−Kρ characteristic of the scaling of a weak single impurity. Further increasing L 96 1 5.2 Spin- 2 fermions 1 L = 103 L = 104 L = 105 G/(2e2/h) 0.8 0.6 0.4 0.2 0 -0.85 -0.8 -0.75 Vg -0.7 -0.65 w 10−2 10−3 103 104 105 106 L Figure 5.31: Upper panel: conductance as a function of gate voltage for the extended Hubbard model at n = 3/4, with U = 1, and U ′ = 0.65, for a noninteracting dot with LD = 1, t′ = 0.1 and different L; lower panel: scaling of the resonance width for the same parameters as in the upper panel corresponding to a small backscattering interaction (open symbols), and for n = 1/4, U = 1, U ′ = 0 corresponding to a large backscattering interaction (filled symbols). the behavior eventually crosses over to the off-resonance power-law suppression of G mentioned above. Due to an exponentially large crossover scale, even for the very large system sizes accessible with our method the complete crossover from one to the other power law can not be seen for a single fixed Vg but follows from one-parameter scaling [Kane and Fisher 1992a,c; Enss et al. 2005]. For a sizeable backscattering amplitude the off-resonance conductance and thus the width first slightly increase for increasing L - becoming larger than for the noninteracting dot - followed by a 97 5 Solution of fRG equations and results crossover to a decrease for exponentially large L, as indicated by the filled symbols in the lower panel. Due to the logarithmic vanishing of the backscattering process this behavior can in general not be observed on experimentally accessible scales, confirming the important role of two-particle backscattering on intermediate length scales. An upper bound of the length of quasi one-dimensional wires realized in experiments is of the order of µm, roughly corresponding to 104 lattice sites [Bockrath et al. 1999; Yao et al. 1999; Auslaender and Fishman 2000; Picciotto et al. 2001]. 1 L = 103 L = 104 L = 105 G/(2e2/h) 0.8 0.6 0.4 0.2 0 -1 -0.5 0 0.5 105 106 conductance Vg 10−3 10−4 103 104 L Figure 5.32: Upper panel: conductance G(Vg ) as in Fig. 5.31 (same parameters), but now with interaction on the dot; lower panel: scaling of G/(2e2 /h) at Vg = 0 outside the plateau (filled symbols), and of 1 − G/(2e2 /h) on the plateau at Vg = −0.685 (open symbols). We now analyze the linear conductance through a quantum dot in the presence of both Kondo effect as well as Luttinger-liquid leads. The upper panel of Fig. 5.32 shows the L dependence of G(Vg ) for the same parameters as in Fig. 5.31, but now 98 1 5.2 Spin- 2 fermions including the interaction on the dot. Note the different scales on the x-axis. For interactions large compared to the hybridization the broad plateau-like resonance induced by the Kondo effect is also present at least for finite Luttinger-liquid leads. The same holds for LD > 1. The width of the plateaus is proportional to the ratio of the local component of the effective interaction at the end of the fRG flow and LD . The differences between the curves for different L are barely visible, in particular the changes of the resonance width are marginal. For generic parameters with sizeable backscattering, the difference between curves computed for different L are even smaller. We note that the plateaus vanish if U, U ′ , and n are chosen such that at the end of the fRG flow the local part of the interaction is small, and the resonance peaks are sharp. To analyze the L dependence at small backscattering in more detail in the lower panel of Fig. 5.32 the scaling of G/(2e2 /h) for a gate voltage outside the plateau (filled symbols) and of 1 − G/(2e2 /h) for a gate voltage on the resonance plateau (open symbols) are shown. Off resonance G follows a power-law with the exponent 2αB and G vanishes for L → ∞. Within every plateau there a value of Vg = Vgr where G = 2e2 /h independently of L. For Vg = Vgr still within the plateau the deviation of G from the unitary limit scales as L1−Kρ , that is, with the weak single-impurity exponent. This shows that any deviation from Vgr acts as an impurity. By analogy with the single-impurity behavior discussed in the previous sections we conclude that in the asymptotic low-energy limit the impurity will effectively grow and in the limit L → ∞ the plateaus will vanish. For infinitely long Luttinger-liquid leads the resonances are infinitely sharp even in the presence of Kondo physics. However, for tunnel barriers with small transmission the plateaus at finite L are well developed and the length scale on which the plateaus start to deteriorate is extremely large. For sizeable two-particle backscattering this scale is enhanced and the plateaus are more pronounced. Also for asymmetric barriers we find (almost) plateau-like resonances. To discuss this in more detail we focus on typical parameters with N = 104 and an asymmetry ∆l /∆R ∼ 2. Then the width is almost unaffected by the asymmetry. For the interaction and filling as in Fig. 5.32 the height within the plateaus varies by a few percent (with maxima at the left and right boundaries) and has average value ∼ 0.85 (2e2/h). With increasing L the variation of the conductance on the plateaus increases while the average value decreases. We expect that for L → ∞ the resonance disappears. Concerning the power-law scaling of the conductance as a function of tempera- 99 5 Solution of fRG equations and results G/(2e2 /h) 100 0 g1⊥ sizeable 0 g1⊥ small 10−1 10−2 exponent 0.5 0 -0.5 -1 10−4 10−3 10−2 T 10−1 100 Figure 5.33: Temperature dependence of the conductance for the extended Hubbard model with L = 104 sites, Hubbard interaction U = 1, for U ′ = 0.65 and n = 3/4 (leading to a small backscattering interaction), and for U ′ = 0.75 and n = 1/4 (leading to a sizable backscattering interaction); dot parameters: t′ = 0.1, LD = 100; upper panel: log-log plot of the conductance, lower panel: effective exponents; the gate potential is chosen at the center of the resonance plateau closest to Vg = 0. ture, G(T ), the resonant tunneling behavior in general extends over a range of gate voltages defined by the width of the resonance plateau, since for all experimentally accessible length scales and for typical asymmetries of the dot-lead hybridizations the plateau-like resonances characteristic for Kondo physics will also be present if the leads are Luttinger liquids. The vanishing of the resonance plateau in the limit of infinite system size is beyond the infrared cutoff scale set by the system size T ∼ πvF /L for the appearance of power-law scaling with interaction-dependent exponents characteristic for Luttinger-liquid behavior. Hence the behavior of G(T ) is similar to the one found for spinless fermions [Enss et al. 2005; Meden et al. 2005]. In Fig. 5.33 the temperature dependence of the conductance for a large dot with LD = 100 and for Vg placed at the center of the resonance plateau closest to Vg = 0 100 1 5.2 Spin- 2 fermions is shown. We distinguish the case of sizable and small backscattering processes 0 g1⊥ . For a small backscattering amplitude we identify temperature regimes in which G(T ) follows distinctive power-law behavior with universal exponents. For T larger than the level spacing of the dot given by πvF /LD a power law with the singleimpurity exponent 2αB is found, arising from two independent single barriers acting as resistors in series. It follows a sequential tunneling regime characterized by the power-law exponent αB − 1, until saturation sets in at T ∼ πvF /L. Small dots with LD of order 1 exhibit only the latter. Similar results are obtained for other Vg within the resonance plateau, outside the plateaus the conductance follows the onresonance behavior down to a scale set by the deviation from resonance. For smaller T we find a crossover to T 2αB . By contrast, for a sizable backscattering amplitude no clear power law can be distinguished, as can be seen from the conductance curve for n = 1/4 and U ′ = 0.75 in Fig. 5.33. The value of Kρ for these parameters almost coincides with the one for another parameter set in the plot, n = 3/4 and U ′ = 0.65, but the behavior of G(T ) is completely different. For weak barriers analogous powerlaws are detected in 1−G/(2e2 /h), described by the single-impurity exponent Kρ −1 at high temperatures and by Kρ + 1 in the sequential tunneling regime, similarly to the spinless case investigated in detailed in Ref. [Enss et al. 2005]. For systems with long-range interactions backscattering is strongly reduced compared to forward scattering. This seems to be the case in carbon nanotubes [Egger and Gogolin 1997; Kane et al. 1997]. Hence, the conductance can be expected to follow the asymptotic power law at accessible temperature scales for sufficiently strong impurities in these systems, as is indicated also by experiments [Yao et al. 1999]. However, the effects due to two-particle backscattering should be observable in systems with a screened Coulomb interaction. Whether Luttingerliquid behavior has convincingly been demonstrated experimentally is still a matter of debate. Nonetheless, the above scenario has to be taken into consideration in the discussion of the influence of boundaries or impurities in quasi one-dimensional conductors. 101 6 Conclusions and outlook The fRG provides a powerful tool to compute the intriguing properties of Luttinger liquids with static impurities. It captures the physics at all energy scales from the Fermi energy to the ultimate low-energy limit. The presented computation scheme extends previous work for spinless fermions [Meden et al. 2002a,b] to 1 spin- 2 fermions, including the vertex renormalization in addition to the renormalization of the effective impurity potential. The underlying approximations are devised for weak short-range interactions and arbitrary impurity potentials. The results agree remarkably well with exact asymptotic results up to intermediate interaction strength, and cover the universal low-energy asymptotics, as well as nonuniversal behavior and crossover phenomena at higher scales. Various observables have been computed for different fermion lattice models: spectral properties of single-particle excitations, the oscillations in the density profile induced by impurities or boundaries, and the temperature dependence of the linear conductance. The comparison to DMRG results, for those observables and system sizes for which such data could be obtained, yields a good agreement at weak coupling. For intermediate interaction strengths with sizable two-particle backscat1 tering and strong impurities the deviations are significantly larger for spin- 2 than for spinless fermions. For the computation of the nonuniversal behavior at intermediate scales the neglected influence of impurities on vertex renormalization, in particular the interplay of impurities and the two-particle backscattering amplitude, is probably important for fermions with spin. We confirm the universality of the open chain fixed point, but it turns out that very large systems are required to reach the fixed point for realistic choices of the impurity and interaction parameters. The spectral properties of single-particle excitations and the Friedel oscillations in the density profile induced by impurities or boundaries present the characteristic asymptotic power laws at low energy or large distance. For the linear conductance through a single impurity in Luttinger liquids connected to noninteracting leads the fRG captures the expected power-law scaling, as well as the complete crossover from the weak to the strong-impurity 102 limit determined by a one-parameter scaling function. For resonant tunneling in a Luttinger liquid with a double barrier enclosing a dot region, depending on the dot parameters several temperature regimes with distinctive power-law behavior of the resonance conductance as well as regimes characterized by non-universal behavior are identified [Enss et al. 2005; Meden et al. 2005]. Including the spin degree of freedom, two-particle backscattering of particles with opposite spin at opposite Fermi points leads to two important effects, not present in the case of spinless fermions. First, the expected decrease of spectral weight and of the conductance at low energy scales is often preceded by an increase, which can be particularly pronounced for the density of states near an impurity or boundary as a function of ω. For the density of states near a boundary this effect has been found already earlier within a Hartree-Fock and DMRG study of the Hubbard model [Meden et al. 2000; Sch¨nhammer et al. 2000], and for the conductance by a o renormalization-group analysis of the g-ology model [Matveev et al. 1993; Yue et al. 1994]. Second, the asymptotic low-energy power laws are usually modified by logarithmic corrections. In the extended Hubbard model the backscattering can be eliminated for a special fine-tuned choice of parameters. Then the results are very similar to those for spinless fermions. For weak and intermediate impurity strengths the asymptotic low-energy behavior is approached only at rather low scales, which are accessible only for very large systems. This slow convergence observed already for spinless fermions holds also in the absence of two-particle backscattering. Interesting further extensions of the fRG for impurities in Luttinger liquids include the investigation of non-equilibrium phenomena and the analysis of disorder. For isolated impurities the influence of impurities on the vertex renormalization is irrelevant for the asymptotic low-energy or long-distance behavior, although it may contribute quantitatively at intermediate scales. For disordered systems with a finite impurity density the influence of the latter on the two-particle vertex is crucial and must be taken into account [Giamarchi 2004]. In principle this is possible by computing the vertex flow with full propagators, which contain the renormalized impurity potential via the self-energy. A further challenging extension concerns the inclusion of inelastic processes. They appear at second order in the interaction and can be included in the flow equations by inserting the two-particle vertex into the flow equation for the self-energy without neglecting its frequency dependence. Finally, a flexible microscopic modeling feasible with the fRG approach allows for a more realistic description of contacts and leads in experimental systems. 103 A Evaluation of vertex flow for spin- 1 2 fermions A.1 Functional RG Here we present a detailed derivation of the flow equations for the two-particle vertex ΓΛ . Starting from the flow equation (4.51) we insert on the right-hand side the parametrization (4.47) for ΓΛ and (4.49) for ΓΛ . The flow of the singlet vertex t s ′ ′ is computed for the three choices of (k1 , k2 , k1 , k2 ) which yield the flow ΓΛ ′ ′ s| k1 ,k2 ;k1 ,k2 Λ Λ Λ of gs2 , gs3, gs4 corresponding to Eqs. (4.42) and (4.43), while the flow of the triplet ′ ′ vertex ΓΛ ′ ′ is evaluated only for (k1 , k2 , k1 , k2 ) = (kF , −kF , kF , −kF ) as in t| k1 ,k2 ;k1 ,k2 Λ (4.41), which yields the flow of gt . For α = s2, s3, s4, t we obtain 1 ∂gα = 2 ∂Λ 4π 2π ω=±Λ 0 dp fα (p, ω) , 2π (A.1) with fs2 (p, ω) = ′ ′ ′ (2Ps + Us − 2µ0Us cos p)2 (2Us + Us − µ0 (Us + Ps ) cos p)2 + 0 0 0 2(iω − ξp )(−iω − ξ−p ) 4(iω − ξp )2 ′ ′ (4 − µ2 )(Us − Ps )2 sin2 p + 6(2 + µ0 cos p)Ut′ (2Us + Us ) 0 − 0 4(iω − ξp )2 + ′ ′ 6(2 + µ0 cos p)µ0 Ut′ (Us + Ps ) cos p − (4 − µ2 )(Us − Ps ) sin2 p 0 0 4(iω − ξp )2 − ′ ′ 3(2 cos p + µ0 )2 U ′ 2 ((µ2 − 2)Us + Us + 2(Us + Ps ) cos p)2 0 t + 0 0 0 4(iω − ξp )2 4(iω − ξp−kF )(iω − ξp+kF ) − ′ ′ 6(µ2 − 2 − 2 cos p)Ut′ ((µ2 − 2)Us + Us + 2(Us + Ps ) cos p) 0 0 0 0 4(iω − ξp−kF )(iω − ξp+kF ) 3(µ2 − 2 − 2 cos p)2 U ′ 2 0 t − 0 0 4(iω − ξp−kF )(iω − ξp+kF ) 104 (A.2) A.1 Functional RG fs3 (p, ω) = ′ ′ ′ (2Us − Us )2 − 16U ′ 2 sin2 p 4(Us + Ps )2 sin2 p − (2Us − Us )2 s − 0 0 0 0 2(iω − ξp )(−iω − ξπ−p ) 2(iω − ξp )(iω − ξπ+p ) ′ ′ 6Ut′ (2(Us + Ps ) sin2 p − 2Us + Us ) + 6U ′ 2 cos2 p t − 0 )(iω − ξ 0 ) (iω − ξp π+p fs4 (p, ω) = (A.3) ′ (4Us cos p + (µ2 − 2)Ps + Us )2 0 0 0 2(iω − ξp−kF )(−iω − ξp+kF ) + ′ ′ ′ (2Us + Us − µ0 (Us + Ps ) cos p + 2(Us − Ps ) sin p sin kF )2 0 2(iω − ξp )2 − ′ ′ 6(2 + µ0 cos p − 2 sin p sin kF )(2Us + Us − µ0 (Us + Ps ) cos p)Ut′ 0 2(iω − ξp )2 − ′ 12(2 + µ0 cos p − 2 sin p sin kF )Ut′ (Us − Ps ) sin p sin kF 0 2(iω − ξp )2 3(2 + µ0 cos p − 2 sin p sin kF )2 U ′ 2 t − 0 )2 2(iω − ξp ft (p, ω) = (A.4) ′ ′ (2Us + Us − µ0 (Us + Ps ) cos p)2 2(4 − µ2 )U ′ 2 sin2 p 0 t − 0 0 0 (iω − ξp )(−iω − ξ−p ) 4(iω − ξp )2 + ′ ′ (4 − µ0 )(Us − Ps )2 sin2 p − 2(2 + µ0 cos p)(2Us + Us )Ut′ 0 4(iω − ξp )2 + ′ ′ 2(µ0 (2 + µ0 cos p)(Us + Ps ) cos p − (4 − µ2 )(Us − Ps ) sin2 p)Ut′ 0 0 4(iω − ξp )2 − ′ ′ 5(µ0 + 2 cos p)2 ((µ2 − 2)Us + Us + 2(Us + Ps ) cos p)2 0 + 0 0 0 4(iω − ξp−kF )(iω − ξp+kF ) 4(iω − ξp )2 U ′ 2 t + ′ ′ 2(µ2 − 2 − 2 cos p)((µ2 − 2)Us + Us + 2(Us + Ps ) cos p)Ut′ 0 0 0 0 4(iω − ξp−kF )(iω − ξp+kF ) 5(µ2 − 2 − 2 cos p)2 U ′ 2 0 t . + 0 0 4(iω − ξp−kF )(iω − ξp+kF ) (A.5) 105 A Evaluation of vertex flow for spin- 1 fermions 2 0 Here ξk = −2 cos k − µ0 with µ0 = −2 cos kF is the bare dispersion relation relative to the bare Fermi level. Since the functions fα (p, ω) can be written as rational functions of cos p and sin p, the p -integral of Eq. (A.1) can be carried out analytically using the substitution z = eip and the residue theorem, in analogy to the spinless case described in Sec. 4.2.3. The resulting differential equations for the momentum Λ space couplings gα read ′ ′ ∂gs2 2(U ′ 2 + Ps2 − 3U ′ 2 + 6Us Ut′ ) + µ2 (2U ′ 2 + Us Ps + 3Ps Ut′ ) s t 0 s = ∂Λ 4π + Re γ(Λ) π(4 − (µ0 + iΛ)2 ) ′ (2Ps + Us + µ0 (µ0 + iΛ)Us )2 iΛ + ′ ′ ′ ′ (µ0 + iΛ)((2Us + Us )2 + 2µ2 (Us + Ps )2 ) + 4µ0(2Us + Us )(Us + Ps ) 0 2(4 − (µ0 + iΛ)2 ) − ′ ′ ′ µ2 (µ0 + iΛ)3 (Us + Ps )2 + 24(µ0 + iΛ)(4Us + 2Us − µ2 (Us + Ps ))Ut′ 0 0 8(4 − (µ0 + iΛ)2 ) 12µ0 (Us − 2Ps )Ut′ − 3((µ0 + iΛ)(µ2 + 4) − 8µ0 )U ′ 2 0 t + 2) 2(4 − (µ0 + iΛ) ′ ′ (µ0 + iΛ)(((µ2 − 2)Us + Us )2 + 4(Us + Ps )2 ) 0 − 2((µ0 + iΛ)2 − µ2 ) 0 − ′ ′ 2µ0 (Us + Ps )((µ2 − 2)Us + Us ) − 6µ0 ((µ2 − 2)Ps − Us )Ut′ 0 0 (µ0 + iΛ)2 − µ2 0 + ′ ′ 3(µ0 + iΛ)((µ2 − 2)((µ2 − 2)Us + Us ) − 4(Us + Ps ))Ut′ 0 0 (µ0 + iΛ)2 − µ2 0 + 3((µ0 + iΛ)((µ2 − 2)2 + 4) − 4µ0 (µ2 − 2))U ′ 2 2 0 0 t + (µ0 + iΛ)(U ′ s + Ps2 ) 2((µ0 + iΛ)2 − µ2 ) 0 − ′ ′ (µ0 + iΛ)(µ2 (Us − Ps )2 − 12µ2 Ps Ut′ − 48Us Ut′ + 24U ′ 2 ) 0 0 t 8 ′ ′ ∂gs3 4U ′ 2 + (Us + Ps )(Us + Ps + 6Ut′ ) − 3U ′ 2 s t =− ∂Λ 2π + Re 106 γ(Λ) π ′ ′ 4U ′ 2 (Us + Ps )(Us + Ps + 6Ut′ ) s + µ0 µ0 + iΛ (A.6) A.1 Functional RG − ′ ′ (2Us − Us )(2Us − Us − 12Ut′ ) 3(µ0 + iΛ)U ′ 2 t + 4(µ0 + iΛ) 4 − (µ0 + iΛ)2 − ′ ′ 4(2Ps − Us )2 + µ0 (µ0 + iΛ)(2Us − Us )(2Us − Us − 12Ut′ ) 4µ0 (4 − (µ0 + iΛ)2 ) (A.7) ′ ′ ∂gs4 8U ′ 2 + (µ2 − 2)(U ′ 2 + Ps2 − 3U ′ 2 + 6Us Ut′ ) + 4Us Ps + 12Ps Ut′ s 0 s t = ∂Λ 4π + Re γ(Λ) π(4 − (µ0 + iΛ)2 ) ′ 2(µ0 + iΛ)Ps (Us + 3Ut′ ) ′ (µ0 + iΛ)(µ2 − 2)(U ′ 2 + Ps2 − 3U ′ 2 + 6Ut′ Us ) 0 s t + 2 − ′ ((µ2 − 2)Ps + Us + ((µ0 + iΛ)µ0 − 4iγ(Λ) sin kF )Us )2 0 µ0 (µ0 + iΛ) µ0 Λ2 + 4iγ(Λ) sin kF − 4 + ′ ′ (µ0 + iΛ)((2Us + Us )2 + µ2 (Us + Ps )2 ) 0 4 − (µ0 + iΛ)2 + ′ ′ ′ ′ 4µ0 (2Us + Us )(Us + Ps ) − 6(µ0 + iΛ)(4Us + 2Us − µ2 (Us + Ps ))Ut′ 0 4 − (µ0 + iΛ)2 + 12µ0 (Us − 2Ps )Ut′ − 3((µ0 + iΛ)(4 + µ2 ) − 8µ0)U ′ 2 0 t 4 − (µ0 + iΛ)2 (A.8) ′ ∂gt (4 − µ2 )(Us Ps − Ps Ut′ − 2U ′ 2 ) 0 t = ∂Λ 4π + Re γ(Λ) π(4 − (µ0 + iΛ)2 ) ′ (µ0 + iΛ)(4 − µ2 )Ps (Us − Ut′ ) 0 2 + ′ ′ (4 − µ2 )(4 − (µ0 + iΛ)2 )U ′ 2 2µ0 (2Us + Us )(Us + Ps ) 0 t − iΛ (4 − (µ0 + iΛ)2 ) − ′ ′ (µ0 + iΛ)((2Us + Us )2 + µ2 (Us + Ps )2 ) 0 2(4 − (µ0 + iΛ)2 ) − ′ ′ ((µ0 + iΛ)(4Us + 2Us − µ2 (Us + Ps )) − 2µ0 (Us − 2Ps ))Ut′ 0 4 − (µ0 + iΛ)2 107 A Evaluation of vertex flow for spin- 1 fermions 2 − ′ 5((µ0 + iΛ)(µ2 + 4) − 8µ0 )U ′ 2 (µ0 + iΛ)((µ2 − 2)Us + Us )2 0 t 0 − 2(4 − (µ0 + iΛ)2 ) 2((µ0 + iΛ)2 − µ2 ) 0 − ′ ′ ′ 2(µ0 + iΛ)(Us + Ps )2 + 2µ0 (Us + Ps )((µ2 − 2)Us + Us ) 0 (µ0 + iΛ)2 − µ2 0 − ′ ′ (µ0 + iΛ)((µ2 − 2)((µ2 − 2)Us + Us ) − 4(Us + Ps ))Ut′ 0 0 (µ0 + iΛ)2 − µ2 0 2µ0 ((µ2 − 2)Ps − Us )Ut′ 5(µ0 + iΛ)((µ2 − 2)2 + 4)U ′ 2 0 t 0 − − (µ0 + iΛ)2 − µ2 2((µ0 + iΛ)2 − µ2 ) 0 0 10µ0(µ2 − 2)U ′ 2 0 t + 2 − µ2 (µ0 + iΛ) 0 , (A.9) 4 . (µ0 + iΛ)2 (A.10) with γ(Λ) = (µ0 + iΛ) 2 1− Using the linear equations (4.48) and (4.50) to replace gα by the renormalized real space interactions on the left-hand side of the flow equations, we obtain a complete ′ set of flow equations for the four renormalized interactions Ut′ , Us , Us , and Ps of the form (4.54). A.2 One-loop g-ology calculation In the low-energy limit the flow of the two-particle vertex ΓΛ , as described in Sec. A.1, reduces to the one-loop flow of the g-ology model, the general effective low-energy model for one-dimensional fermions [S´lyom 1979]. In the g-ology apo proach interaction processes are classified into backward scattering (g1⊥ ), forward scattering involving electrons from opposite Fermi points (g2⊥ ), from the same Fermi points (g4⊥ ), and umklapp scattering (g3⊥ ). All further momentum dependences of the vertex are discarded. The g-ology couplings are related to the momentum space couplings gs2 , gs3, gs4 and gt by 1 g1⊥ = (gs2 − gt ) 2 1 g2⊥ = (gs2 + gt ) 2 108 A.2 One-loop g-ology calculation gs3 2 gs4 = 2 g3⊥ = g4⊥ (A.11) and to the real space couplings by 1 g1⊥ = (µ2 Γs2 + 2Γs3 + Γs4 − (4 − µ2 )Γt ) 0 2 0 1 g2⊥ = (µ2 Γs2 + 2Γs3 + Γs4 + (4 − µ2 )Γt ) 0 2 0 1 g3⊥ = (−4Γs2 − 2Γs3 + Γs4 ) 2 1 g4⊥ = (4Γs2 − (2 − µ2 )Γs3 + Γs4 ) 0 2 (A.12) respectively. The flow equation for ΓΛ (4.51) reduces to the standard one-loop g-ology calculation [S´lyom 1979], once the dependence on the internal momentum p on the o right-hand side of the flow equation is neglected. Applying the above parametrization, we obtain a complete set of flow equations for gi⊥ for i = 1, . . . , 4 of the form 1 ∂g1⊥ = g1⊥ g2⊥ P P (0) + g1⊥ g4⊥ P H(0) + 2g1⊥ (g2⊥ − g1⊥ )P H(2kF ) ∂Λ 2π 1 1 2 ∂g2⊥ 2 2 = (g + g2⊥ )P P (0) + g4⊥ (g1⊥ − g2⊥ )P H(0) + g2⊥ P H(2kF ) ∂Λ 2π 2 1⊥ 1 ∂g3⊥ g3⊥ g4⊥ P P (π) + g3⊥ (2g2⊥ − g1⊥ )P H(π) = ∂Λ 2π ∂g4⊥ 1 2 1 2 2 2 g4⊥ P P (2kF ) + (g1⊥ + 2g1⊥ g2⊥ − 2g2⊥ + g4⊥ )P H(0) , = ∂Λ 2π 2 (A.13) where 1 P P (q) = 2π ω=±Λ 1 P H(q) = 2π ω=±Λ 2π dp 1 1 2π iω − ξp+q/2 −iω − ξ−p+q/2 2π dp 1 1 . 2π iω − ξp+q/2 iω − ξp−q/2 0 0 (A.14) Λ0 Λ0 Λ0 Λ0 The initial conditions are g1⊥ = g3⊥ = U + (µ2 − 2)U ′ and g2⊥ = g4⊥ = U + 2U ′ . 0 The integrals in Eq. (A.14) can be computed analytically using the residue theorem, 109 A Evaluation of vertex flow for spin- 1 fermions 2 as discussed in Sec. A.1; for q = 0 and q = 2kF we obtain P P (0) = − 4 iγ(Λ) Re Λ 4 − (µ0 + iΛ)2 P H(0) = 4 Re P P (2kF ) = − (µ0 + iΛ)γ(Λ) (4 − (µ0 + iΛ)2 )2 4 γ(Λ) Re 2 )(4 − µ (µ + iΛ) + 4iγ(Λ) sin k ) µ0 (4 − (µ0 + iΛ) 0 0 F P H(2kF ) = 4 Re (µ0 + iΛ)γ(Λ) , (4 − (µ0 + iΛ)2 )(µ2 − (µ0 + iΛ)2 ) 0 (A.15) where γ(Λ) is defined by Eq. (A.10). For µ = 0 the above equations reduce to 2 P P (0) = Λ 4 − µ2 0 P H(0) = − P P (2kF ) = 4(2 + µ2 )Λ 0 4 − µ 2 )5 0 ( 8Λ 4 − µ 2 )5 0 ( P H(2kF ) = − 1 Λ 4 − µ2 0 (A.16) in the limit Λ → 0. The resulting flow equations in the low-energy limit are 2 ∂g1⊥ g1⊥ = ∂Λ Λπ 4 − µ2 0 2 g1⊥ ∂g2⊥ = ∂Λ 2Λπ 4 − µ2 0 (A.17) for g1⊥ and g2⊥ , whereas the flow for g3⊥ and g4⊥ vanishes. The solution reads Λ g1⊥ = Λ g2⊥ = U + (µ2 − 2)U ′ 0 1+ U +(µ2 −2)U ′ 0 π √ 4−µ2 0 Λ ln Λ0 U + (µ2 − 2)U ′ 1 0 + U + (6 − µ2 )U ′ , 0 U +(µ2 −2)U ′ Λ 2 1 + √0 ln Λ0 2 π (A.18) 4−µ0 1 yielding the fixed-point couplings g1⊥ = 0, g2⊥ = 2 (U + (6 − µ2 )U ′ ), and g3⊥ = 0 2 ′ ′ U + (µ0 − 2)U , g4⊥ = U + 2U . 110 B Bethe-ansatz calculation of Kρ for the Hubbard model The Luttinger-liquid parameter Kρ can be evaluated exactly from the Bethe ansatz solution for the one-dimensional Hubbard model [Lieb and Wu 1968]. The Bethe ansatz provides the energies of the ground and excited states as solution of specific integral equations. In the description of critical properties the “dressed charge matrix” is introduced [Frahm and Korepin 1990]. This 2 × 2 matrix contains the effective renormalized coupling constants within and between the low-energy charge and spin sectors of the Hilbert space of the Hubbard model, and therefore directly determines all critical exponents. It is defined as Zcc Zcs Zsc Zss Z= . (B.1) In the absence of a magnetic field, Z is completely determined by its first element Zcc as ξ(k0 ) 0 ξ(k0 ) 2 Z= 1 √ 2 , (B.2) where k0 is a cutoff determined by the particle density n [Frahm and Korepin 1990]. ξ(k) obeys the integral equation ξ(k) = 1 + 4 U k0 cos k ′ R −k0 4 (sin k − sin k ′ ) ξ(k ′ ) dk ′ , U (B.3) with the kernel R(x) = 1 2π ∞ 0 cos( xy ) 2 dy . 1 + ey (B.4) The cutoff momentum k0 is defined by k0 ρ(k ′ ) dk ′ = n , (B.5) −k0 111 B Bethe-ansatz calculation of Kρ for the Hubbard model where the integral equation for the ground-state charge distribution function ρ(k) reads ρ(k) = 4 1 + cos k 2π U k0 R −k0 4 (sin k − sin k ′ ) ρ(k ′ ) dk ′ U (B.6) in the limit L → ∞ . The compressibility κ is related to the dressed charge matrix element ξ(k0) by ξ 2 (k0 ) = πvρ κ, where vρ is the charge velocity [Frahm and Korepin 1990]. On the other hand, the Luttinger-liquid parameter Kρ is given by Kρ = πvρ κ/2. Hence Kρ is determined by Kρ = ξ 2 (k0 ) . 2 (B.7) Solving the integral equation for ξ(k0 ) with k0 determined from equations (B.5) and (B.6), yields the exact Luttinger-liquid parameter. For the numerical computation of the kernel R(x) defined in Eq. (B.4), the following expression is more convenient 1 R(x) = π ∞ (−1)l+1 l=1 x2 2l . + (2l)2 (B.8) Similarly the chemical potential µ is determined. A comparison of the fRG results with the exact results allows a quantitative estimate of the effect due to the neglected frequency dependence of the two-particle vertex. The chemical potential can be derived from the ground-state energy via µ= ∂ǫ ∂ǫ = ∂n ∂k0 ∂n ∂k0 −1 , (B.9) where the energy per lattice site is given by k0 ǫ(k) = −2 cos k ρ(k) dk . (B.10) −k0 Using equations (B.5) and (B.6), the solution of the following integral equations ∂ǫ = −4ρ(k0 ) cos k0 − 2 ∂k0 ∂n = 2ρ(k0 ) + ∂k0 112 k0 −k0 k0 cos k −k0 ∂ρ(k) dk , ∂k0 ∂ρ(k) dk ∂k0 (B.11) with 4 4 ∂ρ(k) = ρ(k0 ) cos k R (sin k − sin k0 ) ∂k0 U U + 4 cos k U k0 R −k0 +R 4 (sin k + sin k0 ) U 4 ∂ρ(k ′ ) ′ (sin k − sin k ′ ) dk U ∂k0 (B.12) provides the exact chemical potential. Analogous expressions can be derived for the spinless fermion model [Haldane 1980]. 113 Bibliography The electronic PDF file contains links to the online arXiv and journal references. S. Andergassen, T. Enss, and V. Meden, Kondo physics in transport through a quantum dot with Luttinger liquid leads, cond-mat/0509576. S. Andergassen, T. Enss, V. Meden, W. Metzner, U. Schollw¨ck, and K. Sch¨no o hammer, Functional renormalization group for Luttinger liquids with impurities, Phys. Rev. B 70, 075102 (2004), cond-mat/0403517. S. Andergassen, T. Enss, V. Meden, W. Metzner, U. Schollw¨ck, and K. Sch¨no o hammer, Renormalization group analysis of the one-dimensional extended Hubbard model with a single impurity, cond-mat/0509021. W. Apel and T. M. Rice, Combined effect of disorder and interaction on the conductance of a one-dimensional fermion system, Phys. Rev. B 26, R7063 (1982). O. M. Auslaender and S. Fishman, Correlations in the Adiabatic Response of Chaotic Systems, Phys. Rev. Lett. 84, 1886 (2000). X. Barnab´-Th´riault, A. Sedeki, V. Meden, and K. Sch¨nhammer, A junce e o tion of three quantum wires: restoring time-reversal symmetry by interaction, Phys. Rev. Lett. 94, 136405 (2005a), cond-mat/0411612. X. Barnab´-Th´riault, A. Sedeki, V. Meden, and K. Sch¨nhammer, Junce e o tions of one-dimensional quantum wires - correlation effects in transport, Phys. Rev. B 71, 205327 (2005b), cond-mat/0501742. G. Baym and L. P. Kadanoff, Conservation Laws and Correlation Functions, Phys. Rev. 124, 287 (1961). G. Benfatto and G. Gallavotti, Perturbation Theory of the Fermi Surface in a Quantum Liquid. A General Quasiparticle Formalism and One-Dimensional Systems, J. Stat. Phys. 59, 541 (1990). 114 Bibliography M. Bockrath, D. H. Cobden, J. Lu, A. G. Rinzler, R. Smalley, L. Balents, and P. L. Mceuen, Luttinger-liquid behaviour in carbon nanotubes, Nature 397, 598 (1999). R. T. Clay, A. W. Sandvik, and D. K. Campbell, Possible exotic phases in the one-dimensional extended Hubbard model, Phys. Rev. B 59, 4665 (1999). A. Cohen, K. Richter, and R. Berkovits, Spin and interaction effects on charge distribution and currents in one-dimensional conductors and rings within the HartreeFock approximation, Phys. Rev. B 57, 6223 (1998), cond-mat/9804018. S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambrigde, 1995. I. E. Dzyaloshinskii and A. I. Larkin, Correlation functions for a one-dimensional fermi system with long-range interaction (Tomonaga model), Zh. Eksp. Teor. Fiz. 65, 411 (1973), [Sov. Phys. JETP 38, 202 (1974)]. R. Egger and A. O. Gogolin, Effective Low-Energy Theory for Correlated Carbon Nanotubes, Phys. Rev. Lett. 79, 5082 (1997), cond-mat/9708065. R. Egger and H. Grabert, Friedel Oscillations for Interacting Fermions in One Dimension, Phys. Rev. Lett. 75, 3505 (1995), cond-mat/9509100. S. Eggert and I. Affleck, Magnetic impurities in half-integer-spin Heisenberg antiferromagnetic chains, Phys. Rev. B 46, 10866 (1992). S. Ejima, F. Gebhard, and S. Nishimoto, Tomonaga-Luttinger parameters for doped Mott insulators, Europhys. Lett. 70, 492 (2005), cond-mat/0507508, private communication. T. Enss, Renormalization, Conservation Laws and Transport in Correlated Electron Systems, PhD thesis, University of Stuttgart, Germany, 2005, cond-mat/0504703. T. Enss, V. Meden, S. Andergassen, X. Barnab´-Th´riault, W. Mete e zner, and K. Sch¨nhammer, Impurity and correlation effects on transo port in one-dimensional quantum wires, Phys. Rev. B 71, 155401 (2005), cond-mat/0411310. J. Feldman and E. Trubowitz, Perturbation Theory for Many Fermion Systems, Helv. Phys. Acta 63, 156 (1990). 115 Bibliography P. Fendley, A. W. W. Ludwig, and H. Saleur, Exact Conductance through Point Contacts in the ν = 1/3 Fractional Quantum Hall Effect, Phys. Rev. Lett. 74, 3005 (1995), cond-mat/9408068. H. Frahm and V. E. Korepin, Critical exponents for the one-dimensional Hubbard model, Phys. Rev. B 42, 10553 (1990). A. Furusaki, Resonant tunneling through a quantum dot weakly coupled to quantum wires or quantum Hall edge states, Phys. Rev. B 57, 7141 (1998), cond-mat/9712054. A. Furusaki, Kondo Problems in Tomonaga-Luttinger J. Phys. Soc. Jpn. 74, 73 (2005), cond-mat/0409016. liquids, A. Furusaki and N. Nagaosa, Resonant tunneling in a Luttinger liquid, Phys. Rev. B 47, 3827 (1993a). A. Furusaki and N. Nagaosa, Single-barrier problem and Anderson localization in a one-dimensional interacting electron system, Phys. Rev. B 47, 4631 (1993b). A. Furusaki and N. Nagaosa, Tunneling through a barrier in a TomonagaLuttinger liquid connected to reservoirs, Phys. Rev. B 54, R5239 (1996), cond-mat/9604193. U. Gerland, J. von Delft, T. A. Costi, and Y. Oreg, Transmission Phase Shift of a Quantum Dot with Kondo Correlations, Phys. Rev. Lett. 84, 3710 (2000), cond-mat/9909401. T. Giamarchi, Quantum Physics in One Dimension, Oxford University Press, New York, 2004. T. Giamarchi and H. J. Schulz, Anderson localization and interactions in onedimensional metals, Phys. Rev. B 37, 325 (1988). ´ L. I. Glazman and M. E. Raikh, Resonant Kondo transparency of a barrier with quasilocal impurity states, JETP Lett. 47, 452 (1988). C. J. Halboth and W. Metzner, Renormalization group analysis of the 2D Hubbard model, Phys. Rev. B 61, 7364 (2000), cond-mat/9908471. F. D. M. Haldane, General Relation of Correlation Exponents and Spectral Properties of One-Dimensional Fermi Systems: Application to the Anisotropic S = 1/2 Heisenberg Chain, Phys. Rev. Lett. 45, 1358 (1980). 116 Bibliography F. D. M. Haldane, Effective Harmonic-Fluid Approach to Low-Energy Properties of One-Dimensional Quantum Fluids, Phys. Rev. Lett. 47, 1840 (1981a). F. D. M. Haldane, ‘Luttinger liquid theory’ of one-dimensional quantum fluids. I. Properties of the Luttinger model and their extension to the general 1D interacting spinless Fermi gas, J. Phys. C 14, 2585 (1981b). A. Hewson, The Kondo Problem to Heavy Fermions, Cambridge University Press, Cambridge, 1993. C. Honerkamp, D. Rohe, S. Andergassen, and T. Enss, Interaction flow method for many-fermion systems, Phys. Rev. B 70, 235115 (2004), cond-mat/0403633. C. Honerkamp and M. Salmhofer, The temperature-flow renormalization group and the competition between superconductivity and ferromagnetism, Phys. Rev. B 64, 184516 (2001), cond-mat/0105218. C. Honerkamp, M. Salmhofer, N. Furukawa, and T. M. Rice, Breakdown of the Landau-Fermi liquid in Two Dimensions due to Umklapp Scattering, Phys. Rev. B 63, 035109 (2001), cond-mat/9912358. A. P. Kampf and A. A. Katanin, Competing phases in the extended U-VJ Hubbard model near the Van Hove fillings, Phys. Rev. B 67, 125104 (2003), cond-mat/0212190. C. Kane, L. Balents, and M. P. A. Fisher, Coulomb Interactions and Mesoscopic Effects in Carbon Nanotubes, Phys. Rev. Lett. 79, 5086 (1997), cond-mat/9708054. C. L. Kane and M. P. A. Fisher, Resonant tunneling in an interacting onedimensional electron gas, Phys. Rev. B 46, 7268 (1992a). C. L. Kane and M. P. A. Fisher, Transmission through barriers and resonant tunneling in an interacting one-dimensional electron gas, Phys. Rev. B 46, 15233 (1992b). C. L. Kane and M. P. A. Fisher, Transport in a one-channel Luttinger liquid, Phys. Rev. Lett. 68, 1220 (1992c). N. Kawakami and S.-K. Yang, Luttinger anomaly exponent of momentum distribution in the Hubbard chain, Phys. Lett. A 148, 359 (1990). G. Keller, C. Kopper, and M. Salmhofer, Perturbative renormalization and effective Lagrangians in ϕ4 in four dimensions, Helv. Phys. Acta 65, 32 (1992). 117 Bibliography E. H. Lieb and F. Y. Wu, Absence of Mott Transition in an Exact Solution of the Short-Range, One-Band Model in One Dimension, Phys. Rev. Lett. 135, A1505 (1968). A. Luther and I. Peschel, Single-particle states, Kohn anomaly, and pairing fluctuations in one dimension, Phys. Rev. B 9, 2911 (1974). J. M. Luttinger, An exactly soluble model of a many-fermion system, J. Math. Phys. 4, 1154 (1963). G. D. Mahan, Many-particle physics, Kluwer Academic Publishers, New York, 3. edition, 2000. D. C. Mattis, New wave-operator identity applied to the study of persistent currents in 1D, J. Math. Phys. 15, 609 (1974). D. C. Mattis and E. H. Lieb, Exact solution of a many-fermion system and its associated boson field, J. Math. Phys. 6, 304 (1965). K. A. Matveev, D. Yue, and L. I. Glazman, Tunneling in one-dimensional nonLuttinger electron liquid, Phys. Rev. Lett. 71, 3351 (1993), cond-mat/9306041. V. Meden, S. Andergassen, W. Metzner, U. Schollw¨ck, and K. Sch¨nhammer, o o Scaling of the conductance in a quantum wire, Europhys. Lett. 64, 769 (2003), cond-mat/0303460. V. Meden, T. Enss, S. Andergassen, W. Metzner, and K. Sch¨nhammer, o Correlation effects on resonant tunneling in one-dimensional quantum wires, Phys. Rev. B 71, 041302(R) (2005), cond-mat/0403655. V. Meden, W. Metzner, U. Schollw¨ck, O. Schneider, T. Stauber, and K. Sch¨no o hammer, Luttinger liquids with boundaries: Power-laws and energy scales, Eur. Phys. J. B 16, 631 (2000), cond-mat/0002215. V. Meden, W. Metzner, U. Schollw¨ck, and K. Sch¨nhammer, Scaling behavior o o of impurities in mesoscopic Luttinger liquids, Phys. Rev. B 65, 045318 (2002a), cond-mat/0104336. V. Meden, W. Metzner, U. Schollw¨ck, and K. Sch¨nhammer, A single impurity in a o o Luttinger liquid: How it “cuts” the chain, J. Low Temp. Phys. 126, 1147 (2002b), cond-mat/0109013. 118 Bibliography V. Meden, P. Schmitteckert, and N. Shannon, Orthogonality catastrophe in a one-dimensional system of correlated electrons, Phys. Rev. B 57, 8878 (1998), cond-mat/9707082. V. Meden and U. Schollw¨ck, The conductance of interacting nano-wires, o Phys. Rev. B 67, 193303 (2003a), cond-mat/0210515. V. Meden and U. Schollw¨ck, Persistent currents in mesoscopic rings: A o numerical and renormalization group study, Phys. Rev. B 67, 035106 (2003b), cond-mat/0209588. Y. Meir and N. S. Wingreen, Landauer formula for the current through an interacting electron region, Phys. Rev. Lett. 68, 2512 (1992). W. Metzner, C. Castellani, and C. Di Castro, Fermi Systems with Strong Forward Scattering, Advances in Physics 47, 317 (1998), cond-mat/9701012. W. Metzner and C. Di Castro, Conservation laws and correlation functions in the Luttinger liquid, Phys. Rev. B 47, 16107 (1993). K. Moon, H. Yi, C. L. Kane, S. M. Girvin, and M. P. A. Fisher, Resonant tunneling between quantum Hall edge states, Phys. Rev. Lett. 71, 4381 (1993), cond-mat/9304010. T. R. Morris, The Exact Renormalisation Group and Approximate Solutions, Int. J. Mod. Phys. A 9, 2411 (1994), hep-ph/9308265. Y. V. Nazarov and L. I. Glazman, Resonant Tunneling of Interacting Electrons in a One-Dimensional Wire, Phys. Rev. Lett. 91, 126804 (2003), cond-mat/0209090. J. W. Negele and H. Orland, Quantum Many-Particle Systems, Addison-Wesley, Reading, 1987. T. K. Ng and P. A. Lee, On-site Coulomb repulsion and resonant tunneling, Phys. Rev. Lett. 61, 1768 (1988). A. Oguri, Transmission Probability for Interacting Electrons Connected to Reservoirs, J. Phys. Soc. Jpn 70, 2666 (2001), cond-mat/0106033. R. D. Picciotto, H. L. Stormer, L. N. Pfeiffer, K. W. Baldwin, and K. W. West, Four-terminal resistance of a ballistic quantum wire, Nature 411, 51 (2001). 119 Bibliography J. Polchinski, Renormalization Nucl. Phys. B 231, 269 (1984). and effective lagrangians, D. G. Polyakov and I. V. Gornyi, Transport of interacting electrons through a double barrier in quantum wires, Phys. Rev. B 68, 035421 (2003), cond-mat/0212355. M. Salmhofer, Continuous Renormalization for Fermions and Fermi Liquid Theory, Commun. Math. Phys. 194, 249 (1998), cond-mat/9706188. M. Salmhofer, Renormalization. An Introduction, Springer, Berlin, 1999. M. Salmhofer and C. Honerkamp, Fermionic renormalization group flows: Technique and theory, Prog. Theor. Phys. 105, 1 (2001). K. Sch¨nhammer, Luttinger liquids: the basic concepts, In D. Baeriswyl and L. Deo giorgi, editors, Strong Interactions in Low Dimensions, Kluwer Academic Publishers, 2004, cond-mat/0305035. K. Sch¨nhammer, V. Meden, W. Metzner, U. Scholl-w¨ck, and O. Guno o narsson, Boundary effects on one-particle spectra of Luttinger liquids, Phys. Rev. B 61, 4393 (2000), cond-mat/9903121. H. J. Schulz, Correlation exponents and the metal-insulator transition in the onedimensional Hubbard model, Phys. Rev. Lett. 64, 2831 (1990). F. Sch¨ tz, L. Bartosch, and P. Kopietz, Collective fields in the funcu tional renormalization group for fermions, Ward identities, and the exact solution of the Tomonaga-Luttinger model, Phys. Rev. B 72, 035107 (2004), cond-mat/0409404. R. Shankar, Renormalization group for interacting fermions in d ¿ 1, Physica A 177, 530 (1991). R. Shankar, Renormalization-group approach to interacting Reviews of Modern Physics 66, 129 (1994), cond-mat/9307009. fermions, J. S´lyom, The Fermi gas model of one-dimensional conductors, Adv. Phys. 28, 201 o (1979). K. M. Tam, S. W. Tsai, and D. K. Campbell, Functional Renormalization Group Analysis of the Half-filled One-dimensional Extended Hubbard Model, cond-mat/0505396. 120 Bibliography J. R. Taylor, Scattering theory, John Wiley and Sons, New York, 1972. S. Tomonaga, Remarks on Blochs method of sound waves applied to many-fermion problems, Prog. Theor. Phys. 5, 544 (1950). J. Voit, One-Dimensional cond-mat/9510014. Fermi liquids, Rep. Prog. Phys. 58, 977 (1995), F. J. Wegner and A. Houghton, Renormalization Group Equation for Critical Phenomena, Phys. Rev. A 8, 401 (1973). S. Weinberg, Critical phenomena for field theorists, Erice Subnucl. Phys., 1 (1976). U. Weiss, Quantum Dissipative Systems, World Scientific, Singapore, 2. edition, 1999, and references therein. C. Wetterich, Exact evolution Phys. Lett. B 301, 90 (1993). equation for the effective potential, C. Wieczerkowski, Symanzik improved actions from the viewpoint of the renormalization-group, Commun. Math. Phys. 120, 149 (1988). K. J. Wiese, The Functional Renormalization Group Treatment of Disordered Systems: a Review, cond-mat/0302322. K. G. Wilson, Renormalization Group and Critical Phenomena. II. Phase-Space Cell Analysis of Critical Behavior, Phys. Rev. B 4, 3184 (1971). K. G. Wilson and J. Kogut, The renormalization group and the ǫ expansion, Phys. Rep. 12, 75 (1974). C. N. Yang and C. P. Yang, One-Dimensional Chain of Anisotropic Spin-Spin Interactions. I.+II., Phys. Rev. 150, 321 (1966). Z. Yao, H. W. C. Postma, L. Balents, and C. Dekker, Carbon nanotube intramolecular junctions, Nature 402, 273 (1999). D. Yue, L. I. Glazman, and K. A. Matveev, Conduction of a weakly interacting onedimensional electron gas through a single barrier, Phys. Rev. B 49, 1966 (1994). D. Zanchi and H. J. Schulz, Weakly correlated electrons on a square lattice: A renormalization group theory, Europhys. Lett. 44, 235 (1998), cond-mat/9703189. 121 Bibliography D. Zanchi and H. J. Schulz, Weakly correlated electrons on a square lattice: Renormalization-group theory, Phys. Rev. B 61, 13609 (2000), cond-mat/9812303. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford, 4. edition, 2002. 122