Renormalization group study of the Kondo problem at a junction of several Luttinger wires V. Ravi Chandra1 , Sumathi Rao2 and Diptiman Sen1 arXiv:cond-mat/0608187v1 [cond-mat.str-el] 8 Aug 2006 1 Centre for High Energy Physics, Indian Institute of Science, Bangalore 560012, India 2 Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad 211019, India (Dated: May 11, 2018) We study a system consisting of a junction of N quantum wires, where the junction is characterized by a scalar S-matrix, and an impurity spin is coupled to the electrons close to the junction. The wires are modeled as weakly interacting Tomonaga-Luttinger liquids. We derive the renormalization group equations for the Kondo couplings of the spin to the electronic modes on different wires, and analyze the renormalization group flows and fixed points for different values of the initial Kondo couplings and of the junction S-matrix (such as the decoupled S-matrix and the Griffiths S-matrix). We generally find that the Kondo couplings flow towards large and antiferromagnetic values in one of two possible ways. For the Griffiths S-matrix, we study one of the strong coupling flows by a perturbative expansion in the inverse of the Kondo coupling; we find that at large distances, the system approaches the ferromagnetic fixed point of the decoupled S-matrix. For the decoupled Smatrix with antiferromagnetic Kondo couplings and weak inter-electron interactions, the flows are to one of two strong coupling fixed points in which all the channels are strongly coupled to each other through the impurity spin. But strong inter-electron interactions, with Kρ < N/(N + 2), stabilize a multi-channel fixed point in which the coupling between different channels goes to zero. We have also studied the temperature dependence of the conductance at the decoupled and Griffiths S-matrices. PACS numbers: 73.63.Nm, 72.15.Qm, 73.23.-b, 71.10.Pm I. INTRODUCTION The area of molecular electronics has grown tremendously in recent years as a result of the drive towards smaller and smaller electronic devices [1]. Molecular electronic circuits typically need multi-probe junctions. The first experimental growths of three-terminal nanotube junctions were not well controlled [2]; more recently, new growth methods have been developed whereby uniform Y -junctions have been produced [3, 4, 5]. Transport measurements have also been carried out for the Y -junctions [6], as well as for three-terminal junctions obtained by merging together single-walled nanotubes by molecular linkers [7]. On the theoretical side, there have been several studies of junctions of quantum wires. There have been detailed studies of carbon nanotubes with different proposed structures for the junction [8, 9]. Several groups have analyzed the geometry and stability of the junctions [10, 11]. Junctions of quantum wires have also been studied [12, 13, 14, 15, 16, 17] in terms of one-dimensional wires, with the junction being modeled by a scattering matrix S. These studies include the effects of electronelectron interactions which are often cast in the language of Tomonaga-Luttinger liquid (TLL) theory [18, 19, 20]. Many earlier studies of junctions have only included ‘scalar’ scatterings at the junction. i.e., the S-matrix has been taken to be spin-independent. The response of a junction of quantum wires to a magnetic impurity or an impurity spin at the junction has recently been studied both experimentally [21] and theoretically [22, 23, 24, 25, 26]. As is well-known in three dimensions, an impurity spin can lead to the Kondo effect [27]. [The coupling between the conduction electrons and the impurity spin grows as one goes to lower temperatures; this leads to a larger scattering and therefore a larger resistance as long as the temperature is higher than the Kondo temperature TK . Below TK , the resistance due to scattering from the impurity spin decreases (if the value of the impurity spin S is larger than or equal to half the number of channels N ) because the spin decouples from the electrons.] The Kondo effect for a ‘two-wire junction’ in a TLL wire has been studied by several groups [28, 29, 30, 31, 32, 33, 34, 35]. Using a renormalization group (RG) analysis for weak potential scattering, Furusaki and Nagaosa showed that for an impurity spin of 1/2, there is a stable strong coupling fixed point (FP) consisting of two semi-infinite uncoupled TLL wires and a spin singlet [29]. For strong potential scattering, the above FP is reached when the inter-electron interactions are weak. However, sufficiently strong inter-electron interactions are known to stabilize the two-channel Kondo FP instead [30]. The Kondo effect has also been studied in crossed TLL wires [36] and in multi-wire systems [37, 38]. In this paper, we consider a junction of quantum wires which is characterized by an S-matrix at the junction; further, an impurity spin is coupled to the electrons at the junction. The wires are modeled as semi-infinite TLLs. The details of the model defined in the continuum will be described in Sec. II. In Sec. III, we will discuss how RG equations for the Kondo couplings and for the S-matrix 2 at the junction can be obtained by successively integrating out the electronic modes far from the Fermi energy. We find that the flow of the Kondo couplings involve the S-matrix elements, but the flow of the S-matrix elements do not involve the Kondo couplings (up to second order in the latter). To simplify our analysis, therefore, we concentrate on the FPs of the S-matrix RG equations and study how the Kondo couplings evolve in Sec. IV. For the case of N decoupled wires, we find that for a large range of initial values of the Kondo couplings, the system flows to a multi-channel ferromagnetic (FM) FP lying at zero coupling. This FP is associated with spin-flip scatterings of the electrons from the impurity spin whose temperature dependence will be discussed. Outside this range, the flow is towards a strong antiferromagnetic (AFM) coupling. On the other hand, at the Griffiths S-matrix (defined below), there is no stable FP for finite values of the Kondo couplings, and the system flows towards strong AFM coupling in two possible ways. We also consider the case when the scattering matrix has a chiral form. In this case, we find that the Kondo coupling matrix for the three wire case has three independent degrees of freedom and a single FP at strong coupling. The strong coupling flows will be further discussed in Sec. V where we will consider some lattice models at the microscopic length scale. As in the three-dimensional Kondo problem, we find that there are various possibilities depending on the number of wires N and the spin S of the impurity, such as the under-screened, over-screened and exactly screened cases [39]. We will generally see that a Kondo coupling which is small at high temperatures (small length scales) can become large at low temperatures (large length scales). In Sec. VI, we will show that the vicinity of one of the strong coupling FPs can be studied through an expansion in the inverse of the coupling; we will then find that the large coupling can be re-interpreted as a small coupling in a different model. In Sec. VII, we will study the case of decoupled wires with strong interactions using the technique of bosonization. Analogous to the results of [30], we find that the multi-channel (N ≥ 2) AFM Kondo FP is stabilized for Kρ < N/(N + 2). We will discuss the temperature dependence of the conductance in Sec. VIII at both high and low temperature; we will compare the behaviors of Fermi liquids and TLLs. Sec. IX will contain some concluding remarks. A condensed version of some parts of this paper has appeared elsewhere [26]. Before proceeding further, we would like to emphasize that we have not used bosonization in this paper (except in Sec. VII), although this is a powerful and commonly used method for studying TLLs [18, 19, 20]. In the presence of a junction with a general scattering matrix, it is not known whether the idea of bosonization can be implemented. (Some reasons for the difficulty in bosonizing are explained in Ref. [14]). It is therefore necessary to work directly in the fermionic language. We have adopted the following point of view in this work [14, 40]. We start with non-interacting electrons for which the scattering matrix approach and the Landauer formalism for studying electronic transport [41, 42] are justified. We then assume that the interactions between the electrons are weak, and treat the interactions to first order in perturbation theory to derive the RG equations. This is the approach used in most of this paper. Only in Sec. VII do we use bosonization to discuss the effect of strong interactions for the case of decoupled wires, since that is one of the cases where bosonization can be used. II. MODEL FOR SEVERAL WIRES COUPLED TO AN IMPURITY SPIN We begin with N semi-infinite quantum wires which meet at one site which is the junction; on each wire, the spatial coordinate x will be taken to increase from zero at the junction to ∞ as we move far away from the junction. The incoming and outgoing fields are related by an S-matrix at the junction, which is an N × N unitary matrix whose explicit values depend on the details of the junction. Hence the wave function corresponding to an electron with spin α (α =↑, ↓) and wave number k which is incoming in wire i (i = 1, 2, · · · , N ) is given by ψiαk (x) = e−i(k+kF )x + Sii ei(k+kF )x on wire i , = Sji ei(k+kF )x on wire j = i . (1) Here k is the wave number with respect to the Fermi wave number kF (i.e., k = 0 implies that the energy of the electron is equal to the Fermi energy EF ). We will take k to go from −Λ to Λ, where Λ is a cut-off of the order of kF ; we will eventually only be interested in the long wavelength modes with |k| ≪ Λ. We will use a linearized approximation for the dispersion relation so that the energy of an electron with wave number k is given by vF k with respect to the Fermi energy; here vF is the Fermi velocity, and we are setting = 1. In Eq. (1), we will refer to the waves going as e−ikx as the incoming part ψIiαk , and the waves going as eikx as the outgoing part ψOiαk or ψOjαk . The second quantized annihilation operator corresponding to the wave function in Eq. (1) is given by Ψiαk (x) = ciαk ψiαk (x) , (2) where the wire index i runs from 1 to N , and the total second quantized operator is given by Λ Ψα (x) = i −Λ dk ciαk ψiαk (x) . 2π (3) (Note that it is not possible to quantize the system in terms of N independent fields on each of the wires, because an electron that is incoming on one wire has outgoing components on all the other wires as well). The 3 non-interacting part of the Hamiltonian for the electrons is then given by Λ dk k c† ciαk . iαk 2π H0 = vF −Λ α i α,β J S · Ψ† (x = 0) α σαβ Ψβ (x = 0) , 2 Λ Λ −Λ −Λ i,j α,β Jij S · dk1 dk2 2π 2π c† 1 iαk σαβ cjβk2 , 2 Hint dx dy ρ(x) U (x − y) ρ(y) , (6) (7) where the density ρ is given in terms of the second quantized electron field Ψα (x) as ρ = Ψ† Ψ↑ + Ψ† Ψ↓ . As men↑ ↓ tioned earlier for the wave-functions, the electron field can also be written in terms of outgoing and incoming fields as Ψα (x) = ΨOα (x) eikF x + ΨIα (x) e−ikF x . III. THE RENORMALIZATION GROUP EQUATIONS It is known that the interaction parameters g1 , g2 and g4 satisfy some RG equations [43]; the solutions of the lowest order RG equations are given by [40] ˜ U (2kF ) 1 + Hint = [g1 Ψ† Ψ† ΨOβ ΨIα + g2 Ψ† Ψ† ΨIβ ΨOα Oα Iβ Oα Iβ ˜ U(2kF ) πvF , ln L ˜ U (2kF ) 1 1 ˜ ˜ U (2kF ) + , g2 (L) = U (0) − ˜ 2 2 1 + U(2kF ) ln L πvF ˜ g4 (L) = U (0) , (11) where L denotes the length scale. In general, the couplings g1 , g2 and g4 can have different values on different wires; hence we have to add a subscript i to them. For weak interactions, i.e., when g1i , g2i and g4i are all much less than 2πvF , we can derive the RG equations for the S-matrix at the junction [14, 40]. Let us define a parameter αi = g2i − 2 g1i , 2πvF (12) which is a function of length scale due to Eqs. (11), and a diagonal matrix M whose entries are given by Mii = 1 αi rii . 2 (13) Then the RG equations can be written in the matrix form dS = M − SM † S . d ln L (8) If the range of the interaction U (x) is short (of the order of the Fermi wavelength 2π/kF ), such as that of a screened Coulomb repulsion, the Hamiltonian in (7) can be written as dx For repulsive and attractive interactions, g2 > 0 and < 0 respectively. (We have ignored umklapp scattering terms here; they only arise if the model is defined on a lattice and we are at half-filling). g1 (L) = ∗ where Jij = J(1 + l Sli )(1 + m Smj ) is a Hermitian matrix. In general, however, the impurity spin may also be coupled to the electrons at other sites which are slightly away from the junction; for instance, this may be true if the model is defined on a lattice at the microscopic scale as we will see in Sec. V. It is therefore convenient to take Jij to be an arbitrary Hermitian matrix which is not necessarily related to the entries of the S-matrix in any simple way. Next, let us consider density-density interactions between the electrons in each wire of the form (we will drop the wire index i for the moment) 1 = 2 (10) (5) where σ denotes the Pauli matrices. (For simplicity, we will assume an isotropic spin coupling Jx = Jy = Jz ). Eq. (5) can be written in terms of second quantized operators as Hspin = ˜ g1 = U (2kF ) , ˜ and g2 = g4 = U (0) . (4) If the impurity spin is coupled to the electrons at the junction, that part of the Hamiltonian is given by Hspin = where (14) The FPs of this equation are given by the condition M = SM † S. We use the technique of ‘poor man’s RG’ [39, 44] to derive the renormalization of the S-matrix and the Kondo coupling matrix Jij . Briefly, this involves using the second order perturbation expression for the low energy effective Hamiltonian, α,β 1 + g4 (Ψ† Ψ† ΨOβ ΨOα + Ψ† Ψ† ΨIβ ΨIα )], Oα Oβ Iα Iβ 2 (9) Heff = h |l2 >< l2 |Hpert |h >< h|Hpert |l1 >< l1 | , El − Eh (15) 4 J J + π δ(k1 − k2 + k3 − k4 ) , J g 2 (b) (a) FIG. 1: Pictures of the terms which contribute to the renormalization of the Kondo coupling matrix J to order J 2 and g2 J respectively; g2 denotes the coefficient of the electronelectron interaction. Thin lines and thick lines denote low energy and high energy electrons respectively. where the perturbation Hpert is given by the sum of Hspin and Hint in Eqs. (6) and (9), l1 and l2 denote two energy states, and h denotes high energy states. We now restrict the sum over h in Eq. (15) to run over states for which the energy difference Eh −El lies within an energy shell E and E +dE; we have assumed that the difference between different low energy states is much smaller than E, so that we can simply write El1 = El in the denominator of the above equation. We then see that the change in the effective Hamiltonian dHeff is proportional to dE/E which is equal to −d ln L, where the length scale L is inversely related to the energy scale E. In this way, we get an RG equation for the derivatives with respect to ln L of various parameters appearing in the low energy Hamiltonian. Using this method, we find that the Kondo couplings Jij do not contribute to the renormalization of the Smatrix in Eq. (14) up to second order in Jij . (This is not true beyond second order; however, we will only work to second order here assuming that the Jij are small). On the other hand, the S-matrix does contribute to the renormalization of the Jij through the interaction Hamiltonian in Eq. (9); this is because the relation between the outgoing field on wire i (i.e., ΨOiα ) and the operators cjα involves the S-matrix. For instance, the terms involving g2i in Eq. (9) take the form Λ i,j,l α,β Λ Λ Λ −Λ −Λ −Λ −Λ dk1 dk2 dk3 dk4 2π 2π 2π 2π × π δ(k1 − k2 + k3 − k4 ) g2i ∗ × Sij c† 1 c† 2 ciβk3 Sil clαk4 , iβk jαk where we have used the identity ∞ dx e(−ik1 +ik2 −ik3 +ik4 −ǫ)x 0 i k1 − k2 + k3 − k4 − iǫ 1 = − iP k1 − k2 + k3 − k4 = − (16) (17) with ǫ being an infinitesimal positive number. [In Eq. (16), we have kept only the δ-function term and have dropped the principal part term since the latter can be either positive or negative, and its contribution vanishes when one integrates over the variables ki .] Note that the terms involving g2 in Eq. (16) (as well as those involving g1 and g4 in Eq. (9)) conserve momentum while the Kondo coupling terms in Eq. (6) do not. We will omit the details of the RG calculations here apart from making a few comments below. We find that dJij d ln L 1 [ = 2πvF + − Jik Jkj k 1 g2i Sij 2 1 2 k ∗ Jik Sik + k 1 ∗ g2j Sji 2 Jkj Sjk k ∗ ∗ (g2k − 2g1k ) (Jik Skk Skj + Ski Skk Jkj ) ] , (18) where Sij is the S-matrix at the length scale L. Eq. (18) is the key result of this paper. Note that it maintains the hermiticity of the matrix Jij . Eq. (18) always has a trivial FP at Jij = 0. Let us briefly comment on the origin of the various terms on the right hand side of Eq. (18). The first and second lines arise from Figs. 1 (a) and (b) respectively. (The terms of order J 2 in the first line have been derived in Ref. [22]). The parameters g1i and g4i do not appear in the second line of Eq. (18) since the terms which are proportional to these parameters either do not appear in the numerator of Eq. (15) because they are not allowed by momentum conservation, or they appear in Eq. (15) but their contribution vanishes because the Pauli matrices are traceless. Finally, the third line of Eq. (18) arises as follows. In Ref. [14], the RG equation for the S-matrix was derived. This was based on the idea that due to reflections at the junction (these arise from the diagonal elements of the S-matrix which are the reflection amplitudes), there are Friedel oscillations in the density of the electrons; the amplitudes of these oscilla∗ tions are proportional to Skk and Skk in wire k. We now treat the interactions in the Hartree-Fock approximation [14]; this results in reflections from the Friedel oscillations with a strength proportional to g2k − 2g1k in wire k. Now, an electron going from wire j to i can either (i) first go from wire j to wire k with a transmission amplitude Skj , scatter from the Friedel oscillations in wire k ∗ with an amplitude (g2k − 2g1k )Skk , and finally scatter off the impurity spin from wire k to wire i with amplitude Jik , or (ii) first scatter off the impurity spin from wire j to wire k with amplitude Jkj , scatter from the Friedel oscillations in wire k with an amplitude (g2k − 2g1k )Skk , 5 and finally scatter from wire k to wire i with a transmis∗ sion amplitude Ski . These two processes give rise to the third line of Eq. (18). It is interesting to observe that Eq. (18) remains invariant if we transform Sij → eiφi Sij , where the φi can be arbitrary real numbers. According to Eq. (1), this corresponds to the freedom of redefining the phases of the outgoing waves by different amounts on different wires. IV. ANALYSIS OF THE RENORMALIZATION GROUP EQUATIONS To simplify our analysis, we will make two assumptions. (i) The couplings g1i and g2i have the same value on all the wires, and therefore the subscript i on g1 and g2 can be dropped. (ii) The S-matrix is at a FP of Eq. (14), so that S does not flow with the length scale. We will now consider several possibilities for the Smatrix, and will study the RG flows and FPs of the Kondo couplings Jij in each case. The different possibilities can be realized in terms of quantum wires and quantum dots containing the impurity spin as shown in Fig. 2. (a) (b) FIG. 2: Schematic pictures of the system of wires (shown by solid lines), an impurity spin (shown inside a circle), and the coupling between the spin and the wires (dotted lines). Figures (a) and (b) show the cases of disconnected and Griffiths S-matrices respectively. A. N disconnected wires The S-matrix for N disconnected wires is given by the N × N identity matrix (up to phases). (We will assume that N ≥ 2). A picture of the system is indicated in Fig. 2 (a); the wires are disconnected from each other, and the end of each wire is coupled to the impurity spin. A more microscopic description of the system will be discussed in Sec. V. Let us consider a highly symmetric form of the Kondo coupling matrix in which all the diagonal entries are equal to J1 and all the off-diagonal entries are equal to J2 , with both J1 and J2 being real. (In the language of the three-dimensional N -channel Kondo problem, J2 denotes coupling between different channels). Since the S-matrix is also symmetric under the exchange of any two of the N indices, such a symmetric form of the Kondo matrix will remain intact during the course of the RG flow. In other words, it is natural for us to choose the J matrix to have the same symmetry as the S-matrix, since that symmetry is preserved under the RG flow. Eq. (18) gives the two-parameter RG equations 1 dJ1 2 [J 2 + (N − 1)J2 + 2g1 J1 ] , = d ln L 2πvF 1 dJ2 1 2 [2J1 J2 + (N − 2)J2 − (g2 − 2g1 )J2 ] . = d ln L 2πvF (19) (For N = 2 and g1 = 0, Eq. (19) agrees with the results in Ref. [30]). Since g1 (L = ∞) = 0, Eq. (19) has only one FP at finite values of (J1 , J2 ), namely, the trivial FP at (0, 0). We then carry out a linear stability analysis around this FP. [Given a RG equation of the form dX/d ln L = aX, we will say that the FP at X = 0 is stable if a < 0, unstable if a > 0, and marginal if a = 0. In the marginal case, we look at the next order term; if dX/d ln L = bX 2 and b > 0, we say that the FP at X = 0 is stable on the x < 0 side and unstable on the x > 0 side.] If ν ≡ g2 (L = ∞)/(2πvF ) > 0 (repulsive interactions), the stability analysis shows that the trivial FP is stable to small perturbations in J2 . For small perturbations in J1 , this FP is marginal; a second order analysis shows that it is stable if J1 < 0 and unstable if J1 > 0, i.e., it is the usual ferromagnetic FP which is found for Fermi liquid leads. However, the approach to the FP is quite different when the leads are TLLs. At large length scales, the FP is approached as J1 ∼ −1/ ln L and J2 ∼ 1/Lν . From this, we can deduce the behavior at very low temperatures, namely, J1 ∼ − 1/ ln(TK /T ) , and J2 ∼ (T /TK )ν . (20) where we have introduced the Kondo temperature TK . (This is given as usual by TK ∼ Λe−2πvF /J , where Λ is an energy cut-off of the order of the Fermi energy EF , J is the value of a typical Kondo coupling at the microscopic length scale as explained after Eq. (22), and 1/(2πvF ) is the density of states at EF ). The form in Eq. (20) is in contrast to the behavior of J2 for Fermi liquid leads, i.e., for g1 = g2 = 0. In that case, Eq. (19) can be solved exactly in terms of the linear combinations J1 − J2 and J1 + (N − 1)J2 ; we again find a FP at (J1 , J2 ) = (0, 0), with J1 ∼ − 1/ ln(TK /T ) , and J2 ∼ 1/ ln(TK /T )2 . (21) Note that J2 approaches zero faster than J1 for both Fermi liquid leads and TLL leads; but for the latter case, it goes to zero much faster, i.e., as a power of T . 6 Eq. (21) is valid provided that neither J1 − J2 nor J1 + (N − 1)J2 is exactly equal to zero; if one of them is exactly zero and the other is not, then both J1 and J2 go as 1/ ln(TK /T ). However, having one of the two combinations exactly equal to zero requires a special tuning in a microscopic model, as we will see in Sec. V. In general, therefore, the powers of 1/ ln(TK /T ) in J1 and J2 are different; this does not seem to have been noted in the earlier literature. 4 3 2 J2 −−−−−−> 1 0 −1 −2 −3 −4 −4 −3 −2 −1 0 1 J1 −−−−−−> 2 3 4 FIG. 3: RG flows of the Kondo couplings for three discon˜ ˜ nected wires, with U (0) = U (2kF ) = 0.2(2πvF ). Figure 3 shows a picture of the RG flows for three ˜ ˜ wires for U (0) = U (2kF ) = 0.2(2πvF ). [This gives a value of ν which is comparable to what is found in several experimental systems (see [45] and references therein). In all the pictures of RG flows, the values of Jij are shown in units of 2πvF .] We see that the RG flows take a large range of initial conditions to the FP at (0, 0). For all other initial conditions, we see that there are two directions along which the Kondo couplings flow to large values; these are given by J2 /J1 = 1 and J2 /J1 = −1/(N − 1) (with N = 3). [On a cautionary note, we should remember that the RG equations studied here are only valid at the lowest order in Jij and g2 , i.e., for the case of weak repulsion (or attraction) and small Kondo couplings.] The fact that the Kondo couplings flow to large values along two particular directions can be understood as follows. For values of J1 and J2 much larger than g1 and g2 , one can ignore the terms of order g1 and g2 in Eq. (19). One then obtains the decoupled equations d [J1 − J2 ] 1 (J1 − J2 )2 , ≃ d ln L 2πvF 1 d [J1 + (N − 1)J2 ] (J1 + (N − 1)J2 )2 . ≃ d ln L 2πvF (22) From these equations one can deduce that the couplings can flow to large values in one of two ways, depending on the initial conditions. Either J1 + (N − 1)J2 goes to ∞ much faster than J1 − J2 (this is what happens in the first quadrant of the figures in Figs. 3 and 4), or J1 − J2 goes to ∞ much faster than J1 + (N − 1)J2 (this happens in the fourth quadrant of the figures in Figs. 3 and 4). A third possibility is that J2 remains exactly equal to zero while J1 → ∞; however, this can only happen if one begins with J2 exactly equal to zero. (This also seems to happen if the interactions are strong enough as we will discuss in Sec. VII). We will provide a physical interpretation of the first two possibilities in Sec. V. Eq. (22) has the form dJ/d ln L = J 2 /(2πvF ). If J(d) denotes the value of J at a microscopic length d, and J(d) ≪ 2πvF , then it becomes of order 1 at a length scale L0 given by L0 /d ∼ e2πvF /J(d) ; the corresponding temperature is given by TK ∼ Λe−2πvF /J(d) . Finally, note that the special case with J2 = 0 and g1 = g2 = 0 is equivalent to the Kondo problem in three dimensions with N channels and no coupling between channels [27]. In the three-dimensional case, the RG equation has been derived to fifth order in the Kondo coupling [46]. This reveals a stable FP at a finite value of the coupling J1 = 4πvF , N (23) where 1/(2πvF ) is the density of states at the Fermi energy. Thus the couplings Jij need not really flow to infinity as Fig. 3 would suggest; one may find strong coupling FPs lying at values of order 2πvF if one takes into account terms of higher order in the RG equations. In Sec. VII, we do find a strong coupling FP for sufficiently strong inter-electron interactions. Although we have discussed the case of completely disconnected wires here, the results do not change significantly if we allow a small spin-independent tunneling amplitude of the form Ψ† (xi = 0) Ψj,α (xj = 0) . (24) i,α Htun = τ i=j α This is equivalent to changing the S-matrix slightly away from the identity matrix. Using the RG equation in (14), we find that the parameter τ satisfies the RG equation 1 dτ (g2 − 2g1 ) τ . = − d ln L 2πvF (25) This has the same form as the interaction dependent terms in the RG equation for J2 in (19). Hence, τ also scales at low temperatures as T ν just like J2 in Eq. (20). Thus the contributions of both τ and J2 to the conductance go as (T /TK )2ν . Here and subsequently we have not discussed the case of attractive interactions (g2 < 0). The stability analysis 7 can easily be suitably modified in that case; some of the directions for the RG flows may become stable and others may become unstable if the sign of g2 is reversed. 4 3 J2 −−−−−−> 2 1 B. Griffiths S-matrix for N wires 0 −1 This is the case in which all the N wires are connected to each other and there is maximal transmission, subject to the constraint that there is complete symmetry between the N wires. (We will again assume that N ≥ 2.) A picture of the system is indicated in Fig. 2 (b); the wires are connected to each other at a junction, and the junction is also coupled to the impurity spin. A more microscopic description of the junction will be discussed in Sec. V. The maximally transmitting completely symmetric Smatrix is also called the Griffiths S-matrix and has all the diagonal entries equal to −1+2/N and all the off-diagonal entries equal to 2/N . Since here, too, the S-matrix is fully symmetric in the N wires, we again consider the highly symmetric form of the Kondo coupling matrix as in the previous subsection, with real parameters J1 and J2 as the diagonal and off-diagonal entries respectively. Eq. (18) then gives dJ1 2 1 2 2 [J1 + (N − 1)J2 + 2g1 (1 − )2 J1 = d ln L 2πvF N 1 2 − 4g1 (1 − ) (1 − ) J2 ], N N dJ2 4g1 1 2 2 [2J1 J2 + (N − 2)J2 − = (1 − )J1 d ln L 2πvF N N 2 (26) + (g2 − 2g1 (1 − )2 ) J2 ]. N (For N = 2, i.e., a full line with an impurity spin coupled to one point on the line, Eq. (26) agrees with the equations derived in Ref. [29]). The only FP of Eq. (26) is again the trivial FP at the origin. A linear stability analysis shows that this FP is unstable in one direction (J2 ) and marginal in the other (J1 ) for g2 (L = ∞) > 0. Figure 4 shows a picture of the RG flows for three ˜ ˜ wires for U (0) = U (2kF ) = 0.2(2πvF ). We see that there is no stable FP at finite values of the couplings. The couplings flow to large values along one of the two directions J2 /J1 = 1 and J2 /J1 = −1/(N − 1). The reason for this is the same as that explained around Eq. (22) since the RG equations in (19) and (26) have the same form for large values of J1 and J2 . −2 −3 −4 −4 −3 −2 −1 0 1 J1 −−−−−−> 2 3 4 FIG. 4: RG flows of the Kondo couplings for the Griffiths ˜ ˜ S-matrix for three wires, with U (0) = U (2kF ) = 0.2(2πvF ). C. Chiral S-matrix for three wires We choose a chiral S-matrix of the form   0 0 γ S = γ 0 0 , 0 γ 0 (27) where γ is a complex number satisfying |γ| = 1. (We will see a physical realization of this form in Sec. V. Alternatively, we could have considered an S-matrix which is the transpose of the one given above). Let us consider a Kondo coupling matrix of the form   ∗ J1 J2 J2 ∗ J =  J 2 J1 J2  , (28) ∗ J2 J2 J1 where J1 is real but J2 can be complex. Then Eq. (18) gives 1 dJ1 2 [ J1 + 2|J2 |2 ] , = d ln L 2πvF 1 1 dJ2 ∗ [ 2J1 J2 + (J2 )2 + = g2 J2 ] . (29) d ln L 2πvF 2 [Note that the above equations remain invariant under the transformation J2 → J2 ei2π/3 or J2 e−i2π/3 . We will see in Sec. V. C that a lattice realization of the chiral S-matrix has the same symmetry.] One can again show that the only FP of Eq. (29) is the trivial FP at the origin. A linear stability analysis shows that the trivial FP is unstable in one direction (J2 ) and marginal in the other (J1 ) for g2 (L = ∞) > 0. Figure 5 shows a picture of the RG flows for three wires ˜ ˜ for U (0) = U (2kF ) = 0.2(2πvF ). The upper and lower 8 under RG, only the values φ = 0 and ±2π/3 are stable. Substituting this fact that cos(3φ) → 1 in the second equation in (30), and combining it with the first equation in (29), we obtain the decoupled equations 4 3 | J2 | −−−−−> 1 d [J1 − |J2 |] (J1 − |J2 |)2 , ≃ d ln L 2πvF d [J1 + 2|J2 |] 1 (J1 + 2|J2 |)2 . ≃ d ln L 2πvF 2 1 0 −4 −3 −2 −1 0 J1 −−−−−−> 1 2 3 4 4 Phase of J2 −−−−−−> 3 (31) From this we deduce that J1 + 2|J2 | must flow to ∞ much faster than J1 − |J2 | since J1 + 2|J2 | > J1 − |J2 | to begin with. Note that unlike the disconnected and Griffiths cases, where J1 and J2 flow to large values in two possible ways (with |J2 |/J1 → 1 and −1/(N − 1) respectively), in the chiral case, J1 and J2 flow to large values in only one way, along the direction |J2 |/J1 = 1. 2 V. INTERPRETATION IN TERMS OF LATTICE MODELS 1 0 −1 −2 −3 −4 −4 −3 −2 −1 0 J1 −−−−−−> 1 2 3 4 FIG. 5: RG flows for the chiral S-matrix for three wires, with ˜ ˜ U (0) = U (2kF ) = 0.2(2πvF ). The upper and lower figures show the magnitude and phase respectively of J2 . figures show the way in which the magnitude and phase of J2 evolve. We see that there is no stable FP at finite values of the couplings. The phase of J2 flows towards one of the three values, 0 or ±2π/3; this is consistent with the symmetry of J2 pointed out after Eq. (29). Further, J1 and the magnitude of J2 flow in such a way that J1 + 2|J2 | grows much faster than J1 − |J2 |. We can understand these observations as follows. For values of J1 and J2 much larger than g2 , one can ignore the term of order g2 in Eq. (29). If we write J2 = |J2 |eiφ , we find that dφ 1 |J2 | sin(3φ) , ≃ − d ln L 2πvF 1 d|J2 | [ 2J1 |J2 | + |J2 |2 cos(3φ) ] . (30) ≃ d ln L 2πvF The first equation in (30) shows that φ = 0, ±π/3, ±2π/3 and π are fixed points; however, since |J2 | flows to ∞ We will now see how the different S-matrices and RG flows discussed in Sec. IV can be interpreted in terms of lattice models [29]. This will provide us with physical interpretations of the various kinds of RG flows and FPs. We will concentrate on what the lattice models imply about the structure of the region near the junction, rather than the form of the interactions between the electrons in the bulk of the wires which has already been discussed in Sec. II. (The interactions can be introduced in the lattice model by, for instance, writing a Hubbard term at each site). We will again discuss three different cases here. (The models shown in Fig. 6 and discussed below in detail can be thought of as providing a microscopic picture of the systems shown in Fig. 2). 3 2 2 2 1 3 2 1 1 1 0 1 3 2 1 1 1 2 2 3 3 3 3 (a) 2 3 (b) FIG. 6: Lattice models for some of the S-matrices for three wires. (a) can be a model for the disconnected and Griffiths S-matrices, while (b) can be a model for the chiral S-matrix. 9 A. N disconnected wires This system can be realized by a lattice of the form shown in Fig. 6 (a). N wires meet at a junction which is labeled by the site number 0; all the other sites are labeled as n = 1, 2, · · ·, with n increasing as one goes away from the junction. (The lattice spacing will be set equal to one). We take the Hamiltonian to be of the tight-binding form, with a hopping amplitude equal to −t on all the bonds (where t is real), except for the bonds which connect the sites labeled as n = 1 on each wire to the junction site; we set those hopping amplitudes equal to zero. This is equivalent to removing the junction site from the system; we will therefore not consider that site any further in this subsection. We then obtain a system of disconnected wires with an S-matrix which is equal to −1 times the identity matrix. To show this, we consider a wave which is incoming on wire i with a wave number k, where 0 < k < π. We then find that the corresponding eigenstate of the Hamiltonian has an energy equal to Ek = −2t cos k, and a wave function given by ψik (n) = e−ikn − eikn for n = 1, 2, · · · on wire i , = 0 at the junction and on all wires j = i . (32) We introduce an on-site potential which is equal to µ at all sites. In the absence of interactions, the ground state is one in which all the states with energies going from = −2t up to µ are filled; the Fermi wave number kF is given by µ = −2t cos kF , assuming that µ lies in the range [−2t, 2t]. [We can then redefine all the wave numbers k by subtracting kF from them as indicated after Eq. (1); the redefined wave numbers then run from −Λ to Λ, where Λ is of order kF .] Let us now consider coupling the impurity spin to the sites labeled as n = 1 on the different wires by the following Hamiltonian Hspin = F1 S · + F2 S · Ψ† (i, 1) α i α,β σαβ Ψβ (i, 1) 2 Ψ† (i, 1) α i=j α,β σαβ Ψβ (j, 1) , 2 (33) where Ψα (i, 1) denotes the second quantized electron field at site 1 on wire i with spin α. (Eq. (42) below will provide a justification for this Hamiltonian). In Eq. (33), F1 and F2 denote amplitudes for spin-dependent scattering from the impurity within the same wire and between two different wires respectively. Namely, a spin-up electron coming in through one wire can get scattered by the impurity spin as a spin-down electron either along the same wire (F1 ) or along a different wire (F2 ). We then find that the Kondo coupling matrix Jij in Eq. (6) is as follows: all the diagonal entries are given by J1 and all the off-diagonal entries are given by J2 , where J1 = 4F1 sin2 kF , and J2 = 4F2 sin2 kF (34) for modes with redefined wave numbers lying close to zero. This is precisely the kind of Kondo matrix whose RG flows were studied in Sec. IV. A. The flows of the parameters J1 and J2 considered there can be translated into flows of the parameters F1 and F2 here. In particular, the approach to the FP at (J1 , J2 ) = (0, 0) given by Eq. (20) at low temperatures implies that spin-flip scattering within the same wire or between two different wires will have quite different temperature dependences. The flows to strong coupling shown in Fig. 3 can be interpreted as follows. In the first quadrant of Fig. 3, we see that J1 + (N − 1)J2 goes to ∞ faster than |J1 − J2 |; Eq. (34) then implies that F1 and F2 go to ∞. In the fourth quadrant of Fig. 3, J1 − J2 goes to ∞ faster than |J1 + (N − 1)J2 |; this implies that F1 goes to ∞ and F2 goes to −∞ as −F1 /(N − 1). These flows to strong coupling have the following interpretations. In the first case, F1 and F2 flow to ∞. From Eq. (33), this implies that the impurity spin (of magnitude S) is strongly and antiferromagnetically coupled to only one electronic field, namely, the ‘centre of √ mass’ field given by i Ψ(i, 1)/ N (suppressing the spin labels and the Pauli matrices for the moment). Hence that field and the impurity spin will combine to form an effective spin of S − 1/2. In analogy with the threedimensional Kondo problem, we can say that the impurity spin is under-screened or exactly screened if S > 1/2 or S = 1/2 respectively. In the second case, F1 and F2 = −F1 /(N − 1) go to ∞. Using Eq. (33), we can then show that the impurity spin is strongly and antiferromagnetically coupled to the N − 1 ‘difference’ fields (given √ by the orthogonal combinations [Ψ(1, 1) − Ψ(2, 1)]/ 2, √ [Ψ(1, 1) + Ψ(2, 1) − 2Ψ(3, 1)]/ 6, · · ·). Hence those fields and the impurity spin will combine to give an effective spin of S −(N −1)/2 = S +1/2−N/2. Thus the impurity spin is under-screened, exactly screened or over-screened if 2S + 1 is greater than, equal to or less than N respectively. B. Griffiths S-matrix for N wires This system can again be realized by the lattice shown in Fig. 6 (a) and a tight-binding Hamiltonian. However, we now take the hopping amplitude to be −t on all bonds, except for the bonds which connect the sites labeled as n = 1 on each wire to the junction site; on those bonds, we take the hopping amplitude to be t1 = −t 2/N . The on-site potential is taken to be µ at all sites, including the junction. We then find that the S-matrix is of 10 the Griffiths form for all values of the wave number k. Namely, for a wave which is incoming on wire i with a wave number k, the wave function is given by ψik (n) = e−ikn − (1 − 2 ) eikn on wire i , N 2 ikn e on all wires j = i , N 2 = at the junction site . N = (35) We now consider coupling the impurity spin to the junction site labeled by zero, and the sites labeled as n = 1 on the different wires by the following Hamiltonian Hspin = F3 S · Ψ† (0) α α,β + F4 S · σαβ Ψβ (0) 2 Ψ† (i, 1) α i α,β σαβ Ψβ (i, 1) , 2 (36) where Ψα (0) denotes the second quantized electron field at the junction site with spin α. (Sec. VI will provide a justification for this kind of a coupling). Then the Kondo coupling matrix Jij in Eq. (6) takes the following form: all the diagonal entries are given by J1 and all the offdiagonal entries are given by J2 , where 2 4F3 + 2F4 [ 1 − (1 − ) cos 2kF ] , N2 N 4F4 4F3 + cos 2kF (37) = N2 N J1 = and J2 for modes with wave numbers lying close to zero. The RG flows of this kind of Kondo matrix were studied in Sec. IV. B. In terms of F3 and F4 , the variables in Eq. (22) are given by J1 − J2 = 2F4 (1 − cos 2kF ) , 4F3 and J1 + (N − 1)J2 = + 2F4 (1 + cos 2kF ) . N (38) Since 0 < kF < π, 1 ± cos 2kF lie between 0 and 2. In the first quadrant of Fig. 4, J1 + (N − 1)J2 goes to ∞ faster than |J1 − J2 |; Eq. (38) then implies that F3 goes to ∞ and |F4 | ≪ F3 . In the fourth quadrant of Fig. 4, J1 − J2 goes to ∞ faster than |J1 + (N − 1)J2 |; this implies that F4 goes to ∞ and F3 goes to −∞. These flows to strong coupling have the following interpretations. In the first case, F3 flows to ∞ which means that the impurity spin (of magnitude S) is strongly and antiferromagnetically coupled to an electron spin at the junction site n = 0; hence those two spins will combine to form an effective spin of S − 1/2. (This case will be discussed in detail in Sec. VI). In the second case, F3 goes to −∞ while F4 goes to ∞; hence the impurity spin is coupled strongly and ferromagnetically to an electron spin at the site n = 0, and antiferromagnetically to electron spins at the sites labeled as n = 1 on each of the N wires (see Fig. 6 (a) for the site labels). Hence the impurity spin will combine with those N + 1 spins to form an effective spin of S + 1/2 − N/2. Interestingly, we see that the magnitudes of the effective spins formed in the strong coupling limits in the first and fourth quadrants are the same in the cases of N disconnected wires and the Griffiths S-matrix. C. Chiral S-matrix for three wires This system can be realized by a lattice of the form shown in Fig. 6 (b). The three wires meet at a triangle; the sites on each wire are labeled as n = 1, 2, · · ·. The hopping amplitude is taken to be −t on all the bonds, except for the three bonds on the triangle. On those bonds, we take the hopping amplitude to be complex, and of the form −teiθ in the clockwise direction and −te−iθ in the anticlockwise direction. [We can think of the total phase 3θ of the product of hopping amplitudes around the triangle as being the Aharonov-Bohm phase arising from a magnetic flux enclosed by the triangle. Such a flux breaks time reversal symmetry which makes the Smatrix non-symmetric. Note that since only the value of 3θ modulo 2π has any physical significance, we are free to shift the value of θ by ±2π/3. This changes the phase of the coupling J2 defined below.] We then find that the S-matrix is of the chiral form given by Eq. (27), provided that the wave number k satisfies ei(3θ+k) = − 1 . (39) The phase γ in (27) is then given by ei(θ+k) . [Unlike the disconnected and Griffiths cases, we have not found a lattice model which gives an S-matrix as in (27) for all values of the wave number k.] Given a value of θ, we therefore choose a chemical potential µ = −2t cos kF such that kF satisfies Eq. (39). Since the properties of a fermionic system at low temperatures are governed by the modes near kF , the above prescription produces a system with a chiral S-matrix. We now consider coupling the impurity spin to the three sites of the triangle through the Hamiltonian Hspin = F5 S · Ψ† (i, 1) α i α,β σαβ Ψβ (i, 1) . 2 (40) Then the Kondo coupling matrix Jij in Eq. (6) takes the form given in Eq. (28), where J1 = 2F5 , and J2 = F5 e−i(θ+3kF ) (41) 11 for modes with wave numbers lying close to zero. This is a special case of the Kondo matrix given in Eq. (28). [To obtain the most general form given in (28), we need to introduce another parameter, such as a coupling of the impurity spin to the sites labeled by n = 2 in Fig. 6 (b).] The RG flows of this kind of Kondo matrix were studied in Sec. IV. C. has no terms of order t1 or t1 F4 , and it is given by Heff = F1,eff Seff · + F2,eff Seff · +C i=j VI. EXPANSION AROUND A STRONG COUPLING FIXED POINT In Sec. V, we considered several examples of Smatrices and the RG flows of the Kondo coupling. In most cases, we found that the Kondo couplings flow to large values. We can now ask whether the vicinity of the strong coupling FPs can be studied in some way. We will see that it is possible to do so through an expansion in the inverse of the Kondo coupling [39]. We will consider one example of such an expansion here. Following the discussion given after Eq. (38), let us assume that the RG flows for the case of the Griffiths S-matrix have taken us to a strong coupling FP along the direction J2 /J1 = 1, as shown in the first quadrant of Fig. 4. This implies that the coupling of the impurity spin S to an electron spin at the junction site n = 0 has a large and positive (antiferromagnetic) value F3 , while its coupling to the sites labeled as n = 1 on each of the wires has a finite value F4 which is much less than F3 (the site labels are shown in Fig. 6 (a)). The ground state of the F3 term (namely, just the first term in Eq. (36)) consists of a single electron at site n = 0 which forms a total spin of S −1/2 with the impurity spin. The energy of this spin state is −F3 (S + 1)/2; this lies far below the high energy states in which there is a single electron at site n = 0 which forms a total spin of S + 1/2 with the impurity spin (these states have energy F3 S/2), or the states in which the site n = 0 is empty or doubly occupied (these states have zero energy). We can now do a perturbative expansion in 1/F3 . We take the unperturbed Hamiltonian to be one in which the hopping amplitudes on all the bonds are −t, except for the bonds connecting the sites labeled as n = 1 on the different wires to the junction site; we take those hopping amplitudes to be zero. (This means that the unperturbed Hamiltonian corresponds to the case of N disconnected wires). We also include the spin coupling proportional to F3 in the unperturbed Hamiltonian. We take the perturbation Hpert as consisting of (i) the hopping amplitude t1 on the bonds connecting the sites labeled as n = 1 to the junction site, and (ii) the F4 term in Eq. (36). Using this perturbation, we can find an effective Hamiltonian [39]. [Once again, we use the expression in Eq. (15), where the high energy states are the ones listed in the previous paragraph. We will work up to second order in t1 and F4 .] If S > 1/2, we find that the effective Hamiltonian si i Ψ† (i, 1) α i=j α,β σαβ Ψβ (j, 1) 2 (Seff · si ) (Seff · sj ) + D i 1/2, we note that the last two terms in Eq. (42) are irrelevant as boundary operators if g2 (L = ∞)/(2πvF ) is small; this is because si has the scaling dimension 1 − g2 /(2πvF ) (as one can see from Eq. (19)), and therefore the product si ⊗ sj has the scaling dimension 2(1 − g2 /(2πvF )) which is larger than 1. The first two terms in Eq. (42) have the same form as in Eqs. (33) and (34), where the effective Kondo couplings J1,eff = 4F1,eff sin2 kF , and J2,eff = 4F2,eff sin2 kF (44) are equal, negative and small. We can now study the RG flow of this as in Sec. IV. A. With these initial conditions, 12 Eq. (19) and Fig. 3 show that the Kondo couplings flow to the FP at (J1,eff , J2,eff ) = (0, 0). In this example, therefore, we obtain a picture of the RG flows at both short and large length scales. We start with the Griffiths S-matrix with certain values of the Kondo coupling matrix, and we eventually end at the stable FP of the disconnected S-matrix for repulsive interactions, g2 (L = ∞) > 0. We will not discuss here what happens for the other possible RG flow for the Griffiths S-matrix, in which J1 and J2 become large along the direction J2 /J1 = −1/(N − 1). As we noted in Sec. V B, N + 1 spins get coupled strongly to the impurity spin in that case; an expansion in the inverse coupling is much more involved in that case. For the same reason, we will not discuss expansions in the inverse coupling for the flows to strong coupling for the disconnected and chiral S-matrices. VII. DECOUPLED WIRES WITH STRONG INTERACTIONS In this section, we will briefly discuss what happens if there are N decoupled wires and the interactions are strong. For the decoupled S-matrix, one can ‘unfold’ the electron field in each semi-infinite wire to obtain a chiral electron field in an infinite wire, and then bosonize that chiral field [18, 19, 20]. In the language of bosonization, the interaction parameters are given by Kρ for the charge sector and Kσ for the spin sector. Spin rotation invariance implies that Kσ = 1, while Kρ is related to our parameters gi as follows [20], Φi,ρ and Φi,σ . Close to the junction denoted as xj = 0, we have ηi,↑ i(Φi,ρ /√2Kρ + Φi,σ /√2) e , Ψi,↑ ∼ √ 2πd √ ηi,↓ i(Φi,ρ / 2Kρ − Φi,σ /√2) and Ψi,↓ ∼ √ e , (46) 2πd where we have used the fact that Kσ = 1, and we have not explicitly written the arguments of the fields (xi = 0) for notational convenience. The ηi,a denote Klein factors, and d is a short distance cut-off; these will not play any role below. In bosonic language, the Hamiltonian H = H0 + Hint in Eqs. (4) and (9) is given by H = 1 4π ∞ i dxi [ vρ 0 ∂Φi,ρ ∂xi 2 + vσ ∂Φi,σ ∂xi 2 ], (47) where vρ , vσ denote the charge and spin velocities respectively. The bosonic fields satisfy the commutation relations [ ∂Φi,a (xi ) , Φj,b (xj ) ] = i 2π δab δij δ(xi − xj ) , (48) ∂xi where a, b = ρ, σ. The impurity spin part of the Hamiltonian is given by Ψ† i,α Hspin = J1 S · i α,β σαβ Ψi,β 2 Ψ† i,α + J2 S · i=j α,β σαβ Ψj,β . (49) 2 The spin densities on different wires are given by Kρ = 1 + g4 /πvF + (g1 − 2g2 )/2πvF 1 + g4 /πvF − (g1 − 2g2 )/2πvF → 1 + g1 − 2g2 . 2πvF 1 ∂Φi,σ 1 . [ Ψ† Ψi,↑ − Ψ† Ψi,↓ ] = √ i,↑ i,↓ 2 2 2π ∂xi (45) In the second line of the above equation, we have taken the limit of small gi since we have worked to lowest order in the gi in the earlier sections. From Eq. (11), we see that 2g2 − g1 is invariant under the RG flow. The case of repulsive interactions corresponds to 2g2 − g1 > 0, i.e., Kρ < 1. The case of two decoupled wires (N = 2) has been studied by Fabrizio and Gogolin in Ref. [30]. They showed that if the interactions are weak enough, the Kondo couplings J1 and J2 are both relevant; their results then agree with those discussed in Sec. IV A. But if the interactions are sufficiently strong, i.e., Kρ < 1/2, then J2 is irrelevant and flows to zero. We will now show that their results can be generalized to the case of N wires; one finds that there is again a value of Kρ below which J2 is irrelevant. Following Ref. [30], we can write the spin-up and down Fermi fields Ψi,α in wire i in terms of the charge and spin bosonic fields (50) The other terms take the form √ 2Φi,σ Ψ† Ψi,↓ ∼ e−i i,↑ Ψ† Ψj,↑ ∼ e i,↑ Ψ† Ψj,↓ ∼ e i,↑ , √ √ √ (i/ 2)[−Φi,ρ / Kρ −Φi,σ +Φj,ρ / Kρ +Φj,σ ] √ √ √ (i/ 2)[−Φi,ρ / Kρ −Φi,σ +Φj,ρ / Kρ −Φj,σ ] , , (51) and so on. In (49) and (51), we have not explictly written the arguments of the fields, xi = xj = 0; we will continue to do this wherever convenient. [The bosonic forms of the fermion bilinears in Eqs. (50) and (51) are so different because we are using abelian bosonization. For the same reason, we will find it useful to distinguish between the different components of J1 and J2 , i.e., J1z , J1⊥ , J2z , and J2⊥ .] Let us define N ‘orthonormal’ linear combinations of the spin boson fields, namely, the ‘centre of mass’ combination 1 Φ0 = √ σ N Φi,σ , i (52) 13 and the ‘difference’ fields n 1 [ n(n + 1) Φn = σ + m=1 Φm,σ − n Φn+1,σ ] , (53) where n = 1, 2, · · · , N − 1. We can now write Eq. (49) in the bosonic language. We obtain Hspin = J1z √ Sz 2 2π + − ∂Φi,σ ∂xi √ 0 i J1⊥ [ S + ei 4πd J2z z S πd 2/N Φσ ei n i or < N/(N + 2). For N = 2, this gives the critical value of Kρ to be 1/2 [30], while for N → ∞, the critical value of Kρ approaches 1, i.e., the limit of weak repulsive interactions. We saw in Sec. IV A that a flow to strong coupling is indeed possible along the line J2 = 0, although that line is unstable to small perturbations in J2 . We now see that the line is stabilized (to first order in the couplings) if the interactions are sufficiently strong, i.e., if Kρ < (bn )2 = 1 , ij n + H.c. ] , + H.c.] b n Φn ij σ sin ei J1z = i × sin + n n n ai Φσ J2⊥ [ S+ 2πd N . N +2 (59) If J2 flows to zero and J1 flows to large values, Eq. (33) shows that the impurity spin is coupled strongly and antiferromagnetically to the electron fields Ψ(i, 1) on all the N wires; hence they will combine to form an effective spin of S − N/2. (If S < N/2, the impurity spin is overscreened). This describes a N -channel AFM FP with no coupling between channels [23, 25]. VIII. CONDUCTANCE CALCULATIONS Our calculations for the Kondo couplings can be explicitly applied to various geometries of quantum wires and a quantum dot (containing the impurity spin) shown in Fig. 2, such as (a) a dot coupled independently to each wire (disconnected S-matrix for the wires), so that the conductance can only occur through the dot, or (b) a side-coupled dot (Griffiths S-matrix for the wires), where the conductance can occur directly between the wires. In general, of course, one can have any S-matrix at the junction, so that the conductance can occur both through the dot and directly between the wires. Let us now consider the conductance near the different FPs [24, 35] for the case of weak interactions. In the Griffiths case where the conductance can occur directly 14 between the wires, let us consider the case of small values of J1 , J2 (both much smaller than 2πvF ), and g2 ≫ g1 . At high temperatures, before the Ji ’s have grown very much under RG, we see from Eq. (26) that J1 remains small, while J2 grows due to the term g2 J2 . Namely, J2 ∼ (T /TK )−ν , where ν = g2 (L = ∞)/(2πvF ). The effect of J2 is to scatter the electrons from the impurity spin, and thereby reduce the conductance between any two wires from the maximal value of G0 = (4/N 2 )e2 /h. 2 Since the scattering probability is proportional to J2 , the conductance at high temperatures (T ≫ TK ) is given by G − G0 ∼ − G0 S(S + 1) (T /TK )−2ν . (60) [The factor of S(S + 1) appears for the following reason. Consider an electron coming in through wire i; it can have spin up or down, and the impurity spin can have any value of S z from S to −S. We assign all these 2(2S + 1) states the same probability. As a result of the Kondo coupling J2 , the electron can scatter to a different wire j; as a result, its spin may or may not flip, and the value of S z for the impurity spin can also change by 0 or ±1. If we calculate the probabilities of all the different possible processes and add them, we get a factor of S(S + 1)]. Using Eq. (45), we see that (60) takes the form G − G0 ∼ − G0 S(S + 1) (T /TK )Kρ −1 , (61) where we have used the RG flow to set g1 = 0 and ν = g2 (L = ∞)/(2πvF ). On the other hand, if the leads were Fermi liquids (g1 = g2 = 0), J2 would be given by Eq. (21), and we would get G − G0 ∼ − G0 S(S + 1) . ln(T /TK )4 (62) At low temperatures, the Kondo couplings flow to large values; as discussed at the end of Sec. VI, their behaviors are then governed by the FP at (J1,eff , J2,eff ) = (0, 0) of the disconnected wire case with an effective spin Seff = 2 S − 1/2. In this case, only J2 contributes to the conductance between two different wires. From Eq. (20), we see that the conductance is given by G ∼ G0 Seff (Seff + 1) (T /TK )2ν ∼ G0 Seff (Seff + 1) (T /TK )1−Kρ (63) for T ≪ TK . For Fermi liquid leads, Eq. (21) implies that the conductance is given by G ∼ G0 Seff (Seff + 1) . ln(T /TK )4 35], but Eqs. (62) and (64) differ from the expressions given in earlier papers (like Ref. [35]) for the powers of 1/ ln(T /TK ). (As we had discussed earlier after Eq. (21), we would get the same powers of 1/ ln(T /TK ) as in Ref. [35] if J2 was exactly equal to J1 or −J1 /(N − 1)). The above expressions for the conductance shows that for both Fermi liquid leads and TLL leads (with repulsive interactions), and for both T ≫ TK and T ≪ TK , the conductance increases with the temperature. It is then natural to assume that this would be true for intermediate temperatures as well, so that the conductance increases monotonically with temperature from 0 to G0 ; this would be consistent with the results in Refs. [29, 35]. It may be useful to discuss here why there is no Kondo resonance peak in the conductance at low temperatures in our model, in contrast to what is found in other models (for instance Refs. [24, 49, 50]) and observed experimentally [51, 52]. In our model, once the impurity spin gets very strongly coupled to the junction site in Fig. 6 (b) (due to the flow to large J1 and J2 in the Griffiths case), that site decouples from the wires; this leaves no other pathway for the electrons to transmit from one wire to another. In contrast to this, if the junction region was more complicated (for instance, if there were additional bonds which connect different wires without going through the impurity spin, or there was a dot with several energy levels through which the electron can transmit), then the electron may still be able to transmit even after the impurity quenches the electron on a single site (or level). Hence, it may be possible for the conductance to increase to the unitarity limit at the lowest temperatures; this is known to occur for models with Fermi liquid leads. For TLL leads, however, our analysis remains valid even if there are additional bonds between the wires, because any such direct tunneling amplitudes are irrelevant and renormalize to zero as shown in Eq. (25). Finally, let us briefly consider the case of strong interelectron interactions. For Kρ < N/(N + 2), we saw in Sec. VII that a multi-channel FP gets stabilized in the case of N disconnected wires. To obtain the conductance at this point, we need to study the operators perturbing this point, similar to the analysis in Refs. [25, 34, 35]; this has not yet been done. (64) Thus a measurement of the temperature dependence of the conductance should be able to distinguish between the Fermi liquid and TLL cases at both high and low temperatures. For the case N = 2, the expressions in Eqs. (61) and (63) agree with those given in Refs. [29, IX. CONCLUSIONS To summarize our results, we have studied systems of TLL wires which meet at a junction. The junction is described by a spin-independent S-matrix, and there is an impurity spin which is coupled isotropically to the electrons in the neighborhood of the junction. The Smatrix and the Kondo coupling matrix Jij satisfy certain RG equations. We have studied the RG flows of the Kondo couplings for a variety of FPs of the S-matrices. Although the Kondo couplings generally grow large, one 15 can sometimes study the system through an expansion in the inverse of the coupling. This leads to a new system in which the effective Kondo couplings are weak; the RG flows of these effective couplings can then be studied. For example, at the fully connected or Griffiths Smatrix, we find that for a range of initial conditions, the Kondo couplings can flow to a strong coupling FP along the direction J2 /J1 = 1, where their fate is decided by a 1/J analysis. This analysis then shows that the couplings flow to the FM FP of the disconnected S-matrix lying at (J1,eff , J2,eff ) = (0, 0). For this system, therefore, one obtains a description of the system at both short and large length scales. For the case of disconnected wires and repulsive interactions, there is a range of Kondo couplings which flow towards a multi-channel FM FP at (J1 , J2 ) = (0, 0). At low temperatures, we find spin-flip scattering processes with temperature dependences which are dictated by both the Kondo effect and the inter-electron interactions. It may be possible to observe such scatterings by placing a quantum dot with a spin at a junction of several wires with interacting electrons. For other initial conditions for the disconnected case, the Kondo couplings flow towards the strong coupling FPs at J1 , |J2 | → ∞. In general, this is just the single channel strong coupling AFM FP. But there is a special line where J1 → ∞ and J2 = 0; this is the multichannel AFM FP. The RG equations show that both J1 and J2 are relevant around the weak coupling FP if the interactions are weak. 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