The fermionic limit of the δ-function Bose gas: a pseudopotential approach arXiv:cond-mat/0301210v1 [cond-mat.str-el] 14 Jan 2003 Diptiman Sen 1 Centre for Theoretical Studies, Indian Institute of Science, Bangalore 560012, India Abstract We use first-order perturbation theory near the fermionic limit of the δ-function Bose gas in one dimension (i.e., a system of weakly interacting fermions) to study three situations of physical interest. The calculation is done using a pseudopotential which takes the form of a two-body δ′′ -function interaction. The three cases considered are the behavior of the system with a hard wall, with a point where the strength of the pseudopotential changes discontinuously, and with a region of finite length where the pseudopotential strength is non-zero (this is sometimes used as a model for a quantum wire). In all cases, we obtain exact expressions for the density to first order in the pseudopotential strength. The asymptotic behaviors of the densities are in agreement with the results obtained from bosonization for a Tomonaga-Luttinger liquid, namely, an interaction dependent power-law decay of the density far from the hard wall, a reflection from the point of discontinuity, and transmission resonances for the interacting region of finite length. Our results provide a non-trivial verification of the Tomonaga-Luttinger liquid description of the δ-function Bose gas near the fermionic limit. PACS number: 71.10.Pm 1 E-mail address: diptiman@cts.iisc.ernet.in 1 1. Introduction The δ-function Bose gas in one dimension has been studied extensively ever since Lieb and Liniger solved it using the Bethe ansatz [1]. However, the wave functions which one obtains from the Bethe ansatz are usually quite difficult to work with. It is therefore useful to examine other ways of studying the δ-function Bose gas. With this in mind, we recently developed a way of doing perturbation theory near the fermionic limit of the model [2], which is equivalent to a Fermi gas with weakly attractive interactions. This involves the use of a two-body pseudopotential which takes the form of a δ ′′ -function ; let us denote the strength of the pseudopotential by a parameter g which will be defined below. In Ref. 2, we showed that this perturbative approach correctly reproduces the ground state energy up to order g 2 . In this paper, we will apply the pseudopotential approach to three situations which cannot be solved using the Bethe ansatz; this will illustrate the power of this approach. The δ-function Bose gas is an example of a Tomonaga-Luttinger liquid (TLL) [3, 4]. A TLL is characterized by two quantities, the velocity of the low-energy excitations v (the dispersion relation of these excitations is given by ω = v|k|), and the interaction parameter K. Once these two parameters are known, the low-energy and long wavelength properties of a TLL can be found by the technique of bosonization [3, 4]. In particular, K determines the exponents governing the long distance behavior of various correlation functions. We will show that the results obtained using the pseudopotential approach are in complete agreement with those expected from bosonization. Our study will therefore provide a non-trivial check of the pseudopotential approach and will also confirm the expression for the parameter K. The plan of this paper is as follows. In section 2, we will introduce the δ-function Bose gas and the pseudopotential approach for doing perturbation theory near the fermionic limit. In section 3, we will consider the δ-function Bose gas with a hard wall. We will obtain an expression for the density which is exact to first order in g. We will then find the asymptotic behavior of the density far from the hard wall. The expression of the density will involve logarithmic factors, which, in section 6, will be recognized as being due to an interaction dependent power-law decay of the density which is characteristic of a TLL. In section 4, we will consider a model in which the pseudopotential parameter g changes discontinuously at one point. We will again compute the density exactly to first order in g and show that there are oscillations which can be interpreted as being due to a reflection of the particles from that point. The amplitudes of reflection from the two sides are found to be equal to ±g. In section 5, we will extend the model of section 4 to the case of a region of finite length over which the pseudopotential has a non-zero strength. Finally, in section 6, we will discuss the TLL approach to the δ-function Bose gas, and will compute the Luttinger parameters K and v to first order in g. We will then use bosonization to compute the asymptotic behaviors of the density far from a hard wall and from a point of discontinuity in the Luttinger parameters. We will then see that these results agree precisely with those obtained in the previous sections. We will conclude in section 7 by pointing out some possible directions for future research. 2 2. The δ-function Bose gas The δ-function Bose gas is defined by the Hamiltonian H = − 1 2m 1≤i≤N ∂2 2c + 2 ∂xi m 1≤i N. (0) (B) a state of the type Ψ(0) (n′1 , n′2 ; n1 , n2 ) which differs from the ground state Ψ0 in that only two particles are excited from the levels n1 and n2 to the levels n′1 and n′2 , where 1 ≤ n2 < n1 ≤ N and n′1 > n′2 > N. 5 (0) The matrix elements of (2) between Ψ0 and states of type A are given by Vn′ ;n where An′ ,n (0) Ψ(0) (n′ ; n)|V |Ψ0 ≡ = π2 An′ ;n , 8mcL3 = n′ + n ≤N , 2 n′ − n − 3(n2 + n′2 ) + 10nn′ if n′ + n is even and 1 ≤ ≤N . 2 0 otherwise . (16) 3(n2 + n′2 ) + 10nn′ if n′ + n is even and 1 ≤ = = (0) The matrix elements of (2) between Ψ0 and states of type B will be denoted by Vn′1 ,n′2 ;n1 ,n2 ; however, we do not need the expressions for these for reasons which will soon become clear. Following (11), we see that the ground state wave function up to order 1/c is given by Ψ0 = (0) Ψ0 Ψ(0) (n′ ; n) + n′ >N 1≤n≤N (n2 − Vn′ ;n ′2 )π 2 /(2mL2 ) n Ψ(0) (n′1 , n′2 ; n1 , n2 ) + n′ >n′ >N 1≤n2 > 1) as shown in the appendix. The density turns out to have an expansion in powers of 1/x multiplied by sines, cosines, and logarithms. The leading order terms are given by ρ(x) = ρ0 − sin(2kF x) cos(2kF x) [ 1 − 2g ln(4πeC−1/2 ρ0 x) ] + 3g , 2πx 4x (22) where C = 0.5772 . . . is Euler’s constant. We will see in section 6 that the logarithm is a sign that the power of x in the denominator of sin(2kF x) should really be 1 + 2g (plus terms of higher order in g), rather than 1. Namely, 1 1 = [1 − 2g ln(ρ0 x)] (ρ0 x)1+2g ρ0 x (23) plus terms of higher order in g. 4. The δ-function Bose gas with a discontinuity in the interaction strength In this section, we will consider another application of the pseudopotential approach. We will consider a δ-function Bose gas on a circle, with the interaction parameter in (1) being equal to a large value c for 0 < x < L/2 and ∞ for L/2 < x < L. It is clear from (5) that the chemical potential will then be different in the two halves of the system; this would imply that a state which has the same density everywhere can no longer be close to the ground state. We will compensate for this imbalance by adding a one-body potential in the region 0 < x < L/2 which is equal to δV = 4π 2 ρ3 0 . 3mc (24) This ensures that the chemical potential and the density are the same in the two halves of the system; therefore the ground states of the noninteracting (c = ∞) and interacting 7 (large c) systems are smoothly connected to each other. We will therefore work with the following perturbation to the noninteracting system, V where f (x) = = = − 1 mc 1≤i 0. The bosonized form of this fermionic theory also contains only a left-moving boson field φL defined on the full line; the two fields are related as ˜ ΨL (x) ∼ exp [i √ π {φL (x) + φL (−x)} + i πK {φL (x) − φL (−x)}] . K (57) Now we can compute the fermion density which is equal to an expectation value in the ground state, ρ(x) = Ψ† (x)Ψ(x) , where x > 0. Following (51) and (56), this is given by ˜ ˜ ˜ ˜ Ψ† (x)ΨL (x) + Ψ† (−x)ΨL (−x) L L † −i2kF x ˜ ˜ ˜L ˜ + ΨL (−x)ΨL (x) e + Ψ† (x)ΨL (−x) ei2kF x . ρ(x) = (58) On using the bosonization expression (57), we find that the first two terms on the right hand side of (58) are independent of x; they give rise to a constant which is ρ0 . The last two terms in (58) give √ exp [±i2 πK{φL (−x) − φL (x)}] ∼ 1 . xK (59) Including the factors of e±i2kF x in (58), we find that ρ(x) − ρ0 ∼ sin(2kF x) . xK (60) We thus see that in the presence of a hard wall, the density of a TLL far from the wall has an oscillatory piece whose amplitude decays as 1/xK . If K is close to 1 as in (45), we see that the amplitude has an expansion in powers of g which is given by (1 − 2g lnx)/x. This is exactly what we found in section 3. We now turn to the model of section 4, where the Luttinger parameters are given by (v, K) for x > 0 and by (vF , 1) for x < 0. We can use the Lagrangian density in (55) with these parameters to find the equations of motion for the boson fields. The matching conditions at x = 0 turn out to be [7] vF φ(x = 0−, t) ∂φ(x, t) x=0− ∂x = = 13 φ(x = 0+, t) , v ∂φ(x, t) K ∂x x=0+ . (61) We can now consider what happens when a wave is incident from x = −∞. The equations of motion give φ(x, t) = = exp [ik(x − vF t)] + r− exp [−ik(x + vF t)] for x < 0 , kvF t− exp [i (x − vt)] for x > 0 . v (62) The matching conditions in (61) now lead to the following expressions for the reflection and transmission amplitudes r− = t− = K −1 , K +1 2K . K +1 (63) 2 Note that current conservation is satisfied since vF (1 − r− ) = vt2 and vF = vK. Similarly, for a wave incident from x = ∞, we find that r+ = t+ = 1−K , K +1 2 , K +1 (64) 2 which satisfies current conservation since v(1 − r+ ) = vF t2 . Upon using (45), we see that + the reflection amplitudes r± obtained here agree with those obtained in section 4. We therefore have the interesting result that a discontinuity in the interaction parameters is sufficient to cause scattering, even if there is no other scattering mechanism (like an impurity) present in the system. The calculations presented in section 4 can be viewed as a microscopic derivation of this interesting phenomenon which had earlier been obtained only from bosonization [7]. The results in section 5 have implications for the subject of transport through a quantum wire which is sometimes modeled as a TLL of finite length which is bounded on the two sides by Fermi liquid leads [7, 8, 9]. In these models, therefore, the Luttinger parameters change discontinuously at the contacts between the quantum wire and its leads. For the case of two identical impurities in a TLL, it is known that the transmission resonances are infinitely sharp at zero temperature [9]. Although the model we have studied in section 5 has two points of discontinuity in the interaction parameter, rather than two impurities, it is possible that the structure of the transmission resonance will be found to change significantly if we go up to higher orders in the interaction parameter 1/c. 6. Discussion We have used the pseudopotential approach to study the behavior of a system of fermions with weak attractive interactions. The various situations we have considered are not solvable by the Bethe ansatz [1]. This is because the Bethe ansatz only works in models which are both invariant under translations and have N commuting operators including the Hamiltonian. In such systems, the momenta of the N particles k1 , k2 , · · ·, kN 14 are good quantum numbers, and the wave function can be found exactly as a superposition of N! plane waves. In the absence of translation invariance, there are reflections (at a hard wall or at a point of discontinuity in the interaction strength) due to which the particle momenta are no longer good quantum numbers. The wave functions are then no longer a superposition of a finite number of plane waves, and they cannot be found exactly. When the Bethe ansatz fails, the pseudopotential approach seems to be the only way to obtain exact results near the fermionic limit, although calculational difficulties may restrict its use to low orders in perturbation theory. We have shown how exact expressions for the density can be obtained in certain situations. While the agreement between our results and those obtained from bosonization for the asymptotic behavior of the density is satisfying, we should also emphasize that there are relatively few models with interactions in which something can be computed at all distances. Our methods can be applied to other problems involving weakly interacting fermions in one dimension. For instance, one can study the Kane-Fisher model of a single impurity placed in a TLL [9], and the effect of a junction of three or more semi-infinite wires [6]. While these problems have been studied earlier using bosonization (and other methods which are only valid at long wavelengths [5]), it may be interesting to apply the pseudopotential method to these situations since we may be able to obtain expressions for certain quantities which are valid at all distances. Finally, one can use the pseudopotential method to study dynamical quantities like the conductance of a finite length TLL at finite frequencies. After this work was completed, we found a paper which discusses some properties of the one-dimensional Bose-Hubbard model at low densities [10]; the continuum Hamiltonian which governs that system is essentially the same as the one studied by us. Acknowledgments I thank Siddhartha Lal and Sumathi Rao for numerous discussions about bosonization and quantum wires. I acknowledge financial support from a Homi Bhabha Fellowship and the Council of Scientific and Industrial Research, India through grant no. 03(0911)/00/EMR-II. Appendix In this appendix, we will discuss some methods for computing the various expressions for the density presented above. Let us start with the expression for the hole in (21). Suppose that we have a function f (x) defined by the integral f (x) = u0 0 du cos(axu) h(u) , (65) where h(u) is finite and continuous at u = 0. Then we find ∞ 0 dx f (x) = limα→0 u0 0 du h(u) 15 ∞ 0 2 dx cos(axu) e−αx , π α 1 limα→0 2 π h(0) . 2a = = u0 0 2 u2 /(4α) du h(u) e−a , (66) On applying this to (20), we obtain (21). Next, we discuss how to obtain asymptotic expressions (for ax >> 1) for functions of the type f (x) = g(x) = u0 0 0 u0 du cos(axu) h(u) , du sin(axu) h(u) . (67) If h(u) is finite and continuous for all values of u in the range [0, u0], then the leading order expressions for (67) are of order 1/x, and they are obtained by integrating the functions cos(axu) and sin(axu). Namely, f (x) = g(x) = 1 [ h(u0 ) sin(axu0 ) ] , ax 1 [ h(0) − h(u0 ) cos(axu0 ) ] , ax (68) plus terms of order 1/x2 and higher. These formulae are valid even if u0 = ∞, provided that h(∞) = 0. Now suppose that the function h(u) has a logarithmic divergence at one point. Some examples of this are the integrals [11] 1 0 1 0 du sin(axu) lnu = du cos(axu) lnu = 1 [ ln(ax) + C ] , ax π , − 2ax − (69) plus terms of order 1/x2 and higher. (Here C = 0.5722 . . . is Euler’s constant). From (69), one can show that 1 0 1 0 du sin(axu) ln(1 − u) = du cos(axu) ln(1 − u) = 1 π [ ln(ax) cos(ax) + C cos(ax) − sin(ax) ] , ax 2 π 1 [ ln(ax) sin(ax) + C sin(ax) + cos(ax) ] , − ax 2 (70) plus terms of order 1/x2 and higher. Now consider a function h(u) in (67) which contains a logarithmic divergence at u = 0 ˜ of the form b lnu, where b is some constant. Then we define a new function h(u) = h(u) − b lnu which is finite for all values of u in the range of integration. We can then ˜ integrate over h(u) using (67-68), and over b lnu using (69). Combining the two gives the result of integrating over h(u). Similarly, we can compute the integrals in (67) if h(u) has a logarithmic divergence at u = u0 . Finally, we can also compute integrals over functions 16 h(u) in (67) which have a logarithmic divergence at a point u1 which lies inside the range [0, u0 ]. We simply divide the integrals in (67) into two parts, one over the range [0, u1] and the other over the range [u1 , u0 ]. Then each of these integrals can be computed as explained above. Using all these methods, one can derive the expressions in (22) and (29). References [1] E. H. Lieb and W. Liniger, Phys. Rev. 130 (1963) 1605; E. H. Lieb, ibid 130 (1963) 1616. [2] D. Sen, Int. J. Mod. Phys. A 14 (1999) 1789. [3] A. O. Gogolin, A. A. Nersesyan and A. M. Tsvelik, Bosonization and Strongly Correlated Systems (Cambridge University Press, Cambridge, 1998). [4] F. D. M. Haldane, J. Phys. C 14 (1981) 2585; H. J. Schulz, G. Cuniberti and P. Pieri, in Field Theories for Low-Dimensional Condensed Matter Systems, edited by G. Morandi, A. Tagliacozzo and P. Sodano (Springer, Berlin, 2000), cond-mat/9807366; S. Rao and D. Sen, in Field Theories in Condensed Matter Systems, edited by S. Rao (Hindustan Book Agency, New Delhi, 2001), cond-mat/0005492. [5] D. Yue, L. I. Glazman, and K. A. Matveev, Phys. Rev. B 49, 1966 (1994). [6] S. Lal, S. Rao and D. Sen, Phys. Rev. B 66, 165327 (2002). [7] I. Safi and H. J. Schulz, Phys. Rev. B 52, 17040 (1995); I. Safi and H. J. Schulz, ibid 59, 3040 (1999). [8] D. L. Maslov and M. Stone, Phys. Rev. B 52, 5539 (1995); V. V. Ponomarenko, ibid 52, 8666 (1995); A. Furusaki and N. Nagaosa, ibid 54, 5239 (1996); S. Lal, S. Rao and D. Sen, ibid 65, 195304 (2002), and cond-mat/0104402. [9] C. L. Kane and M. P. A. Fisher, Phys. Rev. B 46, 15233 (1992). [10] M. A. Cazalilla, cond-mat/0301167. [11] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, New York, 1994), pp. 612. 17