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Proceedings of the Japan Academy Series A Mathematical Sciences

Studies on holonomic quantum fields, XVI. Density matrix of impenetrable bose gas

Michio Jimbo, Tetsuji Miwa, Yasuko Mōri, Mikio Sato

Published Jan 1, 1979• 7 citations
Publisher

Abstract

In this article we report the ollowing result concerning system of impenetrable bosons in one dimension at zero temperature: The one particle reduced density matrix p(x) satisfies a non-linear differ- ential equation, an equivalent o a Painlev equation oi the fith kind.This enables us to calculate the small and large x behaviors of p(x) to an arbitrary order.For the statement of the problem see [1] and references cited therein.As mentioned in [2], our calculation is done by relating the problem to the preceding result obtai,ned there concerning the double scaling limit of the XY model.Main results are summarized in 1.Their derivation is briefly described in 2-3.

DOI
10.3792/pjaa.55.317

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In this article we report the ollowing result concerning system of impenetrable bosons in one dimension at zero temperature: The one particle reduced density matrix p(x) satisfies a non-linear differ- ential equation, an equivalent o a Painlev equation oi the fith kind.This enables us to calculate the small and large x behaviors of p(x) to an arbitrary order.For the statement of the problem see [1] and references cited therein.As mentioned in [2], our calculation is done by relating the problem to the preceding result obtai,ned there concerning the double scaling limit of the XY model.Main results are summarized in 1.Their derivation is briefly described in 2-3.
We compute exactly the one particle reduced density matrix ρ (r) of a system of impenetrable bosons in one dimension at zero temperature. We do this by relating ρ (r) to a certain double scaling limit of the transverse correlation function of the one-dimensional spin 1/2 X–Y model. We study the asymptotic behavior of ρ (r) for large r. This expansion contains oscillatory terms which arise due to the intrinsic quantum mechanical nature of the problem. We use these results to discuss the analytic structure of the momentum density function n (k).
We report the large-r\mathcal{r}r and small-r\mathcal{r}r expansions of the one-particle reduced density m‘\stackrel{\mathrm{`}}{\mathrm{m}}m‘atrix $\ensuremath{\rho}(\mathcal{r})$ of a system of impenetrable bosons in one dimension at zero temperature. These expansions were derived from an exact calculation of $\ensuremath{\rho}(\mathcal{r})$ to be reported elsewhere. We find that the large-r\mathcal{r}r expansion of $\ensuremath{\rho}(\mathcal{r})$ contains oscillatory terms which we relate to the analytic structure of the momentum density function n(k)n(k)n(k).
The well-known results concerning a dilute Bose gas with the short-range repulsive interaction should be reconsidered due to a thermodynamic inconsistency of the method being basic to much of the present understanding of this subject. The aim of our paper is to propose another way of treating the dilute Bose gas with an arbitrary strong interaction. Using the reduced density matrix of the second order and a variational procedure, this way allows us to escape the inconsistency mentioned and operate with singular potentials of the Lennard-Jones type. The derived expansion of the condensate depletion in powers of the boson density n=N/V reproduces the familiar result, while the expansion for the mean energy per particle is of the form epsilon=2 pi planck(2)an/m[1+128/(15 square root of [pi]) square root of [na(3)](1-5b/8a)+...], where a is the scattering length and b> or =0 stands for one more characteristic length depending on the shape of the interaction potential (in particular, for the hard spheres a=b). All the consideration concerns the zero temperature.
The simple relationship between the wavefunctions of a system of impenetrable one-dimensional bosons and impenetrable one-dimensional fermions is exploited to derive an expansion of the boson reduced density matrices in terms of the fermion reduced density matrices and vice versa. This expansion is independent of the statistical ensemble used, and of the interparticle potential (subject to the impenetrability condition). The special case of zero impenetrability radius with no other forces is treated in detail, using the grand canonical ensemble. This leads to a natural generalization of a formula previously found for zero temperature only, for the one-particle reduced density matrix.
We present a systematic study of the Green functions of a one-dimensional gas of impenetrable anyons. We show that the one-particle density matrix is the determinant of a Toeplitz matrix whose large N asymptotic is given by the Fisher–Hartwig conjecture. We provide a careful numerical analysis of this determinant for general values of the anyonic parameter, showing in full details the crossover between bosons and fermions and the reorganization of the singularities of the momentum distribution function. We show that the one-particle density matrix satisfies a Painlevé VI differential equation that is then used to derive the small distance and large momentum expansions. We find that the first non-vanishing term in this expansion is always k −4 , that is proved to be true for all couplings in the Lieb–Liniger anyonic gas and that can be traced back to the presence of a delta function interaction in the Hamiltonian.