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Physical Review Letters

One-Particle Reduced Density Matrix of Impenetrable Bosons in One Dimension at Zero Temperature

H. G. Vaidya, Craig A. Tracy

Published Jan 1, 1979• 132 citations
Publisher

Abstract

We report the large-$\mathcal{r}$ and small-$\mathcal{r}$ expansions of the one-particle reduced density $\stackrel{\mathrm{`}}{\mathrm{m}}$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-$\mathcal{r}$ expansion of $\ensuremath{\rho}(\mathcal{r})$ contains oscillatory terms which we relate to the analytic structure of the momentum density function $n(k)$.

DOI
10.1103/physrevlett.42.3

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One-Particle Reduced Density Matrix of Impenetrable Bosons in One Dimension at Zero Temperature.

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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 \ensuremathρ(r)\ensuremath{\rho}(\mathcal{r})\ensuremathρ(r) of a system of impenetrable bosons in one dimension at zero temperature. These expansions were derived from an exact calculation of \ensuremathρ(r)\ensuremath{\rho}(\mathcal{r})\ensuremathρ(r) to be reported elsewhere. We find that the large-r\mathcal{r}r expansion of \ensuremathρ(r)\ensuremath{\rho}(\mathcal{r})\ensuremathρ(r) contains oscillatory terms which we relate to the analytic structure of the momentum density function n(k)n(k)n(k).
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).
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.
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 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.