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Journal of Mathematical Physics

One particle reduced density matrix of impenetrable bosons in one dimension at zero temperature

H. G. Vaidya, Craig A. Tracy

Published Nov 1, 1979• 95 citations
Publisher

Abstract

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).

DOI
10.1063/1.524010

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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).
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.
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.
This work presents the derivation of the large time and distance asymptotic behavior of the field-field correlation functions of impenetrable one-dimensional anyons at finite temperature. In the appropriate limits of the statistics parameter, we recover the well-known results for impenetrable bosons and free fermions. In the low-temperature (usually expected to be the "conformal") limit, and for all values of the statistics parameter away from the bosonic point, the leading term in the correlator does not agree with the prediction of the conformal field theory, and is determined by the singularity of the density of the single-particle states at the bottom of the single-particle energy spectrum.
We consider off-diagonal correlation functions of impenetrable bosons on a lattice. By using the Jordan–Wigner transformation the one-body density matrix is represented as a (Toeplitz) determinant of a matrix of fermionic Green functions. Using the replica method we calculate exactly the full long-range asymptotic behaviour of the one-body density matrix. We discuss how subleading oscillating terms, originating from short-range correlations, give rise to interesting features in the momentum distribution.