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