Rigorous investigation of the reduced density matrix for the ideal Bose gas in harmonic traps by a loop-gas-like approach

TL;DR

This paper rigorously analyzes the reduced density matrix of the ideal Bose gas in harmonic traps using a sum-decomposition method, extending the loop-gas approach to anisotropic traps and exploring generalized Bose-Einstein condensation.

Contribution

It introduces a novel sum-decomposition method for the RDM applicable to anisotropic traps, extending the loop-gas approach to more general trap geometries.

Findings

Identifies additional local density contributions from mesoscopic loops in anisotropic traps.

Connects the RDM behavior to generalized Bose-Einstein condensation.

Provides insights to guide numerical PIMC simulations for anisotropic systems.

Abstract

In this paper, we rigorously investigate the reduced density matrix (RDM) associated to the ideal Bose gas in harmonic traps. We present a method based on a sum-decomposition of the RDM allowing to treat not only the isotropic trap, but also general anisotropic traps. When focusing on the isotropic trap, the method is analogous to the loop-gas approach developed by W.J. Mullin in [38]. Turning to the case of anisotropic traps, we examine the RDM for some anisotropic trap models corresponding to some quasi-1D and quasi-2D regimes. For such models, we bring out an additional contribution in the local density of particles which arises from the mesoscopic loops. The close connection with the occurrence of generalized-BEC is discussed. Our loop-gas-like approach provides relevant information which can help guide numerical investigations on highly anisotropic systems based on the Path Integral Monte Carlo (PIMC) method.