Ground-state Properties of Tonks-Girardeau Gas in One Dimensional Periodic Potential

TL;DR

This paper investigates the ground-state properties of the Tonks-Girardeau gas in a one-dimensional periodic potential, revealing how occupation numbers and momentum distributions behave with system size and potential commensurability.

Contribution

It provides a detailed analysis of how the reduced single-particle density matrix and natural orbitals depend on system size and potential type in the ground state.

Findings

In commensurate cases, the natural orbital and momentum distribution are invariant with N.

Off-diagonal elements decay exponentially with N in commensurate cases.

In incommensurate cases, off-diagonal elements decrease as 1/√N, affecting natural orbitals and momentum distributions.

Abstract

The relations among the occupation number of the lowest natural orbital (ONLNO), momentum distributions (MD) and off-diagonal long-range element (ODLRE) of the reduced single-particle density matrix (RSPDM) are studied while Tonks-Girardeau gas in one dimensional periodic potential is in the ground state. For $N$-body systems of large enough, RSPDM and its lowest natural orbital do not vary with $N$ in overlapped areas in commensurate and incommensurate cases correspondingly. In commensurate case, the ODLRE is exponential attenuation with $N$, which results in that the ONLNO and MD are invariant with $N$. While in contrast, in incommensurate case, the off-diagonal elements are inversely proportional to $\sqrt{N}$, which results in the different behavior of the ONLNO and MD.