The fermionic limit of the delta-function Bose gas: a pseudopotential approach

TL;DR

This paper uses perturbation theory with a pseudopotential to analyze the density behavior of a one-dimensional delta-function Bose gas near the fermionic limit, confirming Tomonaga-Luttinger liquid predictions.

Contribution

It provides exact first-order density expressions for various boundary conditions using a pseudopotential, validating the Tomonaga-Luttinger liquid model near the fermionic limit.

Findings

Density exhibits power-law decay consistent with bosonization

Reflection occurs at points of discontinuity in pseudopotential strength

Transmission resonances are observed in finite-length regions

Abstract

We use first-order perturbation theory near the fermionic limit of the delta-function Bose gas in one dimension (i.e., a system of weakly interacting fermions) to study three situations of physical interest. The calculation is done using a pseudopotential which takes the form of a two-body delta''-function interaction. The three cases considered are the behavior of the system with a hard wall, with a point where the strength of the pseudopotential changes discontinuously, and with a region of finite length where the pseudopotential strength is non-zero (this is sometimes used as a model for a quantum wire). In all cases, we obtain exact expressions for the density to first order in the pseudopotential strength. The asymptotic behaviors of the densities are in agreement with the results obtained from bosonization for a Tomonaga-Luttinger liquid, namely, an interaction dependent power-law decay of the density far from the hard wall, a reflection from the point of discontinuity, and transmission resonances for the interacting region of finite length. Our results provide a non-trivial verification of the Tomonaga-Luttinger liquid description of the delta-function Bose gas near the fermionic limit.