Many-particle Systems in One Dimension in the Harmonic Approximation

TL;DR

This paper analyzes the energetics and structure of one-dimensional many-particle systems with contact interactions using a harmonic approximation, providing insights into ground state energies, thermodynamics, and comparisons with exact results.

Contribution

It introduces an analytical harmonic approximation approach to study one-dimensional many-particle systems, including bosons and fermions, and compares results with known exact solutions.

Findings

Ground state energies scale with particle number

Thermodynamics are similar for bosons and fermions in 1D

Harmonic approximation yields analytically solvable models

Abstract

We consider energetics and structural properties of a many particle system in one dimension with pairwise contact interactions confined in a parabolic external potential. To render the problem analytically solvable, we use the harmonic approximation scheme at the level of the Hamiltonian. We investigate the scaling with particle number of the ground state energies for systems consisting of identical bosons or fermions. We then proceed to focus on bosonic systems and make a detailed comparison to known exact results in the absence of the parabolic external trap for three-body systems. We also consider the thermodynamics of the harmonic model which turns out to be similar for bosons and fermions due to the lack of degeneracy in one dimension.