Gaussian Time-Dependent Variational Principle for Bosons Contact Interaction in one dimension

TL;DR

This paper applies a Gaussian variational approach to a one-dimensional Bosonic system with contact interactions, deriving dynamic equations, calculating ground state energy, and analyzing excitations, with comparisons to exact solutions and other methods.

Contribution

It introduces a Gaussian time-dependent variational method for 1D Bosons with contact interaction, providing dynamic equations and improved ground state estimates.

Findings

Derived non-linear time-dependent equations for variational parameters

Calculated ground state energy and compared with Lieb's exact results

Observed improved accuracy over Bogoliubov scheme as system becomes less dilute

Abstract

We investigate the Dirac time-dependent variational method using a Gaussian trial functional for an infinite one dimensional system of Bosons interacting through a repulsive contact interaction. The method produces a set of non-linear time dependent equations for the variational parameters. By solving the static equations we have calculated the ground state energy per particle. We have also considered small oscillations about equilibrium and obtain mode equations which lead us to a gapless dispersion relation. The existence of an exact numerical solution for the ground state energy and excitations obtained by Lieb allow us to compare with the Gaussian results. We can also, as the system becomes less dilute, see the improvement of the results as compared with the Bogoliubov scheme.