The Functional Integration and the Two-Point Correlation Function of the One-Dimensional Bose Gas in the Harmonic Potential

TL;DR

This paper develops a quantum field-theoretical model for a one-dimensional Bose gas in a harmonic trap, calculating the two-point correlation function and analyzing its asymptotic behavior at low temperatures.

Contribution

It introduces a novel functional integration approach to compute the correlation function of a non-homogeneous Bose gas in a harmonic potential, including one-loop approximation and asymptotic analysis.

Findings

Correlation function exhibits power-law decay at zero temperature.

The critical exponent depends on spatial and thermal variables.

Method provides a new way to analyze non-homogeneous quantum gases.

Abstract

A quantum field-theoretical model which describes spatially non-homogeneous one-dimensional non-relativistic repulsive Bose gas in an external harmonic potential is considered. We calculate the two-point thermal correlation function of the Bose gas in the framework of the functional integration approach. The calculations are done in the coordinate representation. A method of successive integration over the ``high-energy'' functional variables first and then over the ``low-energy'' ones is used. The effective action functional for the low-energy variables is calculated in one loop approximation. The functional integral representation for the correlation function is obtained in terms of the low-energy variables, and is estimated by means of the stationary phase approximation. The asymptotics of the correlation function is studied in the limit when the temperature is going to zero while the volume occupied by non-homogeneous Bose gas infinitely increases. It is demonstrated that the behaviour of the thermal correlation function in the limit described is power-like, and it is governed by the critical exponent which depends on the spatial and thermal arguments.