Dynamical correlation functions for an impenetrable Bose gas with Neumann or Dirichlet boundary conditions

TL;DR

This paper derives Fredholm determinant formulas for time- and temperature-dependent correlation functions of an impenetrable Bose gas with boundary conditions, using Bethe Ansatz and nonlinear PDEs, extending previous ground state results.

Contribution

It introduces new Fredholm determinant formulas for dynamic correlations at finite temperature and time, generalizing earlier ground state formulas for boundary conditions.

Findings

Fredholm determinant formulas for correlation functions at finite temperature and time.

Expression of certain correlation functions in terms of nonlinear PDE solutions.

Extension of ground state correlation formulas to dynamic, finite-temperature cases.

Abstract

We study the time and temperature dependent correlation functions for an impenetrable Bose gas with Neumann or Dirichlet boundary conditions $\langle \psi(x_1,0)\psi^\dagger(x_2,t)\rangle _{\pm,T}$. We derive the Fredholm determinant formulae for the correlation functions, by means of the Bethe Ansatz. For the special case $x_1=0$, we express correlation functions with Neumann boundary conditions $\langle\psi(0,0)\psi^\dagger(x_2,t)\rangle _{+,T}$, in terms of solutions of nonlinear partial differential equations which were introduced in \cite{kojima:Sl} as a generalization of the nonlinear Schr\"odinger equations. We generalize the Fredholm minor determinant formulae of ground state correlation functions $\langle\psi(x_1)\psi^\dagger(x_2)\rangle _{\pm,0}$ in \cite{kojima:K}, to the Fredholm determinant formulae for the time and temperature dependent correlation functions $\langle\psi(x_1,0)\psi^\dagger(x_2,t)\rangle _{\pm,T}$, $t \in {\bf R}$, $T \geq 0$.