Conformal field theory on top of a breathing one-dimensional gas of hard core bosons

TL;DR

This paper extends conformal field theory methods to analyze the dynamics of a one-dimensional gas of hard core bosons in a time-dependent harmonic trap, providing new formulas for correlation functions and validating them numerically.

Contribution

It introduces a novel approach to compute correlation functions in inhomogeneous, dynamical quantum systems, specifically for a breathing gas in a time-dependent trap, with exact formulas and numerical validation.

Findings

Derived new formulas for large-scale correlation functions.

Validated approach with numerical checks on fermionic two-point functions.

Extended previous static results to dynamical, inhomogeneous systems.

Abstract

The recent results of [J. Dubail, J.-M. St\'ephan, J. Viti, P. Calabrese, Scipost Phys. 2, 002 (2017)], which aim at providing access to large scale correlation functions of inhomogeneous critical one-dimensional quantum systems -- e.g. a gas of hard core bosons in a trapping potential -- are extended to a dynamical situation: a breathing gas in a time-dependent harmonic trap. Hard core bosons in a time-dependent harmonic potential are well known to be exactly solvable, and can thus be used as a benchmark for the approach. An extensive discussion of the approach and of its relation with classical and quantum hydrodynamics in one dimension is given, and new formulas for correlation functions, not easily obtainable by other methods, are derived. In particular, a remarkable formula for the large scale asymptotics of the bosonic $n$-particle function $\left< \Psi^\dagger (x_1,t_1) \dots \Psi^\dagger (x_n,t_n) \Psi(x_1',t_1') \dots \Psi(x_n',t_n') \right>$ is obtained. Numerical checks of the approach are carried out for the fermionic two-point function -- easier to access numerically in the microscopic model than the bosonic one -- with perfect agreement.