From the sinh-Gordon field theory to the one-dimensional Bose gas: exact local correlations and full counting statistics

TL;DR

This paper derives exact formulas for local correlations and full counting statistics in the Lieb-Liniger model, connecting integrable quantum field theory with cold atom experiments and non-equilibrium dynamics.

Contribution

It provides explicit analytic expressions for K-body correlation functions in the Lieb-Liniger gas for all states, including non-equilibrium and thermal states, using a novel connection to sinh-Gordon theory.

Findings

Exact formulas for local K-body correlations in Lieb-Liniger model.

Calculation of full counting statistics for particle-number fluctuations.

Application of generalized hydrodynamics to non-homogeneous settings.

Abstract

We derive exact formulas for the expectation value of local observables in a one-dimensional gas of bosons with point-wise repulsive interactions (Lieb-Liniger model). Starting from a recently conjectured expression for the expectation value of vertex operators in the sinh-Gordon field theory, we derive explicit analytic expressions for the one-point $K$-body correlation functions $\langle (\Psi^\dagger)^K(\Psi)^K\rangle$ in the Lieb-Liniger gas, for arbitrary integer $K$. These are valid for all excited states in the thermodynamic limit, including thermal states, generalized Gibbs ensembles and non-equilibrium steady states arising in transport settings. Our formulas display several physically interesting applications: most prominently, they allow us to compute the full counting statistics for the particle-number fluctuations in a short interval. Furthermore, combining our findings with the recently introduced generalized hydrodynamics, we are able to study multi-point correlation functions at the Eulerian scale in non-homogeneous settings. Our results complement previous studies in the literature and provide a full solution to the problem of computing one-point functions in the Lieb Liniger model.