A systematic $1/c$-expansion of form factor sums for dynamical correlations in the Lieb-Liniger model

TL;DR

This paper develops a systematic $1/c$ expansion framework for calculating dynamical correlations in the Lieb-Liniger model, enabling uniform in space and time analysis of correlations in arbitrary eigenstates.

Contribution

It introduces a novel $1/c$ expansion method for form factor sums that is uniform in space and time, applicable to all local operators in the Lieb-Liniger model.

Findings

Derived explicit expressions for density-density correlations at order $1/c^2$.

Reproduced predictions of Luttinger liquid theory and generalized hydrodynamics.

Extended analysis to finite-temperature and non-equilibrium states.

Abstract

We introduce a framework for calculating dynamical correlations in the Lieb-Liniger model in arbitrary energy eigenstates and for all space and time, that combines a Lehmann representation with a $1/c$ expansion. The $n^{\rm th}$ term of the expansion is of order $1/c^n$ and takes into account all $\lfloor \tfrac{n}{2}\rfloor+1$ particle-hole excitations over the averaging eigenstate. Importantly, in contrast to a 'bare' $1/c$ expansion it is uniform in space and time. The framework is based on a method for taking the thermodynamic limit of sums of form factors that exhibit non integrable singularities. We expect our framework to be applicable to any local operator. We determine the first three terms of this expansion and obtain an explicit expression for the density-density dynamical correlations and the dynamical structure factor at order $1/c^2$. We apply these to finite-temperature equilibrium states and non-equilibrium steady states after quantum quenches. We recover predictions of (nonlinear) Luttinger liquid theory and generalized hydrodynamics in the appropriate limits, and are able to compute sub-leading corrections to these.