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/c1/c expansion. The n^\mathrm{th}nth term of the expansion is of order 1/c^n1/cn and takes into account all \lfloor \tfrac{n}{2}\rfloor+1⌊n2⌋+1 particle-hole excitations over the averaging eigenstate. Importantly, in contrast to a "bare" 1/c1/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^21/c2. 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.
Highlights
In contrast to a "bare" 1/c expansion it is uniform in space and time
In Refs [73,75,76] such an expansion in terms of thermodynamic particle-hole excitations was conjectured. It is based an phenomenological assumptions on how partial sums over states in the finite volume combine into thermodynamic form factors
In this work we have introduced and developed an ab initio expansion of dynamical densitydensity correlation functions in the Lieb-Liniger model that can be performed within any energy eigenstate
Summary
The Lieb-Liniger model [1] is a key paradigm of integrable many-particle systems [3]. In Refs [73,75,76] such an expansion in terms of thermodynamic particle-hole excitations was conjectured It is based an phenomenological assumptions on how partial sums over states in the finite volume combine into thermodynamic form factors. In particular we perform non-trivial consistency checks of our formulas, and recover known results from (nonlinear) Luttinger liquid theory and generalized hydrodynamics (GHD) [85, 86, 90]
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