A systematic $1/c$-expansion of form factor sums for dynamical correlations in the Lieb-Liniger model
Etienne Granet, Fabian Essler
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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>/</mml:mi> <mml:mi>c</mml:mi> </mml:mrow> </mml:math> expansion. The n^\mathrm{th} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>n</mml:mi> <mml:mstyle mathvariant="normal"> <mml:mi>t</mml:mi> <mml:mi>h</mml:mi> </mml:mstyle> </mml:msup> </mml:math> term of the expansion is of order 1/c^n <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>/</mml:mi> <mml:msup> <mml:mi>c</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> and takes into account all \lfloor \tfrac{n}{2}\rfloor+1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">⌊</mml:mo> <mml:mstyle displaystyle="false"> <mml:mfrac> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mstyle> <mml:mo stretchy="false" form="postfix">⌋</mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> particle-hole excitations over the averaging eigenstate. Importantly, in contrast to a "bare" 1/c <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>/</mml:mi> <mml:mi>c</mml:mi> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>/</mml:mi> <mml:msup> <mml:mi>c</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> . 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.