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

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.

Systematic strong coupling expansion for out-of-equilibrium dynamics in the Lieb-Liniger model

“…We note that we use a different notation for the time difference τ to avoid confusion with the time t according to which the system evolves after the quench. The analogous two-point function for the density operator was derived in [64] up to order 1/c 2 .…”

“…Our aim is to determine the full time evolution of a number of different observables after the quench in the framework of the systematic 1/c-expansion developed in [64]. We have considered the following one and two-point functions:…”

“…In this work we focus on the second of these problems, namely how to extract the time dependence of local observables after a quantum quench from the spectral representation. We consider the case of a quantum quench to the repulsive Lieb-Liniger model, and bring to bear strong-coupling expansion methods we recently developed in the context of equilibrium response functions [64].…”

Effective free-fermionic form factors and the XY spin chain

“…However, the complete understanding has not been achieved and recently the QTM methods were revisited to address correlation function for the XX spin-chain [38][39][40]. Moreover, new systematic approaches for correlation functions in the Ising model for low density [41] and in the Lieb-Liniger model for the strong-coupling expansions [42] have been proposed. The generalization of Smirnov's form factor axioms for the thermodynamic states has been formulated in Ref.…”

“…As we have mentioned above, this complexity is combinatorial in nature and reflects the fact that each form factor for the thermal states is exponentially small so the number of relevant form factors is exponentially large. This makes direct computation of the corresponding sum for the correlation functions notoriously difficult and force researchers to focus at most on the two particle-hole excitations [34][35][36][37], consider semiclassical approximations [46] or develop other approximation schemes [41,42].…”