Non-Equilibrium Steady State of the Lieb-Liniger model: multiple-integral representation of the time evolved many-body wave-function

Abstract

We continue our study of the emergence of Non-Equilibrium Steady States in quantum integrable models focusing on the expansion of a Lieb-Liniger gas for arbitrary repulsive interaction. As a first step towards the derivation of the asymptotics of observables in the thermodynamic and large distance and time limit, we derive an exact multiple integral representation of the time evolved manybody wave-function. Starting from the known but complicated expression for the overlaps of the initial state of a geometric quench, which are derived from the Slavnov formula for scalar products of Bethe states, we eliminate the awkward dependence on the system size and distinguish the Bethe states into convenient sectors. These steps allow us to express the rather impractical sum over Bethe states as a multiple rapidity integral in various alternative forms. Moreover, we examine the singularities of the obtained integrand and calculate the contribution of the multivariable kinematical poles, which is essential information for the derivation of the asymptotics of interest.

An integrable spin chain with Hilbert space fragmentation and solvable real time dynamics

“…This agrees with the scattering phase found earlier in (15). For the limit of the S-matrix factor between 1-strings and n-strings for n > 1 we find:…”

“…Some elements of the theory can be considered proven, for example key statements about the mean values of current operators (see the review [12]), but it would be desirable to rigorously prove more aspects of the theory. It was shown in the remarkable work [13] that certain statements of GGE and GHD can be checked in the Lieb-Liniger model in a large coupling expansion; up to date this result is one of the most convincing analytic checks of GGE and GHD (for closely related works see [10,11,[14][15][16]). Nevertheless there remains a need for simple toy models, which have genuine interactions in them, and which can lead to exact proofs of the GHD predictions.…”

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“…[96] in the context of a strong coupling expansion. Instead, only partial results [97,98] are currently available for inhomogeneous quench problems.…”

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