Non-Equilibrium Steady State of the Lieb-Liniger model: exact treatment of the Tonks Girardeau limit

Sotiriadis, Spyros

doi:10.48550/arxiv.2007.12683

Cited by 6 publications

(21 citation statements)

References 75 publications

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“…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.…”

Section: Introductionmentioning

confidence: 96%

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

Pozsgay,

Gombor,

Hutsalyuk

et al. 2021

Preprint

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We revisit the so-called folded XXZ model, which was treated earlier by two independent research groups. We argue that this spin-1/2 chain is one of the simplest quantum integrable models, yet it has quite remarkable physical properties. The particles have constant scattering lengths, which leads to a simple treatment of the exact spectrum and the dynamics of the system. The Hilbert space of the model is fragmented, leading to exponentially large degeneracies in the spectrum, such that the exponent depends on the particle content of a given state. We provide an alternative derivation of the Hamiltonian and the conserved charges of the model, including a new interpretation of the so-called "dual model" considered earlier. We also construct a non-local map that connects the model with the Maassarani-Mathieu spin chain, also known as the SU (3) XX model. We consider the exact solution of the model with periodic and open boundary conditions, and also derive multiple descriptions of the exact thermodynamics of the model. We consider quantum quenches of different types. In one class of problems the dynamics can be treated relatively easily: we compute an example for the real time dependence of a local observable. In another class of quenches the degeneracies of the model lead to the breakdown of equilibration, and we argue that they can lead to persistent oscillations. We also discuss connections with the T T and hard rod deformations known from Quantum Field Theories.

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Section: Introductionmentioning

confidence: 96%

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

Pozsgay,

Gombor,

Hutsalyuk

et al. 2021

Preprint

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show abstract

“…In an earlier work [1], we have studied the expansion of a Tonks-Girardeau gas from one half to the entire confining box and demonstrated how the emergence of Non-Equilibrium Steady States (NESS) can be derived from the asymptotics of the many-body wave-function in the combined thermodynamic and large distance and time limit. We avoided using the effectively free fermionic nature of the system, aiming to develop an exact method for the derivation of the asymptotics in the many-body context that would be suitable for generalisation to the genuinely interacting case of the Lieb-Liniger gas at arbitrary repulsive interaction.…”

mentioning

confidence: 99%

“…The above outlined method has not been applied to out-of-equilibrium problems so far. Here, our goal is to generalise the method in the form shown in [1] to the genuinely interacting case of the Lieb-Liniger model at arbitrary interaction c > 0. In particular, we focus on the time evolved many-body wavefunction after the quench and derive a multiple integral representation for it, thus accomplishing the first step of the above general approach.…”

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confidence: 99%

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

Sotiriadis¹

2020

Preprint

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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.

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“…(D1). Taking into account the series of residues at these poles we obtain As a consequence we obtain the expressions for the density (57) and the current (58) in the ray-regime at finite temperature.…”

Section: B3 Summarymentioning

confidence: 99%

“…In Section 5 we obtain the finite time kernel in the limit → +∞. To this aim we have adapted a trick based on contour integrals used in [50] (see also [57,58]). As we show in that section, the study of the limit → +∞ turns out to be more involved in the present continuum setting with a correlated initial condition.…”

Section: Introductionmentioning

confidence: 99%

Quench dynamics of noninteracting fermions with a delta impurity

Gouraud,

Doussal,

Schehr

2022

Preprint

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We study the out-of-equilibrium dynamics of noninteracting fermions in one dimension and in continuum space, in the presence of a delta impurity potential at the origin whose strength g is varied at time t = 0. The system is prepared in its ground state with g = g 0 = +∞, with two different densities and Fermi wave-vectors k L and k R on the two halfspaces x > 0 and x < 0 respectively. It then evolves for t > 0 as an isolated system, with a finite impurity strength g. We compute exactly the time dependent density and current. For a fixed position x and in the large time limit t → ∞, the system reaches a nonequilibrium stationary state (NESS). We obtain analytically the correlation kernel, density, particle current, and energy current in the NESS, and characterize their relaxation, which is algebraic in time. In particular, in the NESS, we show that, away from the impurity, the particle density displays oscillations which are the non-equilibrium analog of the Friedel oscillations. In the regime of "rays", x/t = ξ fixed with x, t → ∞, we compute the same quantities and observe the emergence of two light cones, associated to the Fermi velocities k L and k R in the initial state. Interestingly, we find non trivial quantum correlations between two opposite rays with velocities ξ and −ξ which we compute explicitly. We extend to a continuum setting and to a correlated initial state the analytical methods developed in a recent work of Ljubotina, Sotiriadis and Prosen, in the context of a discrete fermionic chain with an impurity. We also generalize our results to an initial state at finite temperature, recovering, via explicit calculations, some predictions of conformal field theory in the low energy limit.

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Non-Equilibrium Steady State of the Lieb-Liniger model: exact treatment of the Tonks Girardeau limit

Cited by 6 publications

References 75 publications

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

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

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

Quench dynamics of noninteracting fermions with a delta impurity

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