[Paper Review] Interaction quenches in the Lieb-Liniger model

Márton Kormos, Aditya Shashi+2 more|arXiv (Cornell University)|Apr 17, 2012

Theoretical and Computational Physics1 references3 citations

TL;DR

This paper studies interaction quenches in the integrable 1D Lieb-Liniger model using the generalized Gibbs ensemble (GGE) combined with Bethe Ansatz techniques. It shows that infinitely many conserved charges generate non-analytic behavior in the quasimomenta distribution, leading to interaction-dependent Friedel-like oscillations in non-local correlation functions—distinct from thermal predictions and providing a testable signature for ultracold atom experiments.

ABSTRACT

We obtain exact results on interaction quenches in the 1D Bose gas described by the integrable Lieb-Liniger model. We show that in the long time limit integrability leads to significant deviations from the predictions of the grand canonical ensemble and a description within the generalized Gibbs ensemble (GGE) is needed. For a non-interacting initial state and arbitrary final interactions, we find that the presence of infinitely many conserved charges generates a non-analytic behavior in the equilibrated density of quasimomenta. This manifests itself in a dynamically generated Friedel-like oscillation of the non-local correlation functions with interaction dependent oscillation momenta. We also exactly evaluate local correlations and the generalized chemical potentials within GGE.

Motivation & Objective

• To understand the long-time behavior of isolated quantum systems after an interaction quench, particularly in integrable models like the Lieb-Liniger model.
• To determine whether conventional thermal ensembles like the grand canonical ensemble (GCE) describe the steady state, or if a generalized Gibbs ensemble (GGE) is required.
• To compute exact expressions for local correlations and generalized chemical potentials in the GGE framework for arbitrary final interaction strengths.
• To identify experimentally observable signatures of integrability in ultracold atomic systems, such as non-analytic behavior and oscillatory correlation functions.

Proposed method

• Combines the Bethe Ansatz solution of the Lieb-Liniger model with the generalized Gibbs ensemble (GGE) to describe non-equilibrium steady states after an interaction quench.
• Uses the exact spectrum and conserved charges $ Q_m = \sum_j \lambda_j^m $, where $ \lambda_j $ are quasimomenta satisfying the Lieb-Liniger algebraic equations.
• Derives the filling function $ f(\lambda) $ from the solution of the problem of moments, using the relation $ \varepsilon(\lambda) = \log[1/f(\lambda) - 1] $, and solves the nonlinear integral equation for $ \varepsilon(\lambda) $.
• Evaluates non-local correlation functions via form factor expansions and determinant representations, including the $ G(x) $ function for two-point functions.
• Computes generalized chemical potentials $ \beta_{2m} $ by expanding the GGE equation in powers of $ \lambda/\lambda_* $, revealing algebraic decay for large $ m $.
• Uses the semicircle distribution of quasimomenta to derive analytical approximations for correlation functions, fitting them to Bessel functions with exponential decay.

Experimental results

Research questions

• RQ1 Does the generalized Gibbs ensemble (GGE) accurately describe the long-time steady state after an interaction quench in the Lieb-Liniger model, or does the grand canonical ensemble fail due to integrability?
• RQ2 How do infinitely many conserved charges manifest in the non-local correlation functions, and what is the origin of the observed Friedel-like oscillations?
• RQ3 What is the functional form and physical origin of the non-analytic behavior in the quasimomenta distribution at long times?
• RQ4 How do the generalized chemical potentials $ \beta_{2m} $ behave for large $ m $, and what does this imply for the structure of the GGE?
• RQ5 Can the GGE description be used to compute exact local and non-local correlation functions, and how do they differ from thermal predictions?

Key findings

• The long-time steady state after an interaction quench exhibits Friedel-like oscillations in non-local correlation functions, with oscillation momentum $ \lambda_* $ dependent on the final interaction strength $ \gamma $, in stark contrast to the exponential decay in thermal states.
• The density of quasimomenta develops a non-analytic behavior due to the collective effect of infinitely many conserved charges, leading to a 'Fermi edge' in the momentum distribution.
• The two-point correlation function decays as $ \sim 1/x^3 $ with oscillations $ \sim J_1^2(\lambda_* x)/(\lambda_* x)^2 $, indicating power-law decay, while the thermal state exhibits exponential decay.
• The generalized chemical potentials $ \beta_{2m} $ decay algebraically as $ \beta_{2m} \approx 1/(2m \lambda_*^{2m}) = 1/(2m (4n^2\gamma)^m) $ for large $ m $, with no divergence but strong collective effects.
• For large $ \gamma $, the density-density correlation function is approximated by $ g_2(x) \approx 1 - 2n J_1^2(\lambda_* x)/(\lambda_* x)^2 $, confirming oscillatory, power-law decay.
• The inverse temperature $ n^2 \beta_2 $ scales as $ \approx 1/(8\gamma) $ in the $ \gamma \gg 1 $ limit, deviating from the grand canonical prediction and confirming GGE necessity.