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EMS Press eBooks

Recent advances on the Lieb–Liniger δ\deltaδ-Bose gas

Herbert E. Spohn

Published Jun 29, 2022• 0 citations
Publisher

Abstract

From the perspectives of both experiment and theory, the Lieb–Liniger $\delta$-Bose gas remains of current interest. My contribution is focused mostly on generalized hydrodynamics of the model. While in the early days of the subject the relevance of generalized Gibbs ensembles (GGE) and their free energy was not noted, the tools for its computation were already available. For the hydrodynamic scale, in addition the GGE-averaged currents are required. To achieve such a goal, a surprisingly simple method will be explained.

DOI
10.4171/90-2/45

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From the perspectives of both experiment and theory, the Lieb–Liniger δ\deltaδ-Bose gas remains of current interest. My contribution is focused mostly on generalized hydrodynamics of the model. While in the early days of the subject the relevance of generalized Gibbs ensembles (GGE) and their free energy was not noted, the tools for its computation were already available. For the hydrodynamic scale, in addition the GGE-averaged currents are required. To achieve such a goal, a surprisingly simple method will be explained.
The results of benchmarks of the recent theory of generalized hydrodynamics (GHD) in the context of the dynamics of the 1D Bose gas described by the Lieb-Liniger model are reported. Through comparisons against several alternative, well-established computational approaches, the range of applicability of GHD in some of the most challenging out-of-equilibrium scenarios, such as propagation of quantum shock waves and collisional relaxation, is verified and validated and, additionally, some of the methods limitations are pointed out.
Abstract We review the recent theoretical and experimental progress regarding the generalized hydrodynamics (GHD) behavior of the one-dimensional (1D) Bose gas with contact repulsive interactions, also known as the Lieb–Liniger gas. In the first section, we review the theory of the Lieb–Liniger gas, introducing the key notions of the rapidities and of the rapidity distribution. The latter characterizes the Lieb–Liniger gas after relaxation and is at the heart of GHD. We also present the asymptotic regimes of the Lieb–Liniger gas with their dedicated approximate descriptions. In the second section we enter the core of the subject and review the theoretical results of GHD in 1D Bose gases. The third and fourth sections are dedicated to experimental results obtained in cold atom experiments: the experimental realization of the Lieb–Liniger model is presented in section 3, with a selection of key results for systems at equilibrium, and section 4 presents the experimental tests of the GHD theory. In section 5 we review the effects of atom losses, which, assuming slow loss processes, can be described within the GHD framework. We conclude with a few open questions.
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.
The nonequilibrium dynamics of integrable systems are highly constrained by the conservation of certain charges. There is substantial evidence that after a quantum quench they do not thermalize but their asymptotic steady state can be described by a generalized Gibbs ensemble (GGE) built from the conserved charges. Most of the studies on the GGE so far have focused on models that can be mapped to quadratic systems, while analytic treatment in nonquadratic systems remained elusive. We obtain results on interaction quenches in a nonquadratic continuum system, the one-dimensional (1D) Bose gas described by the integrable Lieb-Liniger model. The direct implementation of the GGE prescription is prohibited by the divergence of the conserved charges, which we conjecture to be endemic to any continuum integrable systems with contact interactions undergoing a sudden quench. We compute local correlators for a noninteracting initial state and arbitrary final interactions as well as two-point functions for quenches to the Tonks-Girardeau regime. We show that in the long time limit integrability leads to significant deviations from the predictions of the grand canonical ensemble, allowing for an experimental verification in cold-atom systems.