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arXiv:1309.3471

Equilibration and Generalized GGE in the Lieb Liniger gas

Garry Goldstein, Natan Andrei

Published Sep 13, 2013• 10 citations
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Abstract

We study the nonequilibrium properties of the one dimensional Lieb Liniger model in the finite repulsion regime. Introducing a new version of the Yudson representation applicable to finite size systems and appropriately taking the infinite volume limit we are able to study equilibration of the Lieb Liniger gas in the thermodynamic limit. We provide a formalism to compute various correlation functions for highly non equilibrium initial states. We are able to find explicit analytic expressions for the long time limit of the expectation of the density, density density and related correlation functions. We show that the gas equilibriates to a diagonal ensemble which we show is equivalent to a generalized version of the GGE for sufficiently simple correlation functions, which in particular include density correlations.

DOI
10.48550/arxiv.1309.3471

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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.
In this work we study the properties of the density density correlation function of the 1-D Lieb-Liniger model with infinite repulsion in the GGE regime. The GGE describes the equilibrated system in the long time limit after a quench from a generic initial state. In the case that the initial and hence the final state has low entropy per particle we find that the density density correlation function has a universal form, in particular it depends on a few parameters corresponding to "key" momenta and has power law dependence on the distance. This provides an experimental signature of the GGE which may readily be identified through spectroscopy. These signatures are universal and robust to initial sate preparation.
This PhD thesis is devoted to the study of the long-time and large-distance asymptotic behaviour of the dynamical field--field correlation function of the Lieb--Liniger model in the limit of infinite repulsion (the one-dimensional impenetrable Bose gas). Extending previous works we consider a large class of non-thermal equilibrium conditions in addition to the case of thermal equilibrium that had been studied before. Our starting point is an exact representation of the field--field correlation function as the Fredholm determinant of an integrable integral operators. The integrable operator depends parametrically on time t, distance x and on the filling fraction, which is a function that characterizes the thermal or non-thermal equilibrium conditions. Our work is based on the rigorous asymptotic analysis, using Riemann--Hilbert techniques, of a more general integral operator that depends on two more functional parameters. The long-time and large-distance behaviour of the field--field correlation function is derived for two classes of filling fractions. These classes are characterized by the number of poles on the real axis (a generalization of Fermi points) that, together with the unique saddle point, contribute to the asymptotic expansion. For each class, we derive the long-time and large-distance asymptotic behaviour as a series in x^(-1/2) as x and t go to infinity for a fixed ratio x / t. We provide explicit closed-form expressions for the leading and sub-leading terms, logarithmic corrections, and overall constants in terms of special functions and simple integrals. For the impenetrable Bose gas in thermal equilibrium, we verify the derived asymptotic expansions by comparing them with the existing results in the literature and with numerical data. Our analysis is based solely on the analytic properties of the functions parametrizing the kernel of the integrable integral operator. This generalized approach will enable future applications to the long-time and large-distance asymptotic analysis of the Lieb--Liniger model with finite coupling constant.
The repulsive Lieb-Liniger model can be obtained as the nonrelativistic limit of the sinh-Gordon model: all physical quantities of the latter model (SSS-matrix, Lagrangian, and operators) can be put in correspondence with those of the former. We use this mapping, together with the thermodynamical Bethe ansatz equations and the exact form factors of the sinh-Gordon model, to set up a compact and general formalism for computing the expectation values of the Lieb-Liniger model both at zero and finite temperatures. The computation of one-point correlators is thoroughly detailed and when possible compared with known results in the literature.
We study the density-density correlation function of the 1D Lieb–Liniger model and obtain an exact expression for the small momentum limit of the static correlator in the thermodynamic limit. We achieve this by summing exactly over the relevant form factors of the density operator in the small momentum limit. The result is valid for any eigenstate, including thermal and non-thermal states. We also show that the small momentum limit of the dynamic structure factors obeys a generalized detailed balance relation valid for any equilibrium state.
We present a comprehensive review on the state-of-the-art of the approximate analytic approaches describing the finite-temperature thermodynamic quantities of the Lieb-Liniger model of the one-dimensional (1D) Bose gas with contact repulsive interactions. This paradigmatic model of quantum many-body-theory plays an important role in many areas of physics-thanks to its integrability and possible experimental realization using, e.g., ensembles of ultracold bosonic atoms confined to quasi-1D geometries. The thermodynamics of the uniform Lieb-Liniger gas can be obtained numerically using the exact thermal Bethe ansatz (TBA) method, first derived in 1969 by Yang and Yang. However, the TBA numerical calculations do not allow for the in-depth understanding of the underlying physical mechanisms that govern the thermodynamic behavior of the Lieb-Liniger gas at finite temperature. Our work is then motivated by the insights that emerge naturally from the transparency of closed-form analytic results, which are derived here in six different regimes of the gas and which exhibit an excellent agreement with the TBA numerics. Our findings can be further adopted for characterising the equilibrium properties of inhomogeneous (e.g., harmonically trapped) 1D Bose gases within the local density approximation and for the development of improved hydrodynamic theories, allowing for the calculation of breathing mode frequencies which depend on the underlying thermodynamic equation of state. Our analytic approaches can be applied to other systems including impurities in a quantum bath, liquid helium-4, and ultracold Bose gas mixtures.