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arXiv:1608.07548 · Journal of Statistical Mechanics Theory and Experiment

Non relativistic limit of integrable QFT and Lieb–Liniger models

Alvise Bastianello, Andrea De Luca, Giuseppe Mussardo

Published Dec 1, 2016• 19 citations
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Abstract

Abstract In this paper we study a suitable limit of integrable QFT with the aim to identify continuous non-relativistic integrable models with local interactions. This limit amounts to sending to infinity the speed of light c but simultaneously adjusting the coupling constant g of the quantum field theories in such a way to keep finite the energies of the various excitations. The QFT considered here are Toda field theories and the O ( N ) non-linear sigma model. In both cases the resulting non-relativistic integrable models consist only of Lieb–Liniger models, which are fully decoupled for the Toda theories while symmetrically coupled for the O ( N ) model. These examples provide explicit evidence of the universality and ubiquity of the Lieb–Liniger models and, at the same time, suggest that these models may exhaust the list of possible non-relativistic integrable theories of bosonic particles with local interactions.

DOI
10.1088/1742-5468/aa4f98

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Abstract In this paper we study a suitable limit of integrable QFT with the aim to identify continuous non-relativistic integrable models with local interactions. This limit amounts to sending to infinity the speed of light c but simultaneously adjusting the coupling constant g of the quantum field theories in such a way to keep finite the energies of the various excitations. The QFT considered here are Toda field theories and the O ( N ) non-linear sigma model. In both cases the resulting non-relativistic integrable models consist only of Lieb–Liniger models, which are fully decoupled for the Toda theories while symmetrically coupled for the O ( N ) model. These examples provide explicit evidence of the universality and ubiquity of the Lieb–Liniger models and, at the same time, suggest that these models may exhaust the list of possible non-relativistic integrable theories of bosonic particles with local interactions.
Abstract The aim of this paper is to investigate the non-relativistic limit of integrable quantum field theories with fermionic fields, such as the O ( N ) Gross–Neveu model, the supersymmetric Sinh–Gordon and non-linear sigma models. The non-relativistic limit of these theories is implemented by a double scaling limit which consists of sending the speed of light c to infinity and rescaling at the same time the relevant coupling constant of the model in such a way to have finite energy excitations. For the general purpose of mapping the space of continuous non-relativistic integrable models, this paper completes and integrates the analysis done in Bastianello A et al (2016 J. Stat. Mech . 123104) on the non-relativistic limit of purely bosonic theories.
We consider the one-dimensional Lieb-Liniger model (bosons interacting via 2-body delta potentials) in the infinite coupling constant limit (the so-called Tonks-Girardeau model). This model might be relevant as a description of atomic Bose gases confined in a one-dimensional geometry. It is known to have a fermionic spectrum since the N-body wavefunctions have to vanish at coinciding points, and therefore be symmetrizations of fermionic Slater wavefunctions. We argue that in the infinite coupling constant limit the model is indistinguishable from free fermions, i.e., all physically accessible observables are the same as those of free fermions. Therefore, Bose-Einstein condensate experiments at finite energy that preserve the one-dimensional geometry cannot test any bosonic characteristic of such a model.
We consider the one-dimensional Lieb-Liniger model (bosons interacting via 2-body delta potentials) in the infinite coupling constant limit (the so-called Tonks-Girardeau model). This model might be relevant as a description of atomic Bose gases confined in a one-dimensional geometry. It is known to have a fermionic spectrum since the N-body wavefunctions have to vanish at coinciding points, and therefore be symmetrizations of fermionic Slater wavefunctions. We argue that in the infinite coupling constant limit the model is indistinguishable from free fermions, i.e., all physically accessible observables are the same as those of free fermions. Therefore, Bose-Einstein condensate experiments at finite energy that preserve the one-dimensional geometry cannot test any bosonic characteristic of such a model.
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.