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arXiv:2303.16932 · SciPost Physics

Exact thermodynamics and transport in the classical sine-Gordon model

Rebekka Koch, Alvise Bastianello

Published Oct 5, 2023• 26 citations
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Abstract

We revisit the exact thermodynamic description of the classical sine-Gordon field theory, a well-known integrable model. We found that existing results in the literature based on the soliton-gas picture did not correctly take into account light, but extended, solitons and thus led to incorrect results. This issue is regularized upon requantization: we derive the correct thermodynamics by taking the semiclassical limit of the quantum model. Our results are then extended to transport settings by means of Generalized Hydrodynamics.

DOI
10.21468/scipostphys.15.4.140

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We revisit the exact thermodynamic description of the classical sine-Gordon field theory, a well-known integrable model. We found that existing results in the literature based on the soliton-gas picture did not correctly take into account light, but extended, solitons and thus led to incorrect results. This issue is regularized upon requantization: we derive the correct thermodynamics by taking the semiclassical limit of the quantum model. Our results are then extended to transport settings by means of Generalized Hydrodynamics.
Thermodynamics of the Boltzmann gas consisting of solitons, antisolitons, and breathers of the classical sine-Gordon system is studied by taking their interactions into account. A compact expression for the thermodynamic potential is obtained, which agrees with that obtained from the classical limit of the Bethe-ansatz formulation of the quantum sine-Gordon thermodynamics. We further find that the average number of breathers excited at low temperatures (m\ensuremath{\ll}T\ensuremath{\ll}M, where m and M are the phonon and the soliton masses, respectively) is half the total number of degrees of freedom of the system.
This paper reviews soliton theories for classical statistical mechanics of the massive-Thirring-sine-Gordon model. These theories attempt not only to reproduce the classical transfer matrix method result but also to provide thermodynamic information about exact nonlinear excitations of the model. Three theories are discussed. Firstly, the soliton–gas phenomenology which is successful in the Toda lattice is shown to be not even approximately correct for the sine-Gordon model. This breakdown is due to the incorrect counting of the number of degrees of freedom. Secondly, the soliton–phonon theory which reproduces the WKB result is also shown to be not exact. This non-exactness is ascribed to the fact that the nonlinear phonons are not exact elementary excitations in the model. Thirdly, a non-gas phenomenological soliton–breather theory is derived as a classical limit of quantum statistical mechanics of the model. The exactness of the last theory is demonstrated through an accurate numerical analysis of its basic equations.
Using generalized hydrodynamics (GHD), we develop the Euler hydrodynamics of classical integrable field theory. Classical field GHD is based on a known formalism for Gibbs ensembles of classical fields, that resembles the thermodynamic Bethe ansatz of quantum models, which we extend to generalized Gibbs ensembles (GGEs). In general, GHD must take into account both solitonic and radiative modes of classical fields. We observe that the quasi-particle formulation of GHD remains valid for radiative modes, even though these do not display particle-like properties in their precise dynamics. We point out that because of a UV catastrophe similar to that of black body radiation, radiative modes suffer from divergences that restrict the set of finite-average observables; this set is larger for GGEs with higher conserved charges. We concentrate on the sinh-Gordon model, which only has radiative modes, and study transport in the domain-wall initial problem as well as Euler-scale correlations in GGEs. We confirm a variety of exact GHD predictions, including those coming from hydrodynamic projection theory, by comparing with Metropolis numerical evaluations.
We present the first exact theory and analytical formulas for the large-scale phase fluctuations in the sine-Gordon model, valid in all regimes of the field theory, for arbitrary temperatures and interaction strengths. Our result is based on the ballistic fluctuation theory combined with generalized hydrodynamics, and can be seen as an exact "dressing" of the phenomenological soliton-gas picture first introduced by Sachdev and Young [Phys. Rev. Lett. 78, 2220 (1997)PRLTAO0031-900710.1103/PhysRevLett.78.2220], to the modes of generalized hydrodynamics. The resulting physics of phase fluctuations in the sine-Gordon model is qualitatively different, as the stable quasiparticles of integrability give coherent ballistic propagation instead of diffusive spreading. We provide extensive numerical checks of our analytical predictions within the classical regime of the field theory by using Monte Carlo methods. We discuss how our results are of ready applicability to experiments on tunnel-coupled quasicondensates.