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arXiv:2312.03909 · SciPost Physics

Thermodynamic Bethe Ansatz and generalised hydrodynamics in the sine-Gordon model

Botond C. Nagy, G. Takács, Márton Kormos

Published Jun 4, 2024• 7 citations
arXivHTMLPDF

Abstract

We set up a hydrodynamic description of the non-equilibrium dynamics of sine-Gordon quantum field theory for generic coupling. It is built upon an explicit form of the Bethe Ansatz description of general thermodynamic states, with the structure of the resulting coupled integral equations encoded in terms of graphical diagrams. The resulting framework is applied to derive results for the Drude weights of charge and energy. Quantities associated with the charge universally have fractal dependence on the coupling, which is notably absent from those associated with energy, a feature explained by the different roles played by reflective scattering in transporting these quantities. We then present far-from-equilibrium results, including explicit time evolution starting from bipartite initial states and dynamical correlation functions. Our framework can be applied to explore numerous other aspects of non-equilibrium dynamics, which opens the way to a wide array of theoretical studies and potential novel experimental predictions.

DOI
10.21468/scipostphys.16.6.145

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We set up a hydrodynamic description of the non-equilibrium dynamics of sine-Gordon quantum field theory for generic coupling. It is built upon an explicit form of the Bethe Ansatz description of general thermodynamic states, with the structure of the resulting coupled integral equations encoded in terms of graphical diagrams. The resulting framework is applied to derive results for the Drude weights of charge and energy. Quantities associated with the charge universally have fractal dependence on the coupling, which is notably absent from those associated with energy, a feature explained by the different roles played by reflective scattering in transporting these quantities. We then present far-from-equilibrium results, including explicit time evolution starting from bipartite initial states and dynamical correlation functions. Our framework can be applied to explore numerous other aspects of non-equilibrium dynamics, which opens the way to a wide array of theoretical studies and potential novel experimental predictions.
The sine-Gordon model is a paradigmatic quantum field theory that provides the low-energy effective description of many gapped one-dimensional systems. Despite this fact, its complete thermodynamic description in all its regimes has been lacking. Here, we fill this gap and derive the framework that captures its thermodynamics and serves as the basis of its hydrodynamic description. As a first application, we compute the Drude weight characterizing the ballistic transport of topological charge and demonstrate that its dependence on the value of the coupling shows a fractal structure, similar to the gapless phase of the XXZ spin chain. The thermodynamic framework can be applied to study other features of nonequilibrium dynamics in the sine-Gordon model using generalized hydrodynamics, opening the way to a wide array of theoretical studies and potential novel experimental predictions.
We evaluate the exact energy current and scaled cumulant generating function (related to the large-deviation function) in non-equilibrium steady states with energy flow, in any integrable model of relativistic quantum field theory (IQFT) with diagonal scattering. Our derivations are based on various recent results of D. Bernard and B. Doyon. The steady states are built by connecting homogeneously two infinite halves of the system thermalized at different temperatures Tl, Tr, and waiting for a long time. We evaluate the current J(Tl, Tr) using the exact QFT density matrix describing these non-equilibrium steady states and using Al.B. Zamolodchikov’s method of the thermodynamic Bethe ansatz (TBA). The scaled cumulant generating function is obtained from the extended fluctuation relations which hold in integrable models. We verify our formula in particular by showing that the conformal field theory (CFT) result is obtained in the high-temperature limit. We analyze numerically our non-equilibrium steady-state TBA equations for three models: the sinh-Gordon model, the roaming trajectories model, and the sine-Gordon model at a particular reflectionless point. Based on the numerics, we conjecture that an infinite family of non-equilibrium c-functions, associated to the scaled cumulants, can be defined, which we interpret physically. We study the full scaled distribution function and find that it can be described by a set of independent Poisson processes. Finally, we show that the “additivity ” property of the current, which is known to hold in CFT and was proposed to hold more generally, does not hold in general IQFT, that is J(Tl, Tr) is not of the form f(Tl)−f(Tr).
We evaluate the exact energy current and scaled cumulant generating function (related to the large-deviation function) in non-equilibrium steady states with energy flow, in any integrable model of relativistic quantum field theory (IQFT) with diagonal scattering. Our derivations are based on various recent results of D. Bernard and B. Doyon. The steady states are built by connecting homogeneously two infinite halves of the system thermalized at different temperatures TlT_lTl​, TrT_rTr​, and waiting for a long time. We evaluate the current J(Tl,Tr)J(T_l,T_r)J(Tl​,Tr​) using the exact QFT density matrix describing these non-equilibrium steady states and using Al.B. Zamolodchikov's method of the thermodynamic Bethe ansatz (TBA). The scaled cumulant generating function is obtained from the extended fluctuation relations which hold in integrable models. We verify our formula in particular by showing that the conformal field theory (CFT) result is obtained in the high-temperature limit. We analyze numerically our non-equilibrium steady-state TBA equations for three models: the sinh-Gordon model, the roaming trajectories model, and the sine-Gordon model at a particular reflectionless point. Based on the numerics, we conjecture that an infinite family of non-equilibrium ccc-functions, associated to the scaled cumulants, can be defined, which we interpret physically. We study the full scaled distribution function and find that it can be described by a set of independent Poisson processes. Finally, we show that the "additivity" property of the current, which is known to hold in CFT and was proposed to hold more generally, does not hold in general IQFT, that is J(Tl,Tr)J(T_l,T_r)J(Tl​,Tr​) is not of the form f(Tl)−f(Tr)f(T_l)-f(T_r)f(Tl​)−f(Tr​).
We conduct a comprehensive study of anomalous charge transport in the quantum sine-Gordon model. Employing the framework of generalized hydrodynamics, we compute Drude weights and Onsager matrices across a wide range of coupling strengths to quantify ballistic and diffusive transport, respectively. We find that charge transport is predominantly diffusive at accessible timescales, indicated by the corresponding Onsager matrix significantly exceeding the Drude weight -- contrary to most integrable models where transport is primarily ballistic. Reducing the Onsager matrix to a few key two-particle scattering processes enables us to efficiently examine transport in both low- and high-temperature limits. The charge transport is dictated by nondiagonal scattering of the internal charge degree of freedom: at particular values of the coupling strength with diagonal, diffusive effects amount to merely subleading corrections. However, at couplings approaching these points, the charge Onsager matrix and corresponding diffusive timescale diverge. Our findings relate to similar transport anomalies in XXZ spin chains, offering insights through their shared Bethe ansatz structures.
The sine-Gordon model captures the low-energy effective dynamics of a wealth of one-dimensional quantum systems, stimulating the experimental efforts in building a versatile quantum simulator of this field theory and fueling the parallel development of new theoretical toolkits able to capture far-from-equilibrium settings. In this work, we analyze the realization of the sine-Gordon model from the interference pattern of two one-dimensional quasicondensates: we argue that the emergent field theory is well described by its classical limit, and we develop its large-scale description based on generalized hydrodynamics. We show how, despite the sine-Gordon model being an integrable field theory, trap-induced inhomogeneities cause instabilities of excitations and provide exact analytical results to capture this effect.
The Bethe-ansatz equations for the thermodynamic properties of the quantum sine-Gordon systems are derived in the zero-charge-sector attractive case. For rational values of the coupling parameter $\frac{\ensuremath{\mu}}{\ensuremath{\pi}}$ these reduce to a finite set, solved here numerically for $\ensuremath{\mu}=[\frac{(n\ensuremath{-}1)}{n}]\ensuremath{\pi}$, for several values of nnn, to give the specific heat as a function of temperature. The "soliton" contribution peaks at $\ensuremath{\simeq}0.4$ soliton masses for $\ensuremath{\mu}=\frac{4}{5}\ensuremath{\pi}$, shifting downward for higher $\ensuremath{\mu}$. A detailed analysis of the sine-Gordon limit of the XYZ\mathrm{XYZ}XYZ spin chain is presented, and a non-Lorentz-invariant feature of that limit is noted.