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arXiv:2305.15474 · Physical review. B./Physical review. B

Thermodynamics and fractal Drude weights in the sine-Gordon model

Botond C. Nagy, Márton Kormos, G. Takács

Published Dec 12, 2023• 12 citations
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Abstract

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.

DOI
10.1103/physrevb.108.l241105

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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 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.
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 anomalous nature of spin transport in the XXZ quantum spin chain has been a topic of theoretical interest for some time. Here, the integrability of the underlying dynamics leads to a ballistic component of the spin current, characterized by a spin Drude weight, which measures the degree of divergence of the zero-frequency spin conductivity. However, this quantity had previously proven to be beyond the reach of standard Bethe ansatz techniques. Here, the authors show that a recently developed hydrodynamic formalism for quantum integrable models may be used to compute the spin Drude weight. They also propose a numerical scheme to obtain hydrodynamic predictions for finite-time energy transport. This suggests that the hydrodynamic approach captures completely the ballistic component that dominates transport at long times and distances in the gapless regime of the XXZ model.
Building on the Bethe ansatz and generalized hydrodynamics, we rigorously establish universal relations between Drude weights governing charge, spin, and energy transport and thermodynamic properties for the one-dimensional repulsive two-component continuum Fermi gas - the paradigmatic integrable Yang-Gaudin model. We analytically derive thermodynamic expressions for charge Drude weights: DnnD_{nn}Dnn​, DnmD_{nm}Dnm​, DneD_{ne}Dne​ (responses of particle, magnetization and energy currents to a chemical potential gradient) equal the particle density, magnetization, and temperature times entropy density, respectively. These relations constitute a hallmark of ballistic transport, and can be generalized to other 1D continuum integrable systems. Furthermore, we investigate spin Drude weight (DmmD_{mm}Dmm​, DmeD_{me}Dme​) at zero and low temperatures. They characterize the magnetization and energy current responses to a magnetic field gradient, revealing an essential spin-charge coupling feature in quantum transport. We find that the Drude weights DnmD_{nm}Dnm​, DmmD_{mm}Dmm​, and DmeD_{me}Dme​ across the μ-Hμ\text{-}Hμ-H plane remarkably map out the zero-temperature phase boundary. This work fills a critical gap in the understanding of transport within integrable quantum gases and furnishes a direct theoretical foundation for future ultracold-atom experiments.
In this paper we derive Klümper-Batchelor-Pearce-Destri-de Vega type nonlinear integral equations for describing the thermodynamics in the sine-Gordon model, when a chemical potential coupled to the topological charge is also present in the theory. The equations are valid at any value of the coupling constant and are particularly useful for computing the expectation values of local operators and some currents as functions of the temperature and the chemical potential. The equations are also extended to the case, when an extra term corresponding to the momentum conservation, is added to the thermodynamic potential. The benefits of this description are twofold. On the one hand, it can serve as an appropriate testing ground for other promising theoretical methods, like the method of random surfaces. On the other hand, the efficient computation of vertex operator expectation values, allows one to provide useful theoretical data, for the experimentally measurable coherence factors of a tunnel-coupled cold atomic condensate system.