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The sine-Gordon model is a paradigmatic quantum field theory that providesthe low-energy effective description of many gapped 1D systems. Despite thisfact, its complete thermodynamic description in all its regimes has beenlacking. Here we fill this gap and derive the framework that captures itsthermodynamics and serves as the basis of its hydrodynamic description. As afirst application, we compute the Drude weight characterising the ballistictransport of topological charge and demonstrate that its dependence on thevalue of the coupling shows a fractal structure, similar to the gapless phaseof the XXZ spin chain. The thermodynamic framework can be applied to studyother features of non-equilibrium dynamics in the sine-Gordon model usinggeneralised hydrodynamics, opening the way to a wide array of theoreticalstudies and potential novel experimental predictions.

Thermodynamics and fractal Drude weights in the sine-Gordon model