On Classical Integrability of the Hydrodynamics of Quantum Integrable Systems

Abstract

Recently, a hydrodynamic description of local equilibrium dynamics in quantum integrable systems was discovered. In the diffusionless limit, this is equivalent to a certain "Bethe-Boltzmann" kinetic equation, which has the form of an integro-differential conservation law in $(1+1)$D. The purpose of the present work is to investigate the sense in which the Bethe-Boltzmann equation defines an "integrable kinetic equation". To this end, we study a class of $N$ dimensional systems of evolution equations that arise naturally as finite-dimensional approximations to the Bethe-Boltzmann equation. We obtain non-local Poisson brackets and Hamiltonian densities for these equations and derive an infinite family of first integrals, parameterized by $N$ functional degrees of freedom. We find that the conserved charges arising from quantum integrability map to Casimir invariants of the hydrodynamic bracket and their group velocities map to Hamiltonian flows. Some results from the finite-dimensional setting extend to the underlying integro-differential equation, providing evidence for its integrability in the hydrodynamic sense.

arXiv comment29+3 pages; v3, text revised, refs. added

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• Non-Equilibrium Dynamics and Weakly Broken Integrability
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• Low-Temperature Transport in Out-of-Equilibrium XXZ Chains
• Generalized hydrodynamics: a perspective
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• Kinetic theory of quantum and classical Toda lattices
• Generalized hydrodynamics in box-ball system
• Diffusion and superdiffusion from hydrodynamic projections
• Euler-scale dynamical correlations in integrable systems with fluid motion
• Semiclassical theory of front propagation and front equilibration following an inhomogeneous quantum quench
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• Unusual ergodic and chaotic properties of trapped hard rods
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