Hamiltonian formulation and aspects of integrability of generalised hydrodynamics

TL;DR

This paper demonstrates that generalized hydrodynamics (GHD) for integrable systems can be formulated as a Hamiltonian field theory with a new Poisson structure, revealing its potential integrability and extending previous models.

Contribution

It introduces a Hamiltonian formulation of GHD with spatially extended interaction kernels and a new Poisson bracket, generalizing previous models and suggesting integrability.

Findings

GHD is Hamiltonian with a new Poisson bracket.

The system admits an infinite set of conserved quantities.

The GHD equation can be viewed as a 2+1-dimensional classical field theory.

Abstract

Generalised Hydrodynamics (GHD) describes the large-scale inhomogeneous dynamics of integrable (or close to integrable) systems in one dimension of space, based on a central equation for the fluid density or quasi-particle density: the GHD equation. We consider a new, general form of the GHD equation: we allow for spatially extended interaction kernels, generalising previous constructions. We show that the GHD equation, in our general form and hence also in its conventional form, is Hamiltonian. This holds also including force terms representing inhomogeneous external potentials coupled to conserved densities. To this end, we introduce a new Poisson bracket on functionals of the fluid density, which is seen as our dynamical field variable. The total energy is the Hamiltonian whose flow under this Poisson bracket generates the GHD equation. The fluid density depends on two (real and spectral) variables so the GHD equation can be seen as a $2+1$-dimensional classical field theory. In its $1+1$-dimensional reduction corresponding to the case without external forces, we further show the system admits an infinite set of conserved quantities that are in involution for our Poisson bracket, hinting at integrability of this field theory.