Long-Time Dynamics in Quantum Spin Lattices: Ergodicity and Hydrodynamic Projections at All Frequencies and Wavelengths

Abstract

Obtaining rigorous and general results about the non-equilibrium dynamics of extended many-body systems is a difficult task. In quantum lattice models with short-range interactions, the Lieb–Robinson bound tells us that the spatial extent of operators grows at most linearly in time. But what happens within this light-cone? We discuss rigorous results on ergodicity and the emergence of the Euler hydrodynamic scale in correlation functions, which establish fundamental principles at the root of non-equilibrium physics. One key idea of the present work is that general structures of Euler hydrodynamics, obtained under ballistic scaling, follow independently from the details of the microscopic dynamics, and in particular do not necessitate chaos; they are consequences of “extensivity”. Another crucial observation is that these apply at arbitrary frequencies and wavelengths. That is, long-time, persistent oscillations of correlation functions over ballistic regions of spacetime, which may be of microscopic frequencies and wavelengths, are predicted by a general Euler-hydrodynamic theory that takes the same form as that for smoothed-out correlation functions. This involves a natural extension of notions of conserved quantities and hydrodynamic projection and shows that the Euler hydrodynamic paradigm covers the full frequency-wavelength plane.

Ballistic macroscopic fluctuation theory

“…The idea is that such observables are examples of the "macroscopic observables" introduced by von Neumann in the context of the quantum ergodic theorem [83] (see an analysis of this paper in [84]). An ergodicity result has recently been shown [85], which implies that, in a large class of stationary, homogeneous states of quantum lattices with short-range Hamiltonians, averages of observables over space-time indeed project onto the state average times the identity operator; this further support taking fluid-cell means as classical variables.…”