Abstract
We develop a systematic effective field theory of hydrodynamics for many-body systems on the lattice with global continuous non-Abelian symmetries. Models with continuous non-Abelian symmetries are ubiquitous in physics, arising in diverse settings ranging from hot nuclear matter to cold atomic gases and quantum spin chains. In every dimension and for every flavor symmetry group, the low energy theory is a set of coupled noisy diffusion equations. Independence of the physics on the choice of canonical or microcanonical ensemble is manifest in our hydrodynamic expansion, even though the ensemble choice causes an apparent shift in quasinormal mode spectra. We use our formalism to explain why flavor symmetry is qualitatively different from hydrodynamics with other non-Abelian conservation laws, including angular momentum and charge multipoles.As a significant application of our framework, we study spin and energy diffusion in classical one-dimensional SU(2)-invariant spin chains, including the Heisenberg model along with multiple generalizations. We argue based on both numerical simulations and our effective field theory framework that non-integrable spin chains on a lattice exhibit conventional spin diffusion, in contrast to some recent predictions that diffusion constants grow logarithmically at late times. We show that the apparent enhancement of diffusion is due to slow equilibration caused by (non-Abelian) hydrodynamic fluctuations.
Highlights
Our numerics does show an apparent enhancement of diffusion compatible with (1): we show in Sec. 6.4 that the effect is due to hydrodynamic fluctuations captured by the effective field theory of Sec. 3
In some sense, there is an intuitive tension about how (8) and (9) might both arise in hydrodynamics, where we aim to describe the slow dynamics of the densities of conserved quantities
We observed that there is no non-trivial hydrodynamics with propagating degrees of freedom that can be found by considering possible non-Abelian flavor extension of “fracton hydrodynamics." We begin by a consideration of systems in one dimension, where there are no non-trivial possibilities whatsoever, before discussing higher dimensions, where the only non-trivial possibility is diffusion along subdimensional manifolds
Summary
Hydrodynamics is a universal language for describing thermalization and slow dynamics in chaotic many-body systems, whether they are classical or quantum [1].1 The universality of hydrodynamics arises from its insensitivity to most microscopic details, except for spacetime symmetries and conservation laws. (3) The hydrodynamic modes of any lattice model whose conserved charges are energy and non-Abelian flavor charges are of the form ω = ±Ak2 − iDk2 + · · · , with D > 0 and A nonzero only at finite flavor charge density (Section 3). This conclusion is robust in nonlinear fluctuating hydrodynamics, in all spatial dimensions d, including d = 1.
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