Hydrodynamics in lattice models with continuous non-Abelian symmetries

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.

Stability of Superdiffusion in Nearly Integrable Spin Chains

“…Similar log-diffusion scalings have recently been observed up to long times in various isotropic spin chains, and was interpreted in the framework of the KPZ equation in Ref. [58], though the relation to integrability has remained controversial [66].…”

“…Another interesting question is whether the mechanisms explored here can manifest themselves, e.g., as anomalously large long-time tails in classical hydrodynamics, as suggested in Ref. [66].…”

“…The fate of superdiffusion when integrability is weakly broken is a question of great experimental relevance [23]. Away from integrability, neither exact solutions nor in-finitely stable quasiparticles exist; the theoretical tools we have to deal with this regime are still primitive [58,[60][61][62][63][64][65][66]. A natural framework to address weakly decaying quasiparticles is the Boltzmann equation [67,68], which relies on the matrix elements of generic local operators between eigenstates of the integrable system.…”

Superdiffusion in spin chains

“…Both types of dynamics have the same dynamical exponent z = 2, and thus at the level of linear response they must be linearly superposed to yield a complex diffusion constant. This reasoning has been confirmed by a systematic derivation of fluctuating hydrodynamics for systems with nonabelian global symmetry, within the language of effective field theory 249 . This phenomenon is counterintuitive in the sense that Goldstone physics is not usually believed to hold sway at high temperatures; that Goldstone modes can be important for high-temperature transport is nevertheless consistent with the results described in Secs.…”

“…If some of these relaxation timescales are slow enough, anomalous transport can persist away from the integrable limit. A separate possibility is that the conventional hydrodynamics of systems with nonabelian symmetries might be rich enough to permit anomalous transport; however, recent field-theoretical studies do not seem to support this possibility 249 .…”