Kardar-Parisi-Zhang universality from soft gauge modes

Abstract

The emergence of superdiffusive spin dynamics in integrable classical and quantum magnets is well established by now, but there is no generally valid theoretical explanation for this phenomenon. A fundamental difficulty is that the hydrodynamic fluctuations of conserved quasiparticle modes are purely diffusive. We propose a "hydrodynamic Higgs mechanism" in isotropic integrable magnets, which generates soft gauge modes that are decoupled from the quasiparticle sector. We show that the coarse-grained time evolution of these modes lies in the Kardar-Parisi-Zhang universality class of dynamics.

Integrable matrix models in discrete space-time

“…Based on numerical evidence from the anisotropic Landau-Lifshitz model [76], spin transport depends quite intricately on the value of anisotropy, ultimately responsible for the elusive behavior of the gapless phase in the anisoropic Heisenberg XXZ spin-1/2 chain [128,129] (see [126] for a review). To conclude, we wish to mention that the observed superuniversal nature of the KPZ phenomenon in integrable models with non-abelian Noether currents gives a further (albeit implicit) hint that the notion of hydrodynamic soft modes, employed recently in a phenomenological description of the KPZ phenomenon [41,42], extends beyond the simplest SO(3)-invariant Landau-Lifshitz theory. We postpone the study of these aspects for the future.…”

“…It is remarkable that the same phenomenon is already visible at the classical level, namely in the integrable classical spin chains symmetric under global SO (3) rotations [24,76,77]. In spite of a phenomenological picture based on an effective noisy Burger's equation proposed recently in [41] and further refined in [42], a complete and quantitative understanding of this curious phenomen is still lacking at the moment. More specifically, aside from partial analytical [37,39,78] and numerical evidence [79], it is not very clear what is the precise role of non-abelian symmetries and, particularly, if higher-rank symmetries could potentially alter this picture and possibly unveil new types of transport laws.…”

“…An important source of motivation for this work comes from an ever growing theoretical interest in nonequilibrium phenomena in strongly-correlated quantum systems, nowadays routinely explored in cold-atom experiments using highly-tunable optical lattice setups. Studying systems confined to one spatial dimension is particularly attractive not only because they can exhibit unorthodox phenomena, such as anomalous equilibration [29][30][31][32][33][34] and anomalous transport laws [35][36][37][38][39][40][41][42], but also thanks to a variety of theoretical tools available to study them. In the past few years, our understanding of transport phenomena in low dimensional systems, both in the linear regime and far from equilibrium, has increased quite dramatically.…”

“…This work has been largely inspired by the recent discovery of superdiffusive magnetization transport in the isotropic Heisenberg spin-1/2 chain [66][67][68], subsequently scrutinized in a number of papers [41,42,69], collectively accumulating a convincing numerical evidence for the Kardar-Parisi-Zhang (KPZ) type universality [70] (see also [71,72] and [73][74][75]). It is remarkable that the same phenomenon is already visible at the classical level, namely in the integrable classical spin chains symmetric under global SO (3) rotations [24,76,77].…”

Lecture notes on Generalised Hydrodynamics

“…Example 3.7. The partition function for the hard rod gas of Example 3.4 can be written explicitly, (139) where Θ(•) is Heavyside's step function. That is, it is like a free particle model, but with the constraint that the particles must be pairwise separated by a distance at least a.…”