Turbulent relaxation after a quench in the Heisenberg model

TL;DR

This paper predicts and analyzes turbulent scaling phenomena in the quench dynamics of the two-dimensional Heisenberg model, revealing universal turbulent cascades influenced by anisotropy and initial conditions.

Contribution

It introduces the concept of turbulent cascades in spin systems, providing analytical estimates and numerical validation, and explores the effects of anisotropy and initial conditions on turbulence.

Findings

Universal turbulent scaling observed in isotropic Heisenberg model.

Dual cascade of energy and conserved transverse magnetization in anisotropic case.

Analytical scaling exponents agree with numerical phase space simulations.

Abstract

We predict the emergence of turbulent scaling in the quench dynamics of the two-dimensional Heisenberg model for a wide range of initial conditions and model parameters. In the isotropic Heisenberg model, we find that the spin-spin correlation function exhibits universal scaling consistent with a turbulent energy cascade. When the spin rotational symmetry is broken by an easy-plane exchange anisotropy, we find a dual cascade of energy and an emergent conserved charge associated to transverse magnetization fluctuations. The scaling exponents are estimated analytically and agree with numerical simulations using phase space methods. We also define the space of initial conditions (as a function of energy, magnetization, and spin number $S$) that lead to a turbulent cascade. The universal character of the cascade, insensitive to microscopic details or initial conditions, suggests that turbulence in spin systems can be broadly realized in cold atom and solid-state experiments.