Parafermionic Truncated Wigner Approximation

TL;DR

The paper introduces the parafermionic truncated Wigner approximation ($p$TWA), a semiclassical method for simulating the nonequilibrium dynamics of lattice systems with fractional exchange statistics, extending phase-space approaches to parafermions.

Contribution

It develops a novel phase-space framework for parafermionic systems, enabling efficient semiclassical simulations of their complex quantum dynamics.

Findings

Reproduces key features of exact dynamics such as excitation spreading and disorder effects.

Demonstrates effectiveness in various models including clock and parafermion chains.

Provides a practical tool for studying systems where exact methods are computationally limited.

Abstract

We introduce the parafermionic truncated Wigner approximation ($p$TWA), a semiclassical phase-space framework for simulating the nonequilibrium dynamics of lattice systems with fractional exchange statistics. The method extends truncated Wigner approaches developed for bosonic and fermionic systems to $\mathbb{Z}_n$ Fock parafermions by expressing the Hamiltonian in terms of local Hubbard operators that form a closed Lie algebra. This representation leads to a Lie--Poisson phase-space formulation in which quantum dynamics is approximated by stochastic sampling of initial conditions followed by deterministic semiclassical evolution. We benchmark the approach in several settings, including single-site clock dynamics, the fully connected $\mathbb{Z}_n$ clock model, long-range $\mathbb{Z}_3$ clock chains, and disordered $\mathbb{Z}_3$ Fock parafermion chains. The method reproduces key features of the exact dynamics, including excitation spreading, disorder-induced suppression of transport, and the emergence of long-time imbalance plateaus. Our results demonstrate that $p$TWA provides a practical tool for exploring the dynamics of parafermionic systems in regimes where exact numerical methods are limited by Hilbert-space growth.