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arXiv:2603.29344

Parafermionic Truncated Wigner Approximation

Javad Vahedi, Martin Gärttner

Published Mar 31, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2603.29344

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Phase-space approaches such as the truncated Wigner approximation (TWA) provide an efficient semiclassical framework for performing approximate simulations of the dynamics of open quantum many-body systems outside the reach of exact numerical methods but beyond the mean-field level. For bosonic systems, TWA is known to be equivalent to a Keldysh path-integral formulation truncated at second order in the so-called quantum fluctuations. This semiclassical approach provides an alternative transparent route towards approximate stochastic equations of motion, which can be efficiently solved. Here we establish the corresponding path-integral formulation for interacting open spin-1/21/21/2 systems using the continuous SU(2)\mathrm{SU}(2)SU(2) phase space. We show, in particular, that a consistent treatment of dissipation requires correctly mapping operator products onto the curved spin phase space, leading to stochastic equations that coincide with those obtained from the continuous TWA formulation and thus reproduce the exact dynamics of a single dissipative spin. Our results provide a unified field-theoretic foundation for the TWA to dissipative spin dynamics and offer a systematic starting point for extensions beyond the semiclassical approximation.
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We develop and utilize the SU(3) truncated Wigner approximation (TWA) in order to analyze far-from-equilibrium quantum dynamics of strongly interacting Bose gases in an optical lattice. Specifically, we explicitly represent the corresponding Bose-Hubbard model at an arbitrary filling factor with restricted local Hilbert spaces in terms of SU(3) matrices. Moreover, we introduce a discrete Wigner sampling technique for the SU(3) TWA and examine its performance as well as that of the SU(3) TWA with the Gaussian approximation for the continuous Wigner function. We directly compare outputs of these two approaches with exact computations regarding dynamics of the Bose-Hubbard model at unit filling with a small size and that of a fully connected spin-1 model with a large size. We show that both approaches can quantitatively capture quantum dynamics on a timescale of $\ensuremath{\hbar}/(Jz)$, where JJJ and zzz denote the hopping energy and the coordination number. We apply the two kinds of SU(3) TWA to dynamical spreading of a two-point correlation function of the Bose-Hubbard model on a square lattice with a large system size, which has been measured in recent experiments. Noticeable deviations between the theories and experiments indicate that proper inclusion of effects of the spatial inhomogeneity, which is not straightforward in our formulation of the SU(3) TWA, may be necessary.
We present a semiclassical phase-space method to calculate thermal and ground states of large interacting spin systems. To this end, we extend the recently developed truncated Wigner approximation for spins (TWA) to the imaginary time, termed iTWA. The evolution of the canonical density matrix in imaginary time is mapped to a partial differential equation of its Wigner function. Truncation at the Fokker-Planck level leads to a set of stochastic differential equations, which can be efficiently simulated even for large systems. We show that for general Ising Hamiltonians the approximation becomes exact for large imaginary times subject only to sampling errors. Thus the iTWA is ideal to determine the ground state of spin glasses or to find solutions to quadratic unconstrained binary optimization problems (QUBO) on a controlled approximation level. We illustrate this for MaxCut on random, unweighted 3-regular graphs, encoded in an anti-ferromagnetic Ising Hamiltonian, for which finding the exact ground state and even approximations to it beyond a certain accuracy is know to be NP hard. Furthermore, in order to assess the quality of the method also for general spin models, we analyze the ground-state quantum phase transition of the transverse-field Ising model in one and two spatial dimensions, finding reasonably good agreement with the exact behavior.