Parafermionic Truncated Wigner Approximation

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.

Figures (6)

• Figure 1: Schematic overview of the Parafermionic Truncated Wigner Approximation ($p$TWA). (a) Algebraic Mapping: The nonlinear on-site algebra of $\mathbb{Z}_n$ Fock parafermions $f_j$ is mapped to a local Hubbard operator basis $X_j^{ab} = |a\rangle\langle b|$, which forms a closed $\mathfrak{sl}(n)$ Lie algebra, enabling a well-defined phase-space formulation. (b) Initial State Sampling: Quantum fluctuations are captured by sampling initial conditions $x_j^{ab}(0)$ from either: (1) a multivariate Gaussian distribution matching the first and second quantum moments, or (2) a discrete Wigner distribution $W_\rho(q,p)$ (for prime $n$) using a finite-field phase-space grid. (c) Semiclassical Evolution: Each sampled initial condition is propagated deterministically according to the classical Lie-Poisson equations of motion, $\dot{x}^{ef} = \{ x^{ef}, H_W \}$, where the Poisson brackets are inherited from the Hubbard commutator algebra. (d) Observable Averaging: Time-dependent quantum observables $\langle O(t) \rangle$ are estimated by taking the ensemble average of the corresponding Weyl symbols $O_W$ over $N_{\text{traj}}$ independent classical trajectories.
• Figure 2: Benchmark of the $p$TWA dynamics for the fully connected $\mathbb{Z}_n$ clock model [Eq. (\ref{['eq:Zn_LMG_H']})]. We show the time evolution of the collective magnetization $|m(t)| = \left| \frac{1}{N}\sum_{i=1}^{N}\langle Z_i \rangle \right|$, comparing exact diagonalization (solid lines) with $p$TWA results (dashed lines). The system is initialized in the fully polarized clock state $|0\rangle^{\otimes N}$. Model parameters are $J=1.0$ and $g/J=0.5$. Top panel: convergence with increasing local clock dimension $n=3,4,\dots,7$ at fixed system size $N=30$. Bottom panel: convergence with increasing system size $N=10,20,\dots,80$ at fixed clock dimension $n=5$. In both cases the agreement between $p$TWA and exact dynamics improves systematically as the system approaches the semiclassical limit, either by increasing the local Hilbert-space dimension $n$ or by increasing the number of sites $N$.
• Figure 3: Spatiotemporal propagation of a local excitation in the long-range $\mathbb{Z}_3$ clock chain. We show the excitation probability $P_{\mathrm{exc}}(j,t)=\langle X_j^{11}(t)+X_j^{22}(t)\rangle$ following an initial excitation prepared at the center site $j_0=7$ in a chain of length $L=13$. The left column shows exact quantum dynamics obtained from Krylov-based exact time evolution (ED) Javad2022, while the right column shows semiclassical dynamics computed using the parafermionic truncated Wigner approximation ($p$TWA). Rows correspond to different interaction exponents: $\alpha=3.0$ (top), $\alpha=1.5$ (middle), and $\alpha=0.5$ (bottom). For larger $\alpha$, interactions are effectively short ranged and the excitation spreads outward with a visible light-cone structure. As $\alpha$ decreases and interactions become increasingly long ranged, the propagation becomes more collective and the excitation distribution remains strongly concentrated near the initially excited site. We use open boundary conditions.
• Figure 4: Time evolution of dynamical observables following a local excitation in the long-range $\mathbb{Z}_3$ clock chain of length $L=13$. The system is initialized with a single excitation at the central site $j_0=7$. The left column shows the survival probability $P_{\mathrm{exc}}(j_0,t)$ at the initially excited site, while the right column shows the mean displacement $\bar{r}(t)$, which quantifies the spatial spreading of the excitation. Rows correspond to different interaction exponents: $\alpha=3.0$ (top), $\alpha=1.5$ (middle), and $\alpha=0.5$ (bottom). Black curves denote exact dynamics obtained from exact diagonalization (ED), while red and blue curves show results from the parafermionic truncated Wigner approximation ($p$TWA) using Gaussian and discrete Wigner-function sampling, respectively.
• Figure 5: Disorder-averaged imbalance dynamics $\mathcal{I}(t)$ for the disordered $\mathbb{Z}_3$ Fock parafermion chain with system size $L=12$ and pair-hopping parameter $g=0.3$. The top panel shows $p$TWA results obtained using Gaussian Wigner sampling, while the bottom panel uses discrete Wigner sampling. Dashed curves correspond to $p$TWA trajectories and solid curves to exact diagonalization (ED) benchmarks. Colors indicate the disorder strength $W$. For weak disorder the imbalance decays rapidly toward zero, signaling efficient transport and ergodic relaxation. Inset: time-averaged imbalance $\bar{\mathcal{I}}=\langle \mathcal{I}(t)\rangle_{t\in[2,20]}$ as a function of disorder strength $W$. Black $\times$ markers denote $p$TWA results and red circles show ED benchmarks, revealing a smooth crossover from ergodic transport at weak disorder to strongly suppressed transport at large $W$.