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arXiv:2601.19894

Anomalous transport in non-integrable classical field theories

Matija Koterle, Tomaž Prosen, Tianci Zhou

Published Jan 27, 2026• 0 citations
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Abstract

Anomalous KPZ spin transport is well established in integrable non-Abelian lattice models but has not been investigated in continuum field theories as discretization in numerics generally break the continuum theory's integrability. We show that finite temperature acts as a regulator that can restore anomalous transport over a broad time window. In a family of spin field theories labeled by integer $n$, the $n = 1$ case is the Landau-Lifshitz model, whose numerical data shows spin superdiffusion with Kardar-Parisi-Zhang (KPZ) scaling and, at lower temperature ballistic energy transport, whereas both observables are diffusive at high temperature. The non-integrable $n = 2$ case shows the same crossover. While Lyapunov analysis confirms the model's non-integrability, the structure of spin-density space-time profiles suggests that long-lived soliton-like trajectories exist at low temperature.

DOI
10.48550/arxiv.2601.19894

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Anomalous KPZ spin transport is well established in integrable non-Abelian lattice models but has not been investigated in continuum field theories as discretization in numerics generally break the continuum theory's integrability. We show that finite temperature acts as a regulator that can restore anomalous transport over a broad time window. In a family of spin field theories labeled by integer nnn, the n=1n = 1n=1 case is the Landau-Lifshitz model, whose numerical data shows spin superdiffusion with Kardar-Parisi-Zhang (KPZ) scaling and, at lower temperature ballistic energy transport, whereas both observables are diffusive at high temperature. The non-integrable n=2n = 2n=2 case shows the same crossover. While Lyapunov analysis confirms the model's non-integrability, the structure of spin-density space-time profiles suggests that long-lived soliton-like trajectories exist at low temperature.
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Recent studies report on anomalous spin transport for the integrable Heisenberg spin chain at its isotropic point. Anomalous scaling is also observed in the time evolution of nonequilibrium initial conditions, the decay of current-current correlations, and nonequilibrium steady state averages. These studies indicate a space-time scaling with x∼t^{2/3} behavior at the isotropic point, in sharp contrast to the ballistic form x∼t generically expected for integrable systems. In our contribution we study the scaling behavior for the integrable lattice Landau-Lifshitz spin chain. We report on equilibrium spatiotemporal correlations and dynamics with step initial conditions. Remarkably, for the case with zero mean magnetization, we find strong evidence that the scaling function is identical to the one obtained from the stationary stochastic Burgers equation, alias Kardar-Parisi-Zhang equation. In addition, we present results for the easy-plane and easy-axis regimes for which, respectively, ballistic and diffusive spin transport is observed, whereas the energy remains ballistic over the entire parameter regime.
Finite-temperature spin transport in the quantum Heisenberg spin chain is known to be superdiffusive, and has been conjectured to lie in the Kardar-Parisi-Zhang (KPZ) universality class. Using a kinetic theory of transport, we compute the KPZ coupling strength for the Heisenberg chain as a function of temperature, directly from microscopics; the results agree well with density-matrix renormalization group simulations. We establish a rigorous quantum-classical correspondence between the "giant quasiparticles" that govern superdiffusion and solitons in the classical continuous Landau-Lifshitz ferromagnet. We conclude that KPZ universality has the same origin in classical and quantum integrable isotropic magnets: a finite-temperature gas of low-energy classical solitons.
Although the Bethe ansatz solution of the spin-1/2 Heisenberg model dates back nearly a century, the anomalous nature of its high-temperature transport dynamics has only recently been uncovered. Indeed, numerical and experimental observations have demonstrated that spin transport in this paradigmatic model falls into the Kardar-Parisi-Zhang (KPZ) universality class. This has inspired the significantly stronger conjecture that KPZ dynamics, in fact, occur in all integrable spin chains with non-Abelian symmetry. Here, we provide extensive numerical evidence affirming this conjecture. Moreover, we observe that KPZ transport is even more generic, arising in both supersymmetric and periodically driven models. Motivated by recent advances in the realization of SU(N)-symmetric spin models in alkaline-earth-based optical lattice experiments, we propose and analyze a protocol to directly investigate the KPZ scaling function in such systems.
Finite-temperature spin transport in integrable isotropic spin chains (i.e., spin chains with continuous non-Abelian symmetries) is known to be superdiffusive, with anomalous transport properties displaying remarkable robustness to isotropic integrability-breaking perturbations. Using a discrete-time classical model, we numerically study the crossover to conventional diffusion resulting from both noisy and Floquet isotropic perturbations of strength $\ensuremath{\lambda}$. We identify an anomalously-long crossover timescale ${t}_{★}\ensuremath{\sim}{\ensuremath{\lambda}}^{\ensuremath{-}\ensuremath{\alpha}}$ with $\ensuremath{\alpha}\ensuremath{\approx}6$ in both cases. We discuss our results in terms of a kinetic theory of transport that characterizes the lifetimes of large solitons responsible for superdiffusion.
Anomalous transport in low dimensional spin chains is an intriguing topic that can offer key insights into the interplay of integrability and symmetry in many-body dynamics. Recent studies have shown that spin-spin correlations in spin chains, where integrability is either perfectly preserved or broken by symmetry-preserving interactions, fall in the Kardar-Parisi-Zhang (KPZ) universality class. Similarly, energy transport can show ballistic or diffusive-like behavior. Although such behavior has been studied under equilibrium conditions, no results on nonequilibrium spin transport in classical spin chains has been reported so far. In this work, we investigate both spin and energy transport in classical spin chains (integrable and nonintegrable) when coupled to two reservoirs at two different temperatures/magnetizations. In both the integrable case and the broken-integrability (but spin-symmetry preserving) case, we report anomalous scaling of spin current with system size (J^{s}∝L^{-μ}), with an exponent value that, within error bars, is close to the KPZ universality class value μ≈2/3. On the other hand, it is noteworthy that the energy current remains ballistic (J^{e}∝L^{-η} with η≈0) in the purely integrable case while there is departure from ballistic behavior (η>0) when integrability is broken regardless of spin-symmetry. We also present results on other interesting observables in the nonequilibrium steady state such as the spatial profiles of magnetization and energy, and spin-spin correlations.