Anomalous transport in non-integrable classical field theories

TL;DR

We address whether anomalous KPZ spin transport observed in integrable lattice models persists in continuum field theories where discretization typically breaks integrability. The authors study a family of one-dimensional classical SO(3) spin field theories with Hamiltonians $H^{(n)} = rac{1}{2n} abla oldsymbol{m} vert^2 n dx$, focusing on $n=1$ (integrable Landau-Lifshitz) and $n=2$ (non-integrable) cases, and analyze transport by sampling Gibbs ensembles at inverse temperature $eta$ and evolving the Landau-Lifshitz-type dynamics to compute two-point densities $C_q(x,t)$. They find spin transport with dynamical exponent $z_m=rac{3}{2}$ and KPZ scaling in a finite-temperature window for both models, while energy transport is ballistic at very low temperature and crosses to diffusion at higher $eta$, with LL displaying approximate integrability at low $T$ via reduced Lyapunov exponents and soliton-like trajectories, and $n=2$ showing persistent chaotic dynamics yet slow quasi-particle modes. The results suggest that anomalous transport is robust to integrability breaking in continuum field theories and may be observable in real materials with weak integrability-breaking terms, motivating further theoretical work (e.g., generalized hydrodynamics, low-temperature expansions) and experimental exploration.

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.

Figures (16)

• Figure 1: (a-b) Transport of magnetization in the LL and non-integrable $n=2$ model at finite $\beta$. Both cases exhibit superdiffusive transport with a dynamical exponent $z_m=3/2$. (c-d) Transport of energy in the LL and non-integrable $n=2$ models at finite $\beta$. While both cases exhibit ballistic transport, it is qualitatively different. LL shows large oscillations with revivals at times twice of system size (finite size effect), while the non-integrable model slowly approaches $z_h=1$ only at timescales $t>100$.
• Figure 2: Normalized scaling functions in rescaled coordinates. The blue curves show dynamics generated by LL, while the orange shows dynamics of the $n=2$ model. (a) Magnetization transport (LL at $\beta=2$, $N=1024$, $n=2$ at $\beta=4$, $N=2048$). The dashed lines denote fits of a Gaussian and Prähofer-Spohn (red and black dashes respectively) scaling functions fitted to $C_m(x,t)$ at the latest displayed times. (b) Energy transport (LL at $\beta=50$ with $N=2048$, $n=2$ at $\beta=120$ with $N=16384$). The colored lines show results for $n=2$ model to showcase the data collapse with a ballistic rescaling. (c) Comparison of exponential deviations of trajectories in thermal ensembles for the LL ($\beta\in\{0,3,11\}$, at times corresponding to top axis) and $n=2$ ($\beta\in\{0,4,40\}$ at times corresponding to bottom axis). The choices of $\beta$ fix equal correlation lengths in both models of $\{1,12,44\}$ lattice spacings $a$ respectively. The norm is upper bounded by $2\sqrt{L}$. Integrator tolerance $10^{-10}$. Note that the $x$ coordinates are rescaled for the $n=2$ model by factors 1.25 and 2 on panels (a) and (b) respectively for better visualization.
• Figure 3: Example of trajectories of $m^3(x,t)$ at finite temperature for the Landau-Lifschitz field theory (left panels) and the $n=2$ model (right panels). The top panels are at $\xi \approx 3a$, while the bottom panels are at $\xi \approx44a$. The gauge is fixed such that $\mathbf{m}(0,0)=(0,0,1)$. Parameters of simulation $L=N=256$, integrator tolerance $10^{-8}$.
• Figure 4: Convergence of dynamical exponents $z_m$ for increased $\beta$. The left panel show transport in the LL field theory, while the right panel shows transport in the $n=2$ model. Shaded region shows error of fit.
• Figure 5: Convergence of the dynamical exponent for energy transport $z_h$ for increased $\beta$ in both the LL and $n=2$ models. Note the different $x$ scales for each model.