Anomalous KPZ Transport Achieves Superdiffusion in Non-Integrable Field Theories

Scientists have long sought to understand how energy and information travel in complex physical systems, and new research published this week sheds light on ‘anomalous transport’ in classical field theories. Matija Koterle, Tomaz Prosen (both from the Department of Physics, University of Ljubljana), and Tianci Zhou (Department of Physics, Virginia Tech) et al demonstrate that even in non-integrable field theories , systems previously thought to behave predictably , energy can spread in an unusual, superdiffusive manner. Their work reveals that temperature plays a crucial role in restoring this anomalous behaviour, overcoming limitations inherent in numerical simulations and offering a novel perspective on energy transport in a broad range of physical phenomena , from magnetism to turbulence. This discovery challenges conventional understandings of how these systems evolve and could pave the way for designing materials with tailored transport properties.

Temperature regulates anomalous transport in spin fields

Researchers addressed a long-standing challenge, the difficulty of investigating continuum field theories due to discretization breaking integrability in numerical simulations, by leveraging thermal effects to circumvent this issue. The study unveils a surprising result: a broad time window for anomalous transport emerges when temperature is used to effectively restore integrability. Simultaneously, ballistic energy transport is observed, while at higher temperatures, both spin and energy transport become diffusive. Remarkably, the non-integrable case of n = 2 displays the same crossover from diffusive to superdiffusive behaviour, suggesting a potentially more general phenomenon than previously understood.
The team achieved this by employing a discretized field theory at finite temperature, effectively introducing a symmetric integrability-breaking perturbation from the discretization that is mitigated at lower temperatures. Experiments show that the thermal state introduces a temperature-dependent time window, allowing observation of superdiffusive dynamics if it is an intrinsic feature of the continuum theory. Lyapunov analysis confirms the non-integrability of the n = 2 model, yet the structure of spin-density space-time profiles suggests the existence of long-lived, soliton-like trajectories at low temperatures. These trajectories, reminiscent of those found in discrete non-integrable spin chains, may be responsible for the observed superdiffusive spin transport.

The research establishes a connection between thermal fluctuations and the restoration of integrability, opening avenues for exploring anomalous transport in a wider range of classical field theories. This breakthrough reveals that the discretized field theory, when combined with a thermal regulator, can effectively mimic the behaviour of an integrable system within a specific timescale. The surprising discovery that the non-integrable n = 2 model exhibits a similar crossover raises the possibility that anomalous transport may be more generic in field theories than previously thought, potentially hinting at hidden integrable points within the system. The work opens exciting possibilities for understanding and controlling transport phenomena in various physical systems, from magnetism to hydrodynamics.

Finite Temperature Regulation of KPZ Spin Transport reveals

Experiments employed numerical simulations to track spin dynamics at varying temperatures, meticulously calculating both spin superdiffusion and ballistic energy transport. The team implemented a Metropolis algorithm for numerical integration, alongside detailed correlation function calculations, to characterise the system’s evolution, these computations are fully elaborated in the Supplemental Material. Lyapunov analysis was then performed to confirm the non-integrability of the system, providing a rigorous assessment of its chaotic behaviour. The research harnessed the power of spectral methods, utilising a fourth-order Runge-Kutta scheme with adaptive step size control to ensure accuracy and stability in time evolution.

This technique reveals the structure of spin-density space-time profiles, highlighting the existence of long-lived, soliton-like trajectories at low temperatures, a key finding supported by detailed analysis of the system’s dynamics. Furthermore, the study leveraged the DifferentialEquations. jl package within Julia, a high-performance computing environment, to efficiently solve the governing differential equations and manage the computational demands of the simulations. This innovative approach, combining finite temperature regulation with advanced numerical techniques, enabled the observation of KPZ scaling in a continuum field theory, confirming theoretical predictions and offering new insights into the behaviour of integrable and non-integrable spin systems. The precise measurement of dynamical exponents and the identification of long-lived solitons demonstrate the power of this methodology to unravel complex phenomena in condensed matter physics and provide a foundation for future investigations into anomalous transport.

KPZ Spin Transport Regulated by Finite Temperature exhibits

The team measured magnetization transport for the Landau-Lifshitz model at β = 2 with N = 1024, and for the n = 2 model at β = 4 with N = 2048, fitting Gaussian and Pr ahofer-Spohn scaling functions to Cm(x, t) at the latest displayed times. Results demonstrate that the n = 2 model exhibits a diffusive central peak atop ballistic tails at infinite temperature, while finite temperature reveals anomalous behaviour. Further analysis involved calculating exponential deviations of trajectories in thermal ensembles for both the Landau-Lifshitz model (β ∈{0, 3, 11}) and the n = 2 model (β ∈{0, 4, 40}) at times corresponding to specific correlation lengths of {1, 12, 44} lattice spacings. The norm was upper bounded by 2 √ L, with integrator tolerance maintained at 10−10.

The study found that the spin autocorrelator decays as t−2/3 (zm = 3/2) for both models at finite temperature, indicating anomalous transport. Measurements confirm that the energy autocorrelators become ballistic at lower temperatures, with a crossover to diffusion expected at longer times t∗(β), though observed only at high temperatures β ≲1. The team rescaled space-time profiles of correlation functions, demonstrating clear deviations from Gaussian behaviour and proximity to Pr ahofer-Spohn scaling curves for both n = 1 and n = 2 models. The n = 1 model exhibited an oscillatory step function shape in energy transport, while the n = 2 model displayed two visible peaks and a ballistic exponential tail. This work establishes that even in non-integrable cases, ballistic energy transport emerges due to momentum conservation and a non-zero energy Drude weight.

Temperature regulates superdiffusion in spin field theories

This crossover was observed not only in the integrable Landau-Lifshitz model (n=1) but also in a non-integrable model (n=2), suggesting a surprising robustness of superdiffusion. The findings establish that even non-integrable SO(3) symmetric field theories, similar to the Landau-Lifshitz type, can sustain superdiffusion within a temperature-dependent time window. Despite indicators of chaos, such as a finite Lyapunov exponent and the absence of conserved quantities beyond total spin and energy, the models exhibited slowly moving soliton-like trajectories. The authors acknowledge that their work relies on a momentum space cutoff discretization and that the precise role of continuum field theory requires further clarification. Future research should focus on understanding how momentum conservation might protect these soliton-like modes and potentially developing field theory calculations of transport from a low-temperature expansion. This research suggests that anomalous transport may be observable in real materials where perfect integrability is absent, provided temperatures are not excessively high relative to emergent hydrodynamic scales.