[Paper Review] Dynamical criticality of magnetization transfer in integrable spin chains

Žiga Krajnik, Johannes Schmidt+2 more|arXiv (Cornell University)|Mar 29, 2023

Quantum many-body systems68 references4 citations

TL;DR

This study investigates anomalous magnetization transfer in integrable spin chains using large-scale numerical simulations of a classical anisotropic Landau-Lifshitz model. It reveals that cumulant growth exponents in both the easy-axis and isotropic regimes exceed Gaussian expectations, with the easy-axis case converging to the universal M-Wright distribution of charged single-file systems, while the isotropic case exhibits weakly non-Gaussian statistics, challenging existing hydrodynamic universality descriptions.

ABSTRACT

Recent studies have found that fluctuations of magnetization transfer in integrable spin chains violate the central limit property. Here we revisit the problem of anomalous counting statistics in the Landau-Lifshitz field theory by specializing to two distinct anomalous regimes featuring a dynamical critical point. By performing optimized numerical simulations using an integrable space-time discretization we extract the algebraic growth exponents of time-dependent cumulants which attain their threshold values. The distinctly non-Gaussian statistics of magnetization transfer in the easy-axis regime is found to converge towards the universal distribution of charged single-file systems. At the isotropic point we infer a weakly non-Gaussian distribution, corroborating the view that superdiffusive spin transport in integrable spin chains does not belong to any known dynamical universality class.

Motivation & Objective

• To understand the anomalous full counting statistics of magnetization transfer in integrable spin chains, particularly the breakdown of the central limit property.
• To investigate whether superdiffusive spin transport in integrable systems belongs to a known dynamical universality class.
• To determine whether classical simulations capture universal features relevant to quantum spin chains, especially at the isotropic point.
• To extract scaling exponents of time-dependent cumulants and assess their convergence to threshold values.
• To compare classical results with quantum predictions, particularly the quasi-Gaussian distribution proposed in Ref. [45].

Proposed method

• Numerical simulations of the classical anisotropic Landau-Lifshitz field theory using an integrable space-time discretization to preserve conserved quantities and enable long-time dynamics.
• Computation of time-dependent cumulants of the cumulative spin current up to sixth order to probe non-Gaussianity and scaling behavior.
• Estimation of finite-sample growth exponents ν̂_n from data in the time interval t ∈ [2^16, 2^20] to identify algebraic scaling regimes.
• Use of high-resolution simulations with L = 2^21 lattice sites and N = 5×10^3 independent realizations to access hydrodynamic scaling.
• Comparison of standardized moments (e.g., kurtosis) with Gaussian and quasi-Gaussian expectations to detect deviations.
• Analysis of convergence of the typical distribution of magnetization transfer toward universal forms, including the M-Wright distribution in the easy-axis regime.

Experimental results

Research questions

• RQ1 Does the full counting statistics of magnetization transfer in the easy-axis regime converge to the universal M-Wright distribution observed in charged single-file systems?
• RQ2 To what extent do the cumulant scaling exponents in the isotropic regime deviate from Gaussian expectations, and what does this imply for dynamical universality?
• RQ3 Are the observed non-Gaussian statistics in the isotropic case consistent with the quasi-Gaussian distribution predicted for quantum spin chains?
• RQ4 Can classical simulations of integrable spin chains capture the essential fluctuation statistics relevant to quantum systems, particularly in the absence of quantum entanglement?
• RQ5 Do higher-point temporal correlations in integrable spin chains depend on the underlying symmetry group, suggesting non-universal behavior beyond the dynamical structure factor?

Key findings

• In the easy-axis regime, the time-dependent typical distribution of magnetization transfer converges toward the universal M-Wright distribution characteristic of charged single-file systems, confirming phenomenological predictions.
• At the isotropic point, the kurtosis of the time-dependent distribution shows persistent, small but systematic deviations from Gaussianity, indicating weakly non-Gaussian statistics.
• The estimated growth exponents for cumulants in both regimes coincide with threshold values, confirming a violation of the central limit property in the hydrodynamic limit.
• The finite-sample exponents ν̂_n for n ∈ {2,4,6} in the easy-axis case are consistent with the algebraic scaling ν_n^ea = n/2, indicating superdiffusive scaling with non-Gaussian corrections.
• At the isotropic point, the cumulant exponents ν_n^iso are found to be less than n/2, suggesting subdiffusive-like growth in higher moments, inconsistent with Gaussian or quasi-Gaussian behavior.
• The data do not support the quasi-Gaussian distribution predicted in Ref. [45] for quantum spin chains, implying potential quantum corrections or non-algebraic corrections in the classical limit.