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arXiv:2205.03858 · Physical review. B./Physical review. B

Robustness of Kardar-Parisi-Zhang scaling in a classical integrable spin chain with broken integrability

Dipankar Roy, Abhishek Dhar, Herbert E. Spohn, Manas Kulkarni

Published Mar 20, 2023• 24 citations
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Abstract

Recent investigations have observed superdiffusion in integrable classical and quantum spin chains. An intriguing connection between these spin chains and the Kardar-Parisi-Zhang (KPZ) universality class has emerged. Theoretical developments (e.g., generalized hydrodynamics) have highlighted the role of integrability as well as spin symmetry in KPZ behavior. However, understanding their precise role on superdiffusive transport still remains a challenging task. The widely used quantum spin chain platform comes with severe numerical limitations. To circumvent this barrier, we focus on a classical integrable spin chain which was shown to have a deep analogy with the quantum spin-$\frac{1}{2}$ Heisenberg chain. Remarkably, we find that KPZ behavior prevails even when one considers integrability-breaking but spin-symmetry preserving terms, strongly indicating that spin symmetry plays a central role even in the nonperturbative regime. On the other hand, in the nonperturbative regime, we find that energy correlations exhibit clear diffusive behavior. We also study the classical analog of the out-of-time-ordered correlator and Lyapunov exponents. We find a significant presence of chaos for the integrability-broken cases even though KPZ behavior remains robust. The robustness of KPZ behavior is demonstrated for a wide class of spin-symmetry preserving integrability-breaking terms.

DOI
10.1103/physrevb.107.l100413

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Recent investigations have observed superdiffusion in integrable classical and quantum spin chains. An intriguing connection between these spin chains and the Kardar-Parisi-Zhang (KPZ) universality class has emerged. Theoretical developments (e.g., generalized hydrodynamics) have highlighted the role of integrability as well as spin symmetry in KPZ behavior. However, understanding their precise role on superdiffusive transport still remains a challenging task. The widely used quantum spin chain platform comes with severe numerical limitations. To circumvent this barrier, we focus on a classical integrable spin chain which was shown to have a deep analogy with the quantum spin-12\frac{1}{2}21​ Heisenberg chain. Remarkably, we find that KPZ behavior prevails even when one considers integrability-breaking but spin-symmetry preserving terms, strongly indicating that spin symmetry plays a central role even in the nonperturbative regime. On the other hand, in the nonperturbative regime, we find that energy correlations exhibit clear diffusive behavior. We also study the classical analog of the out-of-time-ordered correlator and Lyapunov exponents. We find a significant presence of chaos for the integrability-broken cases even though KPZ behavior remains robust. The robustness of KPZ behavior is demonstrated for a wide class of spin-symmetry preserving integrability-breaking terms.
Integrable spin chains with a continuous non-Abelian symmetry, such as the one-dimensional isotropic Heisenberg model, show superdiffusive transport with little theoretical understanding. Although recent studies reported a surprising connection to the Kardar-Parisi-Zhang (KPZ) universality class in that case, this view was most recently questioned by discrepancies in full counting statistics. Here, by combining extensive numerical simulations of classical and quantum integrable isotropic spin chains with a framework developed by exact studies of the KPZ class, we characterize various two-point quantities that remain hitherto unexplored in spin chains, and find full agreement with KPZ scaling laws without adjustable parameters. This establishes the partial emergence of the KPZ class in integrable isotropic spin chains. Moreover, we reveal that the KPZ scaling laws are intact in the presence of an energy current, under the appropriate Galilean boost required by the propagation of spacetime correlation.
The Heisenberg spin chain is a canonical integrable model. As such, it features stable ballistically propagating quasiparticles, but spin transport is subballistic at any nonzero temperature: An initially localized spin fluctuation spreads in time t to a width t2/3. This exponent as well as the functional form of the dynamical spin correlation function suggest that spin transport is in the Kardar–Parisi–Zhang (KPZ) universality class. However, the full counting statistics of magnetization is manifestly incompatible with KPZ scaling. A simple two-mode hydrodynamic description, derivable from microscopic principles, captures both the KPZ scaling of the correlation function and the coarse features of the full counting statistics, but remains to be numerically validated. These results generalize to any integrable spin chain invariant under a continuous nonabelian symmetry and are surprisingly robust against moderately strong integrability-breaking perturbations that respect the nonabelian symmetry.
Isotropic integrable spin chains such as the Heisenberg model feature superdiffusive spin transport belonging to an as-yet-unidentified dynamical universality class closely related to that of Kardar, Parisi, and Zhang (KPZ). To determine whether these results extend to more generic one-dimensional models, particularly those realizable in quantum simulators, we investigate spin and energy transport in non-integrable, long-range Heisenberg models using state-of-the-art tensor network methods. Despite the lack of integrability and the asymptotic expectation of diffusion, for power-law models (with exponent 2<α<∞2 < α< \infty2<α<∞) we observe long-lived z=3/2z=3/2z=3/2 superdiffusive spin transport and two-point correlators consistent with KPZ scaling functions, up to times t∼103/Jt \sim 10^3/Jt∼103/J. We conjecture that this KPZ-like transport is due to the proximity of such power-law-interacting models to the integrable family of Inozemtsev models, which we show to also exhibit KPZ-like spin transport across all interaction ranges. Finally, we consider anisotropic spin models naturally realized in Rydberg atom arrays and ultracold polar molecules, demonstrating that a wide range of long-lived, non-diffusive transport can be observed in experimental settings.
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.
Finite temperature spin transport in integrable isotropic spin chains is known to be superdiffusive, with dynamical spin correlations that are conjectured to fall into the Kardar-Parisi-Zhang (KPZ) universality class. However, integrable spin chains have time-reversal and parity symmetries that are absent from the KPZ (Kardar-Parisi-Zhang) or stochastic Burgers equation, which force higher-order spin fluctuations to deviate from standard KPZ predictions. We put forward a nonlinear fluctuating hydrodynamic theory consisting of two coupled stochastic modes: the local spin magnetization and its effective velocity. Our theory fully explains the emergence of anomalous spin dynamics in isotropic chains: it predicts KPZ scaling for the spin structure factor but with a symmetric, quasi-Gaussian, distribution of spin fluctuations. We substantiate our results using matrix-product states calculations.
The universality of dynamical scaling laws constitutes a cornerstone in the theoretical understanding of quantum many-body systems, particularly in non-equilibrium settings. Recent advancements have proposed a phenomenological ansatz based on spontaneous symmetry breaking (SSB) to unify the description of charge transport in open quantum systems. However, it remains unclear under which conditions it fails to capture the emergent hydrodynamics and if it does break down, whether nontrivial dynamics emerge. In this work we show that Kardar-Parisi-Zhang (KPZ) dynamics in an open integrable model (the B3 model), rather than diffusion from SSB, emerges. We find that the B3 model is equivalent to two interacting asymmetric XXZ spin chains and the ansatz can only capture the influence of the inter-chain interactions. When the initial state is appropriate, the asymmetric XXZ structure dominates the dynamics, which gives KPZ scaling behavior even when the hopping rate becomes negative. Our work motivates theory of charge transport in open systems beyond the ansatz based on SSB.