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arXiv:1907.12115 · Physical review. B./Physical review. B

Universal spin dynamics in infinite-temperature one-dimensional quantum magnets

Maxime Dupont, Joel E. Moore

Published Mar 12, 2020• 112 citations
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Abstract

The spin dynamics of strongly interacting one-dimensional quantum magnets at high-temperature displays an emergent coarse-grained hydrodynamic behavior, understood as the result of conserved quantities. Different kinds of emergent hydrodynamics can be characterized by the dynamical exponent $z$ governing the length-time scaling $x\ensuremath{\sim}{t}^{1/z}$. Relying on extensive numerical simulations, the authors extract the exponent $z$ for various microscopic quantum spin-$S$ models. They identify three universal regimes for the spin transport (superdiffusive with $z=3/2$, ballistic with $z=1$, and diffusive with $z=2$), which depend only on the isotropy and integrability of the system, but are independent of its microscopic details.

DOI
10.1103/physrevb.101.121106

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The spin dynamics of strongly interacting one-dimensional quantum magnets at high-temperature displays an emergent coarse-grained hydrodynamic behavior, understood as the result of conserved quantities. Different kinds of emergent hydrodynamics can be characterized by the dynamical exponent zzz governing the length-time scaling x\ensuremath∼t1/zx\ensuremath{\sim}{t}^{1/z}x\ensuremath∼t1/z. Relying on extensive numerical simulations, the authors extract the exponent zzz for various microscopic quantum spin-SSS models. They identify three universal regimes for the spin transport (superdiffusive with z=3/2z=3/2z=3/2, ballistic with z=1z=1z=1, and diffusive with z=2z=2z=2), which depend only on the isotropy and integrability of the system, but are independent of its microscopic details.
We report a systematic study of finite-temperature spin transport in quantum and classical one-dimensional magnets with isotropic spin interactions, including both integrable and nonintegrable models. Employing a phenomenological framework based on a generalized Burgers' equation in a time-dependent stochastic environment, we identify four different universality classes of spin fluctuations. These comprise, aside from normal spin diffusion, three types of superdiffusive transport: the Kardar-Parisi-Zhang universality class and two distinct types of anomalous diffusion with multiplicative logarithmic corrections. Our predictions are supported by extensive numerical simulations on various examples of quantum and classical chains. Contrary to common belief, we demonstrate that even nonintegrable spin chains can display a diverging spin diffusion constant at finite temperatures.
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We study the spin and energy dynamics in one-dimensional spin-12\frac{1}{2}21​ systems induced by local quantum quenches at finite temperatures using a time-dependent density matrix renormalization group method. System sizes are chosen large enough to ensure that the time-dependent data for the accessible time scales represent the behavior in the thermodynamic limit. As a main result, we observe a ballistic spreading of perturbations of the energy density in the integrable spin-12\frac{1}{2}21​ XXZ chain for all temperatures and exchange anisotropies, related to the divergent thermal conductivity in this model and the exact conservation of the energy current. In contrast, the spin dynamics is ballistic in the massless phase, but shows a diffusive behavior at high temperatures in the easy-axis phase in the case of a vanishing background spin density. We extract a quantitative estimate for the spin-diffusion constant from the time dependence of the spatial variance of the spin density, which agrees well with values obtained from current-current correlation functions using an Einstein relation. Interestingly, the diffusion constant approaches a constant value deep in the easy-axis regime. As an example for nonintegrable models, we consider two-leg ladders, for which we observe indications of diffusive energy and spin dynamics. The relevance of our results for recent experiments with quantum magnets and bosons in optical lattices is discussed.
The Family-Vicsek scaling is a fundamental framework for understanding surface growth in non-equilibrium classical systems, providing a universal description of temporal surface roughness evolution. While universal scaling laws are well established in quantum systems, the applicability of Family-Vicsek scaling in quantum many-body dynamics remains largely unexplored. Motivated by this, we investigate the infinite-temperature dynamics of one-dimensional SU(N)SU(N)SU(N) spin chains, focusing on the well-known SU(2)SU(2)SU(2) XXZ model and the SU(3)SU(3)SU(3) Izergin-Korepin model. We compute the quantum analogue of classical surface roughness using the second cumulant of spin fluctuations and demonstrate universal scaling with respect to time and subsystem size. By systematically breaking global SU(N)SU(N)SU(N) symmetry and integrability, we identify distinct transport regimes characterized by the dynamical exponent zzz: (i) ballistic transport with z=1z=1z=1, (ii) superdiffusive transport with the Kardar-Parisi-Zhang exponent z=3/2z=3/2z=3/2, and (iii) diffusive transport with the Edwards-Wilkinson exponent z=2z=2z=2. Notably, breaking integrability always drives the system into the diffusive regime. Our results demonstrate that Family-Vicsek scaling extends beyond classical systems, holding universally across quantum many-body models with SU(N)SU(N)SU(N) symmetry.