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arXiv:2305.02359 · SciPost Physics

Universal aspects of high-temperature relaxation dynamics in random spin models

Tian-Gang Zhou, Wei Zheng, Pengfei Zhang

Published Apr 8, 2025• 2 citations
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Abstract

Universality is a crucial concept in modern physics, allowing us to capture the essential features of a system’s behavior using a small set of parameters. In this work, we unveil universal spin relaxation dynamics in anisotropic random Heisenberg models with infinite-range interactions at high temperatures. Starting from a polarized state, the total magnetization can relax monotonically or decay with long-lived oscillations, determined by the sign of a universal single function A=-\xi_1^2+\xi_2^2-4\xi_2\xi_3+\xi_3^2 A = − ξ 1 2 + ξ 2 2 − 4 ξ 2 ξ 3 + ξ 3 2 . Here (\xi_1,\xi_3,\xi_3) ( ξ 1 , ξ 3 , ξ 3 ) characterizes the anisotropy of the Heisenberg interaction. Furthermore, the oscillation shows up only for A>0 A > 0 , with frequency \Omega \propto \sqrt{A} Ω ∝ A . This result is derived from the Kadanoff-Baym equation under the melon diagram approximation, which is consistent with numerical solutions. Furthermore, we verify our theory and approximation using exact diagonalization, albeit for a small system size of N=8 N = 8 . Our study sheds light on the universal aspect of quantum many-body dynamics beyond low energy limit.

DOI
10.21468/scipostphys.18.4.120

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Universality is a crucial concept in modern physics, allowing us to capture the essential features of a system’s behavior using a small set of parameters. In this work, we unveil universal spin relaxation dynamics in anisotropic random Heisenberg models with infinite-range interactions at high temperatures. Starting from a polarized state, the total magnetization can relax monotonically or decay with long-lived oscillations, determined by the sign of a universal single function A=-\xi_1^2+\xi_2^2-4\xi_2\xi_3+\xi_3^2 A = − ξ 1 2 + ξ 2 2 − 4 ξ 2 ξ 3 + ξ 3 2 . Here (\xi_1,\xi_3,\xi_3) ( ξ 1 , ξ 3 , ξ 3 ) characterizes the anisotropy of the Heisenberg interaction. Furthermore, the oscillation shows up only for A>0 A > 0 , with frequency \Omega \propto \sqrt{A} Ω ∝ A . This result is derived from the Kadanoff-Baym equation under the melon diagram approximation, which is consistent with numerical solutions. Furthermore, we verify our theory and approximation using exact diagonalization, albeit for a small system size of N=8 N = 8 . Our study sheds light on the universal aspect of quantum many-body dynamics beyond low energy limit.
Universality of SU(∞)SU(\infty)SU(∞) relaxation dynamics for SU(nf)SU(n_f)SU(nf​)-symmetric spin-models
Spin-models, where the NNN spins interact pairwise with a SU(nf)SU(n_f)SU(nf​) symmetry preserving hamiltonian, famously simplify in the large nfn_fnf​, NNN limits, as derived by Sachdev and Ye when exploring mean-field behavior of spin-glasses. We present numerical evidence that for a large class of models, the large nfn_fnf​ limit is not necessary: the same dynamical equations can describe the relaxation processes at high temperatures for a set of classical models inspired from mean-field treatments of interacting dense neutrino gases, up to times set by the radius of convergence of the perturbation series for the correlation function. After a simple rescaling of time, the dynamics display a surprising universality, being identical for any value of nfn_fnf​ as long as the rank of the coupling matrix is small. As a corollary of our results, we find that the direct interaction approximation originating from the study of stochastic flows in fluid turbulence should be thought of as only a short-time approximation for generic random coupling systems.
Using exact diagonalization techniques we study the dynamical response of the anisotropic disordered Heisenberg model for systems of S=1/2 spins with infinite range random exchange interactions at temperature T=0. The model can be considered as a generalization, to the quantum case, of the well known Sherrington-Kirkpatrick classical spin-glass model. We also compute and study the behavior of the Edwards Anderson order parameter and energy per spin as the anisotropy evolves from the Ising to the Heisenberg limits.
An important step towards a comprehensive understanding of far-from-equilibrium dynamics of quantum many-body systems is the identification of unifying features that are independent of microscopic details of the system. We experimentally observe such robust features in the magnetization relaxation dynamics of disordered Heisenberg XX, XXZ, and Ising Hamiltonians. We realize these Heisenberg spin models with tunable anisotropy parameter and power-law interactions in an ensemble of Rydberg atoms by encoding the spin in suitable Rydberg state combinations. We consistently observe stretched-exponential relaxation of magnetization for all considered spin models, collapsing onto a single curve after appropriate rescaling of time. This robust short-time relaxation behavior is explained by a perturbative treatment that exploits the strong disorder in pairwise couplings, which leads to a description in terms of approximately independent pairs of spins. In numerical simulations of small systems, we show that these pairs of spins constitute approximate local integrals of motion, which remain at least partially conserved on a timescale exceeding the duration of the relaxation dynamics of the magnetization. Published by the American Physical Society 2024
Recent experimental achievements in controlling ultracold gases in optical lattices open a new perspective on quantum many-body physics. In these experimental setups, it is possible to study coherent time evolution of isolated quantum systems. These dynamics reveal new physics beyond the low-energy properties that are usually relevant in solid-state many-body systems. In this paper, we study the time evolution of antiferromagnetic order in the Heisenberg chain after a sudden change of the anisotropy parameter, using various numerical and analytical methods. As a generic result, we find that the order parameter, which can show oscillatory or non-oscillatory dynamics, decays exponentially except for the effectively non-interacting case of the XX limit. For weakly ordered initial states, we also find evidence for an algebraic correction to the exponential law. The study is based on numerical simulations using a numerical matrix product method for infinite system sizes (iMPS), for which we provide a detailed description and an error analysis. Additionally, we investigate in detail the exactly solvable XX limit. These results are compared to approximative analytical approaches including an effective description by the XZ model as well as by mean-field, Luttinger-liquid and sine-Gordon theories. The comparison reveals which aspects of non-equilibrium dynamics can, as in equilibrium, be described by low-energy theories and which are the novel phenomena specific to quantum quench dynamics. The relevance of the energetically high part of the spectrum is illustrated by means of a full numerical diagonalization of the Hamiltonian.
The dynamics of the classical XYZ model with uniform interaction is investigated by the recursion method and, in part, by exact analysis. The time evolution is anharmonic for arbitrary N (number of spins); only the cases N=2 and \ensuremath{\infty} are completely integrable. For the special (uniaxially symmetric) equivalent-neighbor XXZ model, the nonlinearities in the equations of motion disappear in the limit N\ensuremath{\rightarrow}\ensuremath{\infty}, and the spin autocorrelation functions are determined exactly for infinite temperature: The function 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​(t)Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 exhibits a Gaussian decay to a nonzero constant, and the function 〈Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​(t)Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​〉 decays to zero, algebraically or like a Gaussian, depending on the amount of uniaxial anisotropy. For the general XYZ case, the T=\ensuremath{\infty} dynamical behavior includes four different universality classes, categorized according to the decay law of the spectral densities at high frequencies. That decay law governs the growth rate of the sequence of recurrents that determine the relaxation function in the continued-fraction representation. The four universality classes may serve as prototypes for a classification of the dynamics of classical and quantum many-body systems in general.
We study spin transport in the one-dimensional anisotropic S=1S=1S=1 Heisenberg model. Particular emphasis is given to dynamics at infinite temperature, where current autocorrelations and spatiotemporal correlation functions are obtained by means of an efficient pure-state approach based on the concept of typicality. Our comprehensive numerical analysis unveils that high-temperature spin transport is diffusive in the easy-axis regime for strong exchange anisotropies. This finding is based on the combination of numerous signatures, such as (i) Gaussian spreading of correlations, (ii) a time-independent diffusion coefficient, (iii) power-law decay of equal-site correlations, (iv) exponentially decaying long-wavelength modes, and (v) Lorentzian line shapes of the dynamical structure factor. Moreover, we provide evidence that some of these signatures are not exclusively restricted to the infinite-temperature limit but can persist at lower temperatures as well, where we complement our results by additional quantum Monte Carlo simulations of large systems. In contrast to the easy-axis regime, we show that in the case of an isotropic chain, the signatures (i)--(v) are much less pronounced or even entirely absent, suggesting the existence of anomalous spin transport despite the nonintegrability of the model. Eventually, upon introducing a random on-site magnetic field, we observe a breakdown of diffusion and distinctly slower dynamics. In particular, our results exhibit qualitative similarities to disordered spin-1/21/21/2 chains and might be consistent with the onset of many-body localization in the S=1S=1S=1 model for sufficiently strong disorder.