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arXiv:1911.09917 · Physical Review Research

Decay of spin-spin correlations in disordered quantum and classical spin chains

Jonas Richter, Dennis Schubert, Robin Steinigeweg

Published Feb 6, 2020• 25 citations
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Abstract

The real-time dynamics of equal-site correlation functions is studied for one-dimensional spin models with quenched disorder.Focusing on infinite temperature, we present a comparison between the dynamics of models with different quantum numbers S = 1 2 , 1, 3 2 , as well as of chains consisting of classical spins.Based on this comparison as well as by analyzing the statistics of energy-level spacings, we show that the putative many-body localization transition is shifted to considerably stronger values of disorder for increasing S. In this context, we introduce an effective disorder strength W eff , which provides a mapping between the dynamics for different spin quantum numbers.For small W eff , we show that the real-time correlations become essentially independent of S, and are moreover very well captured by the dynamics of classical spins.Especially for S = 3 2 , the agreement between quantum and classical dynamics is remarkably observed even for very strong values of disorder.This behavior also reflects itself in the corresponding spectral functions, which are obtained via a Fourier transform from the time to the frequency domain.As an aside, we also comment on the self-averaging properties of the correlation function at weak and strong disorder.Our work sheds light on the correspondence between quantum and classical dynamics at high temperatures and extends our understanding of the dynamics in disordered spin chains beyond the well-studied case of S = 1 2 .

DOI
10.1103/physrevresearch.2.013130

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The real-time dynamics of equal-site correlation functions is studied for one-dimensional spin models with quenched disorder.Focusing on infinite temperature, we present a comparison between the dynamics of models with different quantum numbers S = 1 2 , 1, 3 2 , as well as of chains consisting of classical spins.Based on this comparison as well as by analyzing the statistics of energy-level spacings, we show that the putative many-body localization transition is shifted to considerably stronger values of disorder for increasing S. In this context, we introduce an effective disorder strength W eff , which provides a mapping between the dynamics for different spin quantum numbers.For small W eff , we show that the real-time correlations become essentially independent of S, and are moreover very well captured by the dynamics of classical spins.Especially for S = 3 2 , the agreement between quantum and classical dynamics is remarkably observed even for very strong values of disorder.This behavior also reflects itself in the corresponding spectral functions, which are obtained via a Fourier transform from the time to the frequency domain.As an aside, we also comment on the self-averaging properties of the correlation function at weak and strong disorder.Our work sheds light on the correspondence between quantum and classical dynamics at high temperatures and extends our understanding of the dynamics in disordered spin chains beyond the well-studied case of S = 1 2 .
We present a comprehensive comparison of spin and energy dynamics in quantum and classical spin models on different geometries, ranging from one-dimensional chains, over quasi-one-dimensional ladders, to two-dimensional square lattices. Focusing on dynamics at formally infinite temperature, we particularly consider the autocorrelation functions of local densities, where the time evolution is governed either by the linear Schr\odinger equation in the quantum case, or the nonlinear Hamiltonian equations of motion in the case of classical mechanics. While, in full generality, a quantitative agreement between quantum and classical dynamics can therefore not be expected, our large-scale numerical results for spin-1/21/21/2 systems with up to N=36N = 36N=36 lattice sites in fact defy this expectation. Specifically, we observe a remarkably good agreement for all geometries, which is best for the nonintegrable quantum models in quasi-one or two dimensions, but still satisfactory in the case of integrable chains, at least if transport properties are not dominated by the extensive number of conservation laws. Our findings indicate that classical or semi-classical simulations provide a meaningful strategy to analyze the dynamics of quantum many-body models, even in cases where the spin quantum number S=1/2S = 1/2S=1/2 is small and far away from the classical limit S→∞S \to \inftyS→∞.
We present a comprehensive comparison of spin and energy dynamics in quantum and classical spin models on different geometries, ranging from one-dimensional chains, over quasi-one-dimensional ladders, to two-dimensional square lattices. Focusing on dynamics at formally infinite temperature, we particularly consider the autocorrelation functions of local densities, where the time evolution is governed either by the linear Schr\"odinger equation in the quantum case or the nonlinear Hamiltonian equations of motion in the case of classical mechanics. While, in full generality, a quantitative agreement between quantum and classical dynamics can therefore not be expected, our large-scale numerical results for spin-1/21/21/2 systems with up to N=36N=36N=36 lattice sites in fact defy this expectation. Specifically, we observe a remarkably good agreement for all geometries, which is best for the nonintegrable quantum models in quasi-one or two dimensions, but still satisfactory in the case of integrable chains, at least if transport properties are not dominated by the extensive number of conservation laws. Our findings indicate that classical or semiclassical simulations provide a meaningful strategy to analyze the dynamics of quantum many-body models, even in cases where the spin quantum number S=1/2S=1/2S=1/2 is small and far away from the classical limit $S\ensuremath{\rightarrow}\ensuremath{\infty}$.
The presence of frozen uncorrelated random on-site potential in interacting quantum systems can induce a transition from an ergodic phase to a localized one, the so-called many-body localization. Here we numerically study the effects of autocorrelated disorder on the static and dynamical properties of a one-dimensional many-body quantum system which exhibits many-body localization. Specifically, by means of some standard measures of energy level repulsion and localization of energy eigenstates, we show that a strong degree of correlations between the on-site potentials in the one-dimensional spin-1/2 Heisenberg model leads to suppression of the many-body localization phase, while level repulsion is mitigated for small disorder strengths, although energy eigenstates remain well extended. Our findings are also remarkably manifested in the time domain, on which we put the main emphasis, as shown by the time evolution of experimentally relevant observables, like the return probability and the spin autocorrelation function.
We provide a detailed comparison between the dynamics of high-temperature spatiotemporal correlation functions in quantum and classical spin models. In the quantum case, our large-scale numerics are based on the concept of quantum typicality, which exploits the fact that random pure quantum states can faithfully approximate ensemble averages, allowing the simulation of spin-1/21/21/2 systems with up to 40 lattice sites. Due to the exponentially growing Hilbert space, we find that for such system sizes even a single random state is sufficient to yield results with extremely low noise that is negligible for most practical purposes. In contrast, a classical analog of typicality is missing. In particular, we demonstrate that to obtain data with a similar level of noise in the classical case, extensive averaging over classical trajectories is required, no matter how large the system size. Focusing on (quasi-)one-dimensional spin chains and ladders, we find remarkably good agreement between quantum and classical dynamics. This applies not only to cases where both the quantum and classical models are nonintegrable but also to cases where the quantum spin-1/21/21/2 model is integrable and the corresponding classical $s\ensuremath{\rightarrow}\ensuremath{\infty}$ model is not. Our analysis is based on the comparison of space-time profiles of the spin and energy correlation functions, where the agreement is found to hold not only in the bulk but also in the tails of the resulting density distribution. The mean-squared displacement of the density profiles reflects the nature of emerging hydrodynamics and is found to exhibit similar scaling for quantum and classical models.
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.
Within the broad theme of understanding the dynamics of disordered quantum many-body systems, one of the simplest questions one can ask is, given an initial state, how does it evolve in time on the associated Fock-space graph, in terms of the distribution of probabilities thereon? A detailed quantitative description of the temporal evolution of out-of-equilibrium disordered quantum states and probability transport on the Fock space is our central aim here. We investigate it in the context of a disordered quantum spin chain, which hosts a disorder-driven many-body localization transition. Real-time dynamics/probability transport is shown to exhibit a rich phenomenology, which is markedly different between the ergodic and many-body localized phases. The dynamics is, for example, found to be strongly inhomogeneous at intermediate times in both phases, but while it gives way to homogeneity at long times in the ergodic phase, the dynamics remain inhomogeneous and multifractal in nature for arbitrarily long times in the localized phase. Similarly, we show that an appropriately defined dynamical lengthscale on the Fock-space graph is directly related to the local spin autocorrelation, and as such sheds light on the (anomalous) decay of the autocorrelation in the ergodic phase, and lack of it in the localized phase.