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arXiv:1603.06588 · Physical review. B./Physical review. B

Spin transport of weakly disordered Heisenberg chain at infinite temperature

Ilia Khait, Snir Gazit, Norman Y. Yao, Assa Auerbach

Published Jun 17, 2016• 81 citations
arXivHTMLPDF

Abstract

We study the disordered Heisenberg spin chain, which exhibits many-body localization at strong disorder, in the weak to moderate disorder regime. A continued fraction calculation of dynamical correlations is devised, using a variational extrapolation of recurrents. Good convergence for the infinite chain limit is shown. We find that the local spin correlations decay at long times as $C\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\beta}}$, whereas the conductivity exhibits a low-frequency power law $\ensuremath{\sigma}\ensuremath{\sim}{\ensuremath{\omega}}^{\ensuremath{\alpha}}$. The exponents depict subdiffusive behavior $\ensuremath{\beta}<1/2,\phantom{\rule{0.28em}{0ex}}\ensuremath{\alpha}>0$ at all finite disorders and convergence to the scaling result $\ensuremath{\alpha}+2\ensuremath{\beta}=1$ at large disorders.

DOI
10.1103/physrevb.93.224205

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We study the disordered Heisenberg spin chain, which exhibits many body localization at strong disorder, in the weak to moderate disorder regime. A continued fraction calculation of dynamical correlations is devised, using a variational extrapolation of recurrents. Good convergence for the infinite chain limit is shown. We find that the local spin correlations decay at long times as C∼t−βC \sim t^{-\beta}C∼t−β, while the conductivity exhibits a low frequency power law σ∼ωα\sigma \sim \omega^{\alpha}σ∼ωα. The exponents depict sub-diffusive behavior $ \beta 0 $ at all finite disorders, and convergence to the scaling result, α+2β=1\alpha+2\beta = 1α+2β=1, at large disorders.
We study the transport and equilibration properties of a classical Heisenberg chain, whose couplings are random variables drawn from a one-parameter family of power-law distributions. The absence of a scale in the couplings makes the system deviate substantially from the usual paradigm of diffusive spin hydrodynamics and exhibit a regime of subdiffusive transport with an exponent changing continuously with the parameter of the distribution. We propose a solvable phenomenological model that correctly yields the subdiffusive exponent, thereby linking local fluctuations in the coupling strengths to the long-time, large-distance behavior. It also yields the finite-time corrections to the asymptotic scaling, which can be important in fitting the numerical data. We show how such exponents undergo transitions as the distribution of the coupling gets wider, marking the passage from diffusion to a regime of slow diffusion, and finally to subdiffusion.
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.
Using numerical diagonalization techniques, we explore the effect of local and bond disorder on the finite-temperature spin and thermal conductivities of the one-dimensional anisotropic spin-1/2 Heisenberg model. High-temperature results for local disorder show that the dc conductivities are finite, apart from the uncorrelated XYXYXY case where dc transport vanishes. Moreover, at strong disorder, we find finite dc conductivities at all temperatures TTT except at T=0T=0T=0. The low-frequency conductivities are characterized by a nonanalytic cusp shape. Similar behavior is found for bond disorder.
We theoretically study the quench dynamics for an isolated Heisenberg spin chain with a random on-site magnetic field, which is one of the paradigmatic models of a many-body localization transition. We use the time-dependent variational principle as applied to matrix product states, which allows us to controllably study chains of a length up to L=100L=100L=100 spins, i.e., much larger than $L\ensuremath{\simeq}20$ that can be treated via exact diagonalization. For the analysis of the data, three complementary approaches are used: (i) determination of the exponent $\ensuremath{\beta}$ which characterizes the power-law decay of the antiferromagnetic imbalance with time; (ii) similar determination of the exponent ${\ensuremath{\beta}}_{\mathrm{\ensuremath{\Lambda}}}$ which characterizes the decay of a Schmidt gap in the entanglement spectrum; and (iii) machine learning with the use, as an input, of the time dependence of the spin densities in the whole chain. We find that the consideration of the larger system sizes substantially increases the estimate for the critical disorder Wc{W}_{c}Wc​ that separates the ergodic and many-body localized regimes, compared to the values of Wc{W}_{c}Wc​ in the literature. On the ergodic side of the transition, there is a broad interval of the strength of disorder with slow subdiffusive transport. In this regime, the exponents $\ensuremath{\beta}$ and ${\ensuremath{\beta}}_{\mathrm{\ensuremath{\Lambda}}}$ increase, with increasing LLL, for relatively small LLL but saturate for $L\ensuremath{\simeq}50$, indicating that these slow power laws survive in the thermodynamic limit. From a technical perspective, we develop an adaptation of the ``learning by confusion'' machine-learning approach that can determine Wc{W}_{c}Wc​.
In this work, we study the transport properties of the anisotropic Heisenberg model in a disordered magnetic field and in the presence of dephasing due to external degrees of freedom. Without dephasing, the model can display, depending on parameter values, the whole range of possible transport regimes: ideal ballistic conduction, diffusive, or ideal insulating behavior. We show that the presence of dephasing induces normal diffusive transport in a wide range of parameters. We also analyze the dependence of spin conductivity on the dephasing strength. In addition, by analyzing the decay of the spin–spin correlation function, we find a long-range order for finite chain sizes. All our results for a one-dimensional spin chain at infinite temperature can be equivalently rephrased for strongly interacting disordered spinless fermions.