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arXiv:1312.5319 · Physical Review Letters

Spin-Current Autocorrelations from Single Pure-State Propagation

Robin Steinigeweg, Jochen Gemmer B, Wolfram Brenig

Published Mar 26, 2014• 127 citations
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Abstract

We demonstrate that the concept of quantum typicality allows for significant progress in the study of real-time spin dynamics and transport in quantum magnets. To this end, we present a numerical analysis of the spin-current autocorrelation function of the antiferromagnetic and anisotropic spin-1/2 Heisenberg chain as inferred from propagating only a single pure state, randomly chosen as a "typical" representative of the statistical ensemble. Comparing with existing time-dependent density-matrix renormalization group data, we show that typicality is fulfilled extremely well, consistent with an error of our approach, which is perfectly under control and vanishes in the thermodynamic limit. In the long-time limit, our results provide for a new benchmark for the enigmatic spin Drude weight, which we obtain from chains as long as L=33 sites, i.e., from Hilbert spaces of dimensions almost O(104) larger than in existing exact-diagonalization studies.

DOI
10.1103/physrevlett.112.120601

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We demonstrate that the concept of quantum typicality allows for significant progress in the study of real-time spin dynamics and transport in quantum magnets. To this end, we present a numerical analysis of the spin-current autocorrelation function of the antiferromagnetic and anisotropic spin-1/2 Heisenberg chain as inferred from propagating only a single pure state, randomly chosen as a "typical" representative of the statistical ensemble. Comparing with existing time-dependent density-matrix renormalization group data, we show that typicality is fulfilled extremely well, consistent with an error of our approach, which is perfectly under control and vanishes in the thermodynamic limit. In the long-time limit, our results provide for a new benchmark for the enigmatic spin Drude weight, which we obtain from chains as long as L=33 sites, i.e., from Hilbert spaces of dimensions almost O(104) larger than in existing exact-diagonalization studies.
We use the concept of typicality to study the real-time dynamics of spin and energy currents in spin-1/21/21/2 models in one dimension and at nonzero temperatures. These chains are the integrable XXZ chain and a nonintegrable modification due to the presence of a staggered magnetic field oriented in the zzz direction. In the framework of linear response theory, we numerically calculate autocorrelation functions by propagating a single pure state, drawn at random as a typical representative of the full statistical ensemble. By comparing to small-system data from exact diagonalization (ED) and existing short-time data from time-dependent density matrix renormalization group, we show that typicality is satisfied in finite systems over a wide range of temperature and is fulfilled in both integrable and nonintegrable systems. For the integrable case, we calculate the long-time dynamics of the spin current and extract the spin Drude weight for large systems outside the range of ED. We particularly provide strong evidence that the high-temperature Drude weight vanishes at the isotropic point. For the nonintegrable case, we obtain the full relaxation curve of the energy current and determine the heat conductivity as a function of magnetic field, exchange anisotropy, and temperature.
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.
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We investigate transport in several translationally invariant spin-12\frac{1}{2}21​ chains in the limit of high temperatures. We concretely consider spin transport in the anisotropic Heisenberg chain, the pure Heisenberg chain within an alternating field, and energy transport in an Ising chain which is exposed to a tilted field. Our approach is essentially based on a connection between the evolution of the variance of an inhomogeneous nonequilibrium density and the current-autocorrelation function at finite times. Although this relationship is not restricted to the case of diffusive transport, it allows to extract a quantitative value for the diffusion constant in that case. By means of numerically exact diagonalization we indeed observe diffusive behavior in the considered spin chains for a range of model parameters and confirm the diffusion coefficients which were obtained for these systems from nonequilibrium bath scenarios.
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We investigate the transport properties of the Holstein model using the numerically exact quantum typicality (QT) approach. Roughly speaking, QT exploits the fact that even a single, randomly chosen pure state can effectively represent the full statistical ensemble in a high-dimensional Hilbert space. This allows us to compute frequency-dependent mobilities, representative of the thermodynamic limit, that are well converged with respect to all numerical parameters. Our results are compared against other numerically exact methods, and used to analyze the contribution of vertex corrections to frequency-dependent mobility. The promising accuracy and efficiency of the QT approach suggest its applicability to a broader class of Hamiltonians.