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arXiv:1811.02806 · Physical review. B./Physical review. B

Magnetization and energy dynamics in spin ladders: Evidence of diffusion in time, frequency, position, and momentum

Jonas Richter, Fengping Jin, Lars Knipschild, J. Herbrych, Hans A. De Raedt, Kristel F. L. Michielsen, Jochen Gemmer B, Robin Steinigeweg

Published Apr 23, 2019• 23 citations
arXivHTMLPDF

Abstract

The dynamics of magnetization and energy densities are studied in the two-leg spin-$1/2$ ladder. Using an efficient pure-state approach based on the concept of typicality, we calculate spatiotemporal correlation functions for large systems with up to 40 lattice sites. In addition, two subsequent Fourier transforms from real to momentum space as well as from the time to frequency domain yield the respective dynamical structure factors. Summarizing our main results, we unveil the existence of genuine diffusion for both spin and energy. In particular, this finding is based on four distinct signatures which can all be equally well detected: (i) Gaussian density profiles, (ii) time-independent diffusion coefficients, (iii) exponentially decaying density modes, and (iv) Lorentzian line shapes of the dynamical structure factor. The combination of (i)--(iv) provides a comprehensive picture of high-temperature dynamics in this archetypal nonintegrable quantum model.

DOI
10.1103/physrevb.99.144422

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The dynamics of magnetization and energy densities are studied in the two-leg spin-1/21/21/2 ladder. Using an efficient pure-state approach based on the concept of typicality, we calculate spatiotemporal correlation functions for large systems with up to 40 lattice sites. In addition, two subsequent Fourier transforms from real to momentum space as well as from the time to frequency domain yield the respective dynamical structure factors. Summarizing our main results, we unveil the existence of genuine diffusion for both spin and energy. In particular, this finding is based on four distinct signatures which can all be equally well detected: (i) Gaussian density profiles, (ii) time-independent diffusion coefficients, (iii) exponentially decaying density modes, and (iv) Lorentzian line shapes of the dynamical structure factor. The combination of (i)--(iv) provides a comprehensive picture of high-temperature dynamics in this archetypal nonintegrable quantum model.
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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 study the dynamics of spin currents in the spin-12\frac{1}{2}21​ XX ladder at finite temperature. Within linear response theory, we numerically calculate autocorrelation functions for quantum systems larger than what is accessible with exact diagonalization using the concept of dynamical quantum typicality. While the spin Drude weight vanishes exponentially quickly with increasing system size, we show that this model realizes standard diffusive dynamics. Moreover, we unveil the existence of three qualitatively different dependencies of the spin-diffusion coefficient on the rung-coupling strength, resulting from a crossover from exponential to Gaussian dissipation as the rung coupling increases, in agreement with analytical predictions. We further discuss the implications of our results for experiments with cold atomic gases.
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.
Understanding (i) the emergence of diffusion from truly microscopic principles continues to be a major challenge in experimental and theoretical physics. At the same time, isolated quantum many-body systems have experienced an upsurge of interest in recent years. Since in such systems the realization of a proper initial state is the only possibility to induce a nonequilibrium process, understanding (ii) the largely unexplored role of the specific realization is vitally important. Our work reports a substantial step forward and tackles the two issues (i) and (ii) in the context of typicality, entanglement as well as integrability and nonintegrability. Specifically, we consider the spin-1/2 XXZ chain, where integrability can be broken due to an additional next-nearest neighbor interaction, and study the real-time and real-space dynamics of nonequilibrium magnetization profiles for a class of pure states. Summarizing our main results, we show that signatures of diffusion for strong interactions are equally pronounced for the integrable and nonintegrable case. In both cases, we further find a clear difference between the dynamics of states with and without internal randomness. We provide an explanation of this difference by a detailed analysis of the local density of states.