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arXiv:1408.6837 · Physical Review B

Spin and energy currents in integrable and nonintegrable spin- 1 2 chains: A typicality approach to real-time autocorrelations

Robin Steinigeweg, Jochen Gemmer B, Wolfram Brenig

Published Mar 9, 2015• 64 citations
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Abstract

We use the concept of typicality to study the real-time dynamics of spin and energy currents in spin-$1/2$ 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 $z$ 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.

DOI
10.1103/physrevb.91.104404

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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.
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We study the spin and energy dynamics in one-dimensional spin-12\frac{1}{2}21​ systems induced by local quantum quenches at finite temperatures using a time-dependent density matrix renormalization group method. System sizes are chosen large enough to ensure that the time-dependent data for the accessible time scales represent the behavior in the thermodynamic limit. As a main result, we observe a ballistic spreading of perturbations of the energy density in the integrable spin-12\frac{1}{2}21​ XXZ chain for all temperatures and exchange anisotropies, related to the divergent thermal conductivity in this model and the exact conservation of the energy current. In contrast, the spin dynamics is ballistic in the massless phase, but shows a diffusive behavior at high temperatures in the easy-axis phase in the case of a vanishing background spin density. We extract a quantitative estimate for the spin-diffusion constant from the time dependence of the spatial variance of the spin density, which agrees well with values obtained from current-current correlation functions using an Einstein relation. Interestingly, the diffusion constant approaches a constant value deep in the easy-axis regime. As an example for nonintegrable models, we consider two-leg ladders, for which we observe indications of diffusive energy and spin dynamics. The relevance of our results for recent experiments with quantum magnets and bosons in optical lattices is discussed.
We study the Heisenberg-Kitaev spin chain in order to uncover the interplay between two qualitatively different integrable points in the physics of heat transport in one dimension. Focusing on high temperatures and using analytical as well as numerical approaches within linear response theory, we explore several directions in parameter space including exchange-coupling ratios, anisotropies, and external magnetic fields. We show the emergence of purely ballistic energy transport at all integrable points, manifest in pronounced Drude weights and low-frequency suppression of regular-conductivity contributions. Moreover, off integrability, we find extended quantum chaotic regions with vanishing Drude weights and well-defined dc conductivities. In the vicinity of the Kitaev point, we observe clear signatures of the topological gap in the response function. This gap coexists with a nonzero Drude weight in the Kitaev chain.
Using the Bethe ansatz method and generalized hydrodynamics (GHD), properties of the spin transport is analyzed for the integrable XXZ chain with the anisotropy 0 ≤ ∆ < 1.Our approach starts from reconstructing the thermodynamic Bethe ansatz (TBA) equations for the integrable XXZ chain with arbitrary spin.We obtain the fusion hierarchy of the quantum transfer matrices (T -functions) in the arbitrary spin case, and we derive functional relations (T -and Y -systems) satisfied by the T -functions and certain combinations of them (Y -functions).Through analytical arguments, the Y -system is reduced to a set of non-linear integral equations, equivalent to the TBA equations.At first, we use the TBA equations for some higher-spin chains in order to examine the regular zero frequency contribution to the spin current correlation (spin dc conductivity) for the spin-1/2 XXZ chain.In the high temperature limit, we write down the dressed scattering kernels by one quasi-particle bare energies and exactly obtain the spin dc conductivity at zero magnetic field, which is proportional to the inverse temperature β at the β → 0 limit.We find that the high temperature proportionality constant σ reg 0 of the spin dc conductivity is discontinuous at all rational numbers of the anisotropy parameter p 0 = π/ cos -1 ∆ ≥ 2 with the gap increasing larger than the second power of growing magnetization on one quasiparticle.The isotropic ∆ = 1 point is exceptional.Close to this point, σ reg 0 slowly increases in proportion to the first power of the one quasi-particle magnetization.On the other hand, σ reg 0 is proportional to the second power of the same when p 0 approaches irrational numbers.At second, we examine the finite-temperature spin Drude weight for the integrable XXZ chain with arbitrary spin, where GHD methods are employed.We calculate the spin Drude weight at zero magnetic field, and we find a characteristic fractal-like structure of the Drude weight at arbitrary spin across a broad range of temperatures, similar to the spin-1/2 case.In our approach, the solutions to the TBA equations (i.e., the Y -functions) can be explicitly written in terms of the T -functions, thus allowing for a systematic calculation of the hightemperature limit of the Drude weight, as well as the spin dc conductivity in the spin 1/2 case. B Proof of the equivalence between(2.43) and (2.73) C Delivation of the TBA equations for the integrable XXZ chain with arbitrary spin C.1 Free energy . . . . . . . . . .
We study the real-time dynamics of spin chains driven out of thermal equilibrium by an initial temperature gradient ${T}_{L}\ensuremath{\ne}{T}_{R}$ using density matrix renormalization group methods. We demonstrate that the nonequilibrium energy current saturates fast to a finite value if the linear-response thermal conductivity is infinite, i.e., if the Drude weight DDD is nonzero. Our data suggest that a nonintegrable dimerized chain might support such dissipationless transport (D>0D>0D>0). We show that the steady-state value JE{J}_{E}JE​ of the current for arbitrary ${T}_{L}\ensuremath{\ne}{T}_{R}$ is of the functional form ${J}_{E}=f({T}_{L})\ensuremath{-}f({T}_{R})$, i.e., it is completely determined by the linear conductance. We argue for this functional form, which is essentially a Stefan-Boltzmann law in this integrable model; for the XXX ferromagnet, fff can be computed via the thermodynamic Bethe ansatz in good agreement with the numerics. Inhomogeneous systems exhibiting different bulk parameters as well as Luttinger liquid boundary physics induced by single impurities are discussed briefly.
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.