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arXiv:1812.02701 · Physical Review Letters

Kinetic Theory of Spin Diffusion and Superdiffusion in X X Z Spin Chains

Sarang Gopalakrishnan, Romain Vasseur

Published Mar 26, 2019• 234 citations
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Abstract

We address the nature of spin transport in the integrable XXZ spin chain, focusing on the isotropic Heisenberg limit. We calculate the diffusion constant using a kinetic picture based on generalized hydrodynamics combined with Gaussian fluctuations: we find that it diverges, and show that a self-consistent treatment of this divergence gives superdiffusion, with an effective time-dependent diffusion constant that scales as D(t)∼t^{1/3}. This exponent had previously been observed in large-scale numerical simulations, but had not been theoretically explained. We briefly discuss XXZ models with easy-axis anisotropy Δ>1. Our method gives closed-form expressions for the diffusion constant D in the infinite-temperature limit for all Δ>1. We find that D saturates at large anisotropy, and diverges as the Heisenberg limit is approached, as D∼(Δ-1)^{-1/2}.

DOI
10.1103/physrevlett.122.127202

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We address the nature of spin transport in the integrable XXZ spin chain, focusing on the isotropic Heisenberg limit. We calculate the diffusion constant using a kinetic picture based on generalized hydrodynamics combined with Gaussian fluctuations: we find that it diverges, and show that a self-consistent treatment of this divergence gives superdiffusion, with an effective time-dependent diffusion constant that scales as D(t)∼t^{1/3}. This exponent had previously been observed in large-scale numerical simulations, but had not been theoretically explained. We briefly discuss XXZ models with easy-axis anisotropy Δ>1. Our method gives closed-form expressions for the diffusion constant D in the infinite-temperature limit for all Δ>1. We find that D saturates at large anisotropy, and diverges as the Heisenberg limit is approached, as D∼(Δ-1)^{-1/2}.
Understanding the physics of the integrable spin-1/2 XXZ chain has witnessed substantial progress, due to the development and application of sophisticated analytical and numerical techniques. In particular, infinite-temperature magnetization transport has turned out to range from ballistic, over superdiffusive, to diffusive behavior in different parameter regimes of the anisotropy. Since integrability is rather the exception than the rule, a crucial question is the change of transport under integrability-breaking perturbations. This question includes the stability of superdiffusion at the isotropic point and the change of diffusion constants in the easy-axis regime. In our work, we study this change of diffusion constants by a variety of methods and cover both, linear response theory in the closed system and the Lindblad equation in the open system, where we throughout focus on periodic boundary conditions. In the closed system, we compare results from the recursion method to calculations for finite systems and find evidence for a continuous change of diffusion constants over the full range of perturbation strengths. In the open system weakly coupled to baths, we find diffusion constants in quantitative agreement with the ones in the closed system in a range of nonweak perturbations, but disagreement in the limit of weak perturbations. Using a simple model in this limit, we point out the possibility of a diverging diffusion constant in such an open system.
We address spin transport in the easy-axis Heisenberg spin chain subject to different integrability-breaking perturbations. We find subdiffusive spin transport characterized by dynamical exponent z = 4 up to a timescale parametrically long in the anisotropy. In the limit of infinite anisotropy, transport is subdiffusive at all times; for finite anisotropy, one eventually recovers diffusion at late times but with a diffusion constant independent of the strength of the perturbation and solely fixed by the value of the anisotropy. We provide numerical evidence for these findings, and we show how they can be understood in terms of the dynamical screening of the relevant quasiparticle excitations and effective dynamical constraints. Our results show that the diffusion constant of near-integrable diffusive spin chains is generically not perturbative in the integrability-breaking strength.
Generalized hydrodynamics predicts universal ballistic transport in integrable lattice systems when prepared in generic inhomogeneous initial states. However, the ballistic contribution to transport can vanish in systems with additional discrete symmetries. Here we perform large scale numerical simulations of spin dynamics in the anisotropic Heisenberg XXZ spin 1/2 chain starting from an inhomogeneous mixed initial state which is symmetric with respect to a combination of spin reversal and spatial reflection. In the isotropic and easy-axis regimes we find non-ballistic spin transport which we analyse in detail in terms of scaling exponents of the transported magnetization and scaling profiles of the spin density. While in the easy-axis regime we find accurate evidence of normal diffusion, the spin transport in the isotropic case is clearly super-diffusive, with the scaling exponent very close to 2/3, but with universal scaling dynamics which obeys the diffusion equation in nonlinearly scaled time.
The isotropic Heisenberg chain represents a particular case of an integrable many-body system exhibiting superdiffusive spin transport at finite temperatures. Here, we show that this model has distinct properties also at finite magnetization $m\ensuremath{\ne}0$, even upon introducing the SU(2) invariant perturbations. Specifically, we observe nonmonotonic dependence of the diffusion constant ${\mathcal{D}}_{0}(\mathrm{\ensuremath{\Delta}})$ on the spin anisotropy $\mathrm{\ensuremath{\Delta}}$, with a pronounced maximum at $\mathrm{\ensuremath{\Delta}}=1$. The latter dependence remains true also in the zero magnetization sector, with superdiffusion at $\mathrm{\ensuremath{\Delta}}=1$ that is remarkably stable against isotropic perturbation (at least in finite-size systems), consistent with recent experiments with cold atoms.
Based on the ballistic macroscopic fluctuation theory, the integration of the spin correlation function (spin conductivity) is analyzed for the spin-1/2 XXZ chain in the critical regime. In the time when the magnetization of an infinite spin chain fluctuates from an initial state with a wavelength as long as the infinite length NNN, the equal-time two-point spin correlation function is scaled up to O(1/N)O(1/N)O(1/N). In the state where the ballistic spin transport decays at high temperature TTT, the diffusive transport remains on a large scale. We show that the spin conductivity is proportional to 1/T1/T1/T in the limit T→∞T\to\inftyT→∞ and its high temperature proportionality constant diverges in the case where one-quasiparticle magnetization is infinitely large. This analysis informs that the superdiffusive spin transport is driven by the 1/N1/N1/N-scaled long-range spin correlation and sheds a light on the dynamic scaling in spin transport at the isotropic point.
Abstract The gapped spin- 12{1\over 2}21​ XXZ chain with spin anisotropy Δ>1\Delta > 1Δ>1 is a strongly correlated quantum system of great interest in a variety of physical contexts. It is well established that, at zero magnetic field, the spin-diffusion constant associated with the regular part of the spin conductivity is finite both at very low and at high temperatures. It is therefore expected that the spin-diffusion constant remains finite at all temperatures. However, a direct calculation of the spin-diffusion constant over the full temperature range is typically intractable. In this paper, we review recent results showing that, at zero magnetic field, the spin-diffusion constant is finite for all temperatures when Δ>1\Delta > 1Δ>1 , and diverges in the limit Δ→1\Delta \rightarrow 1Δ→1 . Consequently, at zero magnetic field, finite-temperature spin transport is diffusive for Δ>1\Delta > 1Δ>1 , but becomes superdiffusive as Δ→1\Delta \rightarrow 1Δ→1 . These results are in agreement with predictions from the generalized hydrodynamics framework. However, the precise values of the spin-diffusion constant remains unknown except in certain temperature limits.