Anomalous and regular transport in spin 1/2 chains: AC conductivity

Abstract

We study magnetization transport in anisotropic spin-$1/2$ chains governed by the integrable XXZ model with and without integrability-breaking perturbations at high temperatures ($T\to \infty$) using a hybrid approach that combines exact sum-rules with judiciously chosen Ansätze. In the integrable XXZ model we find (i) super-diffusion at the isotropic (Heisenberg) point, with frequency dependent conductivity ${\sigma }^{\prime }(\omega \to 0)\sim |\omega {|}^{\alpha }$, where $\alpha =-3/7$ in close numerical agreement with recent $t$-DMRG computations; (ii) a continuously drifting exponent from $\alpha =-1^{+}$ in the XY limit of the model to $\alpha >0$ within the Ising regime; and (iii) a diffusion constant saturating in the XY coupling deep in the Ising limit. We consider two kinds of integrability breaking perturbations --- a simple next-nearest-neighbor spin-flip term ($J_2$) and a three-spin assisted variant ($t_2$), natural in the fermion particle representation of the spin chain. In the first case we discover a remarkable sensitivity of ${\sigma }^{\prime }(\omega )$ to the sign of $J_2$, with enhanced low frequency spectral weight and a pronounced upward shift in the magnitude of $\alpha$ for $J_2>0$. Perhaps even more surprising, we find sub-diffusion ($\alpha >0$) over a range of $J_2<0$. By contrast, the effects of the \enquote{fermionic} three-spin perturbation are sign symmetric; this perturbation produces a clearly observable hydrodynamic relaxation. At large strength of the integrability breaking term $J_2\to \pm \infty$ the problem is effectively non-interacting (fermions hopping on odd and even sublattices) and we find $\alpha \to -1$ behavior reminiscent of the XY limit of the XXZ chain. Exact diagonalization studies largely corroborate these findings at mid-frequencies.

2-column format, 17 pages, 58 references, 8 figures

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Cited by in corpus (18)

• Finite-temperature transport in one-dimensional quantum lattice models
• Hydrodynamic Diffusion in Integrable Systems
• Kinetic theory of spin diffusion and superdiffusion in XXZ spin chains
• Super-diffusion in one-dimensional quantum lattice models
• Diffusive hydrodynamics from integrability breaking
• Dynamics of magnetization at infinite temperature in a Heisenberg spin chain
• Stability of superdiffusion in nearly integrable spin chains
• Dynamics and Transport at the Threshold of Many-Body Localization
• Anomalous relaxation and the high-temperature structure factor of XXZ spin chains
• Anomalous low-frequency conductivity in easy-plane XXZ spin chains
• Hydrodynamics of nonintegrable systems from a relaxation-time approximation
• Asymmetry in energy versus spin transport in certain interacting, disordered systems
• Superdiffusion from nonabelian symmetries in nearly integrable systems
• Ballistic-to-diffusive transition in spin chains with broken integrability
• Many-body localization transition in a frustrated XY chain
• Less is more: more scattering leading to less resistance
• Open-system spin transport and operator weight dissipation in spin chains
• Infinite temperature spin dynamics in the asymmetric Hatsugai-Kohmoto model