Universality Classes of Spin Transport in One-Dimensional Isotropic Magnets: The Onset of Logarithmic Anomalies

We report a systematic study of finite-temperature spin transport in quantum and classical one-dimensional magnets with isotropic spin interactions, including both integrable and nonintegrable models. Employing a phenomenological framework based on a generalized Burgers’ equation in a time-dependent stochastic environment, we identify four different universality classes of spin fluctuations. These comprise, aside from normal spin diffusion, three types of superdiffusive transport: the Kardar-Parisi-Zhang universality class and two distinct types of anomalous diffusion with multiplicative logarithmic corrections. Our predictions are supported by extensive numerical simulations on various examples of quantum and classical chains. Contrary to common belief, we demonstrate that even nonintegrable spin chains can display a diverging spin diffusion constant at finite temperatures.

Figure 1

Universality classes of spin dynamics in rotationally invariant spin systems. The dynamical exponents $z$ and $b$ characterizing the time-asymptotic behavior of the spin dynamical correlations $⟨\overrightarrow{S}(x,t)·\overrightarrow{S}(0,0)⟩\sim t^{-1/z}{\log }^{-b}(t)$, as predicted by the generalized noisy Burgers’ equation for the hydrodynamic evolution of the torsion component ${\tau }_t+[{\tau }^n+D{\tau }_x+\sqrt{\gamma (t)}\eta {]}_x=0$ with a time-dependent noisy environment $\gamma (t)\sim t^{-\zeta }$.

Figure 2

Left: Spin diffusion constant as a function of $\log (t)$, extracted from numerical simulations of $C_0(t)$, shown for a few representative classical nonintegrable lattice spin models at infinite temperature. The dashed curves are fitting lines as functions of $\log (t)$. The Heisenberg and next-to-nearest-neighbor Hamiltonians belong to the ${\mathbf{\text{D}}}_{\log }^{(2)}$ class, with diverging diffusion $\mathfrak{D}(t)\to (\log t{)}^{4/3}$, whereas the nonintegrable lattice discretizations of $H^{(n)}$ (denoted by $H_{\text{lattice}}^{(n)}$) exhibit normal diffusion $\mathbf{\text{D}}$ for $n\ge 3$, with $\mathfrak{D}(t)\to \text{const}$. Right: Plot of the spin diffusion constant $\mathfrak{D}(\delta )$ as a function of the interaction anisotropy $\delta$ for the classical anisotropic chain $H_{\delta }$ and the quantum nonintegrable spin-1 chain ${\widehat{H}}_{\delta }$ for two values of inverse temperature $\beta =1/T$. Dashed lines are fitting curves $\mathfrak{D}(\delta )=-a^2\log (c^2\delta )$, with fitting parameters $a$ and $c$.