Superdiffusion in One-Dimensional Quantum Lattice Models

We identify a class of one-dimensional spin and fermionic lattice models that display diverging spin and charge diffusion constants, including several paradigmatic models of exactly solvable, strongly correlated many-body dynamics such as the isotropic Heisenberg spin chains, the Fermi-Hubbard model, and the $t\text{−}J$ model at the integrable point. Using the hydrodynamic transport theory, we derive an analytic lower bound on the spin and charge diffusion constants by calculating the curvature of the corresponding Drude weights at half-filling, and demonstrate that for certain lattice models with isotropic interactions some of the Noether charges exhibit superdiffusive transport at finite temperature and half-filling.

Figure 1

$XXZ$ chain at infinite temperature: the black curve shows diffusion bound [Eq. (5),] $D^{(m)}\ge 2{\mathcal{C}}^{(m)}(0)/(\beta v_{LR})$, and the blue points display numerical values of the spin diffusion constant obtained by time-dependent density matrix renormalization group in Refs. [21, 27]. The logarithmic divergence close to $\mathrm{\Delta }=1$ is indicated by the dashed line. Notice that the bound does not vanish even in the Ising limit $\mathrm{\Delta }\to \infty$, in contrast to the dissipative case [68]. Inside the gapless interval we display $\mathrm{\Delta }=\cos \pi /\ell$ with $\ell =3,\dots ,10$.

Figure 2

Large-$s$ scaling of $\log {\mathrm{\Gamma }}_s=\log {\int }_{-\infty }^{\infty }du{p}_s^{\prime }(u){[v_s^{\text{eff}}(u)]}^2$, confirming the asymptotic of Eq. (12) for the isotropic Heisenberg chain for various temperatures, showing that the large-$s$ scaling is independent of $\beta$.