Coexistence of anomalous and normal diffusion in integrable Mott insulators

We study the finite-momentum spin dynamics in the one-dimensional $XXZ$ spin chain within the Ising-type regime at high temperatures using density autocorrelations within linear-response theory and real-time propagation of nonequilibrium densities. While for the nonintegrable model results are well consistent with normal diffusion, the finite-size integrable model unveils the coexistence of anomalous and normal diffusion in different regimes of time. In particular, numerical results show a Gaussian relaxation at smallest nonzero momenta which we relate to nonzero stiffness in a grand canonical ensemble. For larger but still small momenta normal-like diffusion is recovered. Similar results for the model of impenetrable particles also help to resolve rather conflicting conclusions on transport in integrable Mott insulators.

Figure 1

Decay of spin densities $S_q\left(t\right)$ in the anisotropic Heisenberg model with $\Delta =1.5$ shown for smallest nonzero $q=q_1$ on a chain of length $L=26$ and ${\Delta }_2=0$, $0.5$, corresponding to integrable and nonintegrable models, respectively. LR results are obtained via the MCLM method at $\beta =0$ while TP results are obtained at $\beta =0.4/J$.

Figure 2

Spin decay rate ${\mathcal{D}}_q\left(t\right)$ at anisotropy $\Delta =2.0,{\Delta }_2=0$ calculated for the smallest finite $q_1$ in chains of different $L=14,20,24,26,30$. Dashed curves indicate straight lines $\propto 1.0t{J}^2/L$ while $t^{*}=L/\left(3J\right)$ is marked for $L=30$. The inset emphasizes the $L$ dependence at short times.

Figure 3

Decay of densities $S_q$ vs scaled time $t{q}^2$ at anisotropy $\Delta =2.0$, calculated within LR via MCLM on chains with $L=24,28$ sites for different wave vectors $q=q_1,2q_1$, respectively. Dashed curve indicates exponential decay with $\mathcal{D}=0.4J$. The position of the marginal time $t^{*}$ ($=t_{1,2}^{*}$) is also marked for $L=28$.

Figure 4

Decay of spin density autocorrelations $S_q\left(t\right)$ in the $\tilde{t}$ model at $n=1/2$ (quarter filling), zero magnetization, and high temperatures, depicted for the two smallest momenta $q>0$ in a chain of length $L=20$. One solid curve indicates an exponential decay with a constant rate $\mathcal{D}=0.76\tilde{t}$. Inset: Rate ${\mathcal{D}}_q\left(t\right)$ at the smaller of both $q$ for the same parameter set. The dashed curve indicates a line $\propto 0.1t{\tilde{t}}^2$.