Tunable Superdiffusion in Integrable Spin Chains Using Correlated Initial States

Although integrable spin chains host only ballistically propagating particles, they can still feature diffusive charge transfer. This diffusive charge transfer originates from quasiparticle charge fluctuations inherited from the initial state’s magnetization Gaussian fluctuations. We show that ensembles of initial states with quasi-long-range correlations lead to superdiffusive charge transfer with a tunable dynamical exponent. We substantiate our prediction with numerical simulations and discuss how finite time and finite size effects might cause deviations.

Figure 1

Folded $XXZ$ automaton. (a) Circuit geometry for the $XXZ$ automaton follows a staircase pattern. (b) The rules for the updates in the automaton. One can see that they conserve the number of domain walls. (c) A highlighted magnon trajectory traversing ballistically through domains.

Figure 2

Charge variance versus stability parameter. (a) Variance of the charge transfer as a function of time shown for stability parameter $\alpha =1.4$, compared to initial states drawn from an infinite temperature, i.e., uncorrelated, ensemble. Note that the fluctuations for $\alpha =1.4$ grow faster than those in the thermal ensemble indicating superdiffusive behavior. (b) Dynamical exponents $z_{\mathrm{var}}$ versus time obtained via log derivatives for two different values of $\alpha$. (c) $z_{\mathrm{var}}$ obtained via log-derivatives at a late time $t=500$. The orange point $\alpha =2$ is obtained by sampling states from an infinite temperature ensemble. One sees that the prediction $z_{\mathrm{var}}={\alpha }^2/2$ is roughly consistent with the numerics. The data are averaged over ${10}^8$ initial states and the scaling parameter $c=1$.

Figure 3

Dynamical exponent finite size effects. (a) One sees that at short times the dynamical exponent of the variance of the charge differs for different scaling parameters but at longer times converges to the same value. Here, data were averaged ${10}^7$ initial states. (b) The domain wall dynamical exponent as a function of time features much stronger finite sizes, as the exponent does not appear to converge until $\sim {10}^3$, which is an order of magnitude larger than the variance of the charge transfer. Here, results were averaged over ${10}^6$ realizations.