Nonlocal emergent hydrodynamics in a long-range quantum spin system

Generic short-range interacting quantum systems with a conserved quantity exhibit universal diffusive transport at late times. We employ nonequilibrium quantum field theory and semiclassical phase-space simulations to show how this universality is replaced by a more general transport process in a long-range XY spin chain at infinite temperature with couplings decaying algebraically with distance as $r^{-\alpha }$. While diffusion is recovered for $\alpha >1.5$, longer-ranged couplings with $0.5<\alpha \le 1.5$ give rise to effective classical Lévy flights, a random walk with step sizes drawn from a distribution with algebraic tails. We find that the space-time-dependent spin density profiles are self-similar, with scaling functions given by the stable symmetric distributions. As a consequence, for $0.5<\alpha \le 1.5$, autocorrelations show hydrodynamic tails decaying in time as $t^{-1/(2\alpha -1)}$ and linear-response theory breaks down. Our findings can be readily verified with current trapped ion experiments.

Figure 1

Hydrodynamic tails in the spin autocorrelator. (a) For long-range coupling exponents $\alpha >0.5$, autocorrelations decay algebraically at late times with an exponent that depends on $\alpha$. By contrast, for $\alpha \le 0.5$, hydrodynamic tails are absent. (b) The exponents ${\beta }_{\alpha }$ of the hydrodynamic tail obtained from two different approaches (symbols) agree with the predictions from classical Lévy flights in the thermodynamic limit (dashed curve). Deviations at large $\alpha$ are due to finite-time corrections to scaling which can also be understood from Lévy flights.

Figure 2

Short-time dynamics. We compare spin-2PI results with second-order perturbation theory, Eq. (8). (a) The collapse of the autocorrelator for different exponents $\alpha$ shows that the short-time evolution is independent of $\alpha$ and $L$ when the Hamiltonian is rescaled with ${\mathcal{N}}_{\alpha ,L}$. (b) The unequal-time correlation function for $\alpha \in \{0.75,1,1.5,2\}$ (from top to bottom), shows algebraic tails that are entirely captured by second-order perturbation theory. We used a moving average over five to ten lattice sites to smoothen the results.

Figure 3

Emergent self-similar time evolution. The correlation function $C(j,t)$ obtained from spin-2PI for chains of lengths $L=201$ [(e)–(h) $\alpha =2]$, $L=301$ [(a)–(d) $\alpha =1]$. (a), (e) $C(j,t)$ multiplied by $t^{1/(2\alpha -1)}$ to account for the overall decay expected from Lévy flights shows a diffusive cone for $\alpha =2$, whereas for $\alpha =1$ a ballistic light cone emerges. The contour lines for $\alpha =1,2$ correspond to values $t^{1/(2\alpha -1)}C(j,t)=0.03,{10}^{-4}$, respectively. (b), (f) Rescaling of linearly spaced time slices for $23\le Jt\le 84$ ($\alpha =1$) and $42\le Jt\le 226$ ($\alpha =2$) (lines become darker as time increases) for the same data as in (a) and (e) agrees well with the scaling function expected from classical Lévy flights [Eq. (4)]. The only fitting parameter is the generalized diffusion coefficient. (d), (h) Rescaled time slices ($2\le Jt\le 28$) on a double-logarithmic scale reveal for $\alpha =1$ the heavy tail $\sim y^{-2}$ expected from Lévy flights [Eq. (4)], where the dashed-dotted line is the same fit as in (b). The tail $\sim y^{-4}$ (thick black line) for $\alpha =2$ ($8\le Jt\le 85$) is a finite-time effect also present in classical Lévy flights. (c), (g) Unscaled data.

Figure 4

Generalized diffusion constant. The $\alpha$ dependence of the diffusion constant obtained from fits with the scaling function of Lévy flights [Eq. (4)]. The qualitative behavior follows the Lévy flight prediction $D_{\alpha }\sim c_{\alpha }$ for $\alpha <1.5$.