Persistence of Locality in Systems with Power-Law Interactions

Motivated by recent experiments with ultracold matter, we derive a new bound on the propagation of information in $D$-dimensional lattice models exhibiting $1/r^{\alpha }$ interactions with $\alpha >D$. The bound contains two terms: One accounts for the short-ranged part of the interactions, giving rise to a bounded velocity and reflecting the persistence of locality out to intermediate distances, whereas the other contributes a power-law decay at longer distances. We demonstrate that these two contributions not only bound but, except at long times, qualitatively reproduce the short- and long-distance dynamical behavior following a local quench in an $XY$ chain and a transverse-field Ising chain. In addition to describing dynamics in numerous intractable long-range interacting lattice models, our results can be experimentally verified in a variety of ultracold-atomic and solid-state systems.

Figure 1

Illustration of the quench observable $Q_r(t)$. At time $t=0$, one spin (shown in blue) is perturbed by a unitary operator $U$. The effect on another spin (shown in red) a distance $r$ away, characterized by the expectation value of an operator $A$, is measured at a later time $t$.

Figure 2

(a) Illustration of the causal region (shaded) resulting from Eq. (2) for the case $v_1=v_2=v$ and $\mu =1/2$. The boundary switches from linear to logarithmic at a critical $r_c$ satisfying $r_c\sim \alpha \log r_c$. (b) The decay of the signal outside the causal region changes from exponential to algebraic at $r_c$. (c) The Hastings-Koma bound [8] leads to a logarithmically bounded causal region that expands with decreasing interaction range (increasing $\alpha$). (d) The new bound in Eq. (2) gives rise to a causal region that contracts with decreasing interaction range, converging to a linear light cone for $\alpha \to \infty$.

Figure 3

(a) Schematic representation of two hopping processes contributing to the first two terms in Eq. (3). The full amplitude ${\mathcal{J}}_n(i,j)$ is obtained by summing the contribution from all possible $n$th order paths. (b) The largest-magnitude terms contributing to the bound [Eq. (9)] on ${\mathcal{J}}_n(i,j)$ for $n=2,3$.

Figure 4

$Q_r(t)$ following a local quench in (a) a $1/r^{\alpha }$ $XY$ chain (periodic boundary conditions, $N=501$, $t=5$) and (b) a $1/r^{\alpha }$ TFIM chain (open boundary conditions, $N=23$, $B_z=0.5$, $t=1$).