Lieb-Robinson Light Cone for Power-Law Interactions

The Lieb-Robinson theorem states that information propagates with a finite velocity in quantum systems on a lattice with nearest-neighbor interactions. What are the speed limits on information propagation in quantum systems with power-law interactions, which decay as $1/r^{\alpha }$ at distance $r$? Here, we present a definitive answer to this question for all exponents $\alpha >2d$ and all spatial dimensions $d$. Schematically, information takes time at least $r^{\min \{1,\alpha -2d\}}$ to propagate a distance $r$. As recent state transfer protocols saturate this bound, our work closes a decades-long hunt for optimal Lieb-Robinson bounds on quantum information dynamics with power-law interactions.

Figure 1

The gap in the Lieb-Robinson literature in $d>1$ dimensions. The red solid lines represent the exponent $\gamma$ of the Lieb-Robinson light cone $t\gtrsim r^{\gamma }$ in literature. The green solid lines correspond to the light cone exponents of best-known information-propagating protocols. Accordingly, the green region corresponds to attainable light cone exponents, whereas the red region is forbidden by the known bounds. Our result (red dashed line) closes the gap in our understanding of the Lieb-Robinson light cone.

Figure 2

We study the structure of $V_{2,I}$ by dividing the lattice into hypercubes of length ${\ell }_2$ (labeled by $x$, $y$, and $z$ for example). In the interaction picture, how much each $e^{{\mathcal{L}}_{1}t}{h}_{ij}$ contributes to the pairwise “effective interaction” between two hypercubes depends on how strongly the support of $e^{{\mathcal{L}}_{1}t}{h}_{ij}$ (represented by the shaded area) overlaps with the hypercubes. Because of the bound in Eq. (2), the evolved operator $e^{{\mathcal{L}}_{1}t}{h}_{ij}$ is largely confined to the light cones induced by ${\mathcal{L}}_1$ around $i$ and $j$ (the smallest disks around $i$ and $j$). The component of $e^{{\mathcal{L}}_{1}t}{h}_{ij}$ supported outside this light cone is exponentially suppressed with distance (represented by lighter shade). Consequently, the effective interaction between the hypercubes $x$ and $z$ is exponentially smaller than the one between $x$ and $y$.