Nearly Linear Light Cones in Long-Range Interacting Quantum Systems

In nonrelativistic quantum theories with short-range Hamiltonians, a velocity $v$ can be chosen such that the influence of any local perturbation is approximately confined to within a distance $r$ until a time $t\sim r/v$, thereby defining a linear light cone and giving rise to an emergent notion of locality. In systems with power-law ($1/r^{\alpha }$) interactions, when $\alpha$ exceeds the dimension $D$, an analogous bound confines influences to within a distance $r$ only until a time $t\sim (\alpha /v)\log r$, suggesting that the velocity, as calculated from the slope of the light cone, may grow exponentially in time. We rule out this possibility; light cones of power-law interacting systems are bounded by a polynomial for $\alpha >2D$ and become linear as $\alpha \to \infty$. Our results impose strong new constraints on the growth of correlations and the production of entangled states in a variety of rapidly emerging, long-range interacting atomic, molecular, and optical systems.

Figure 1

(a) In a short-range interacting system, perturbing a single spin at $t=r=0$ can only influence another spin (green connection) if it falls within a causal region bounded by a linear light cone ($t\sim r$) [1]. (b) Existing bounds for power-law interacting systems [12, 17] result in a logarithmic light cone ($t\sim \log r$) at large distances and times, and thus the maximum velocity grows exponentially in time. (c) We show that light cones of power-law interacting systems are bounded by a polynomial which becomes increasingly linear for shorter-range interactions.

Figure 2

Schematic illustration of a Lieb-Robinson-type bound. (a) Heisenberg picture. The time evolution of an operator $A$ is bounded by a series in which repeated applications of the Hamiltonian connect site $i$ to site $j$. (b) Interaction picture. A similar series can be used to bound the dynamics induced by the interaction-picture time-evolution operator, but now the operator $A$ and the interaction terms in the Hamiltonian are spread out over a light-cone radius of the short-range Hamiltonian.

Figure 3

(a) Schematic representation of the term $\dots \parallel {\mathcal{W}}_{{\xi }_1}\parallel D({\xi }_1,{\xi }_2)\parallel {\mathcal{W}}_{{\xi }_2}\parallel \dots$ in Eq. (12). Each green line represents a single term in the interaction-picture Hamiltonian, and the operators at the endpoints are supported over a ball or radius $R(t)+m\chi$ (gray disks). (b) In deriving a bound, the additional summations over the sizes of the supports of each operator generate exponentially decaying connections between successive terms [Eq. (13)].