Lieb-Robinson bounds on $n$-partite connected correlation functions

Lieb and Robinson provided bounds on how fast bipartite connected correlations can arise in systems with only short-range interactions. We generalize Lieb-Robinson bounds on bipartite connected correlators to multipartite connected correlators. The bounds imply that an $n$-partite connected correlator can reach unit value in constant time. Remarkably, the bounds also allow for an $n$-partite connected correlator to reach a value that is exponentially large with system size in constant time, a feature which stands in contrast to bipartite connected correlations. We provide explicit examples of such systems.

Figure 1

A typical three-body system. Each dot represents one site. There are three relevant length scales $r_{12}$, $r_{23}$, and $r_{31}$. Which length scale will define the three-body Lieb-Robinson bound?

Figure 2

A geometry where $n$ sites (blue dots) are divided into two cliques such that the clique size $a$ is much smaller than the distance $R$ between cliques.

Figure 3

$n$-qubit cluster states represented by one-dimensional graphs of $n$ vertices. (a) Only consecutive vertices are connected by edges of length 1. (b) Some edges are longer than 1 but interactions are still local.

Figure 4

Time evolution of the $n$-point connected correlator $u_2(Z_1,Z_2,\cdots ,Z_n)={\left[{sin}^2\left(2t\right)\right]}^{n-1}$ of the state in Fig. 3 for different $n$. Here we plot the time-dependent correlator for a few values of $n$. In the limit of large $n$ the correlator remains zero for most of the time before briefly jumping to 1 at $t=\frac{\pi }{4}$.