Lieb-Robinson Bounds and the Speed of Light from Topological Order

We apply the Lieb-Robinson bounds technique to find the maximum speed of interaction in a spin model with topological order whose low-energy effective theory describes light [see X.-G. Wen, Phys. Rev. B 68, 115413 (2003)]. The maximum speed of interactions in two dimensions is bounded from above by less than $e$ times the speed of emerging light, giving a strong indication that light is indeed the maximum speed of interactions. This result does not rely on mean field theoretic methods. In higher spatial dimensions, the Lieb-Robinson speed is conjectured to increase linearly with the dimension itself. The implications for the horizon problem in cosmology are discussed.

Figure 1

A $2D$-dimensional rotor lattice. To every plaquette $p$ is associated a rotor operator $W_p$ as a function of the variables ${\theta }_{ij}$. The graph $G$ is the one drawn in thin black lines. The graph $G^{\prime }$ is the graph with black and blue (lighter, bigger) dots as vertices and blue thin lines as edges. The red dashed line shows a path of length $n=22$ from the point $P$ to the point $Q$ which are at a distance $2d(P,Q)=8$ on $G^{\prime }$ or $d(P,Q)=4$ on $G$. These paths contain alternating link and plaquette operators.