Geometry and entanglement entropy of surfaces in loop quantum gravity

In loop quantum gravity, the area element of embedded spatial surfaces is given by a well-defined operator. We further characterize the quantized geometry of such surfaces by proposing definitions for operators quantizing scalar curvature and mean curvature. By investigating their properties, we shed light on the nature of the geometry of surfaces in loop quantum gravity. We also investigate the entanglement entropy across surfaces in the case where spin network edges are running within the surface. We observe that, on a certain class of states, the entropy gradient across a surface is proportional to the mean curvature. In particular, the entanglement entropy is constant for small deformations of a minimal surface in this case.

Figure 1

Vertex configurations.

Figure 2

Three possible configurations for circles on a surface in a given spin network state.

Figure 3

Vertex type for the discussion of intrinsic curvature.

Figure 4

This plot shows the Ricci scalar $R$ in units of $l_P^{-2}$ in the case where $j^d=0$. In this case, we necessarily have $j^u=j^{u+d}$, which allowed us to reduce the plot by one dimension.

Figure 5

The Ricci scalar $R$ was plotted for a fixed value of $j^d=9$ as a function of $j^u$ and $j^{u+d}$. It shows the behavior for $j^u$ growing large compared to $j^d$.

Figure 6

The coverage angle $\alpha$ was plotted for a fixed value of $j^d=60$ as a function of $j^u$ and $j^{u+d}$. The range was chosen such that the maximal $j^u$ is still of the same order of magnitude as $j^d$.

Figure 7

Defect angle $\lambda$ plotted for a fixed value of $j^d=60$ as a function of $j^u$ and $j^{u+d}$. The range was chosen such that the maximal $j^u$ is still of the same order of magnitude as $j^d$.

Figure 8

This plot shows the scalar extrinsic curvature in units of $l_P^{-1}$ for the case where $j^d=0$. Here, we necessarily have $j^u=j^{u+d}$ which allowed us to reduce the plot by one dimension.

Figure 9

This plot shows the scalar extrinsic curvature in units of $l_P^{-1}$ for fixed $j^d=9$.

Figure 10

The discrete entropy gradient at the three-valent vertex is plotted against its mean curvature. For (a) $j^d=9$ is chosen and $j^u$ ranges from 9 to 70. In (b) we have set $j^d=60$ and $j^u=60,\dots ,80$. There are different branches, corresponding to the possible choices of $j^{u+d}$. In blue we plotted a line through the origin with a slope of 2. One observes that for small mean curvatures there is a linear regime, whereas for larger values of $H$ the dependence gets highly nonlinear.