Entanglement Entropy of 2D Conformal Quantum Critical Points: Hearing the Shape of a Quantum Drum

The entanglement entropy of a pure quantum state of a bipartite system $A\cup B$ is defined as the von Neumann entropy of the reduced density matrix obtained by tracing over one of the two parts. In one dimension, the entanglement of critical ground states diverges logarithmically in the subsystem size, with a universal coefficient that for conformally invariant critical points is related to the central charge of the conformal field theory. We find that the entanglement entropy of a standard class of $z=2$ conformal quantum critical points in two spatial dimensions, in addition to a nonuniversal “area law” contribution linear in the size of the $AB$ boundary, generically has a universal logarithmically divergent correction, which is completely determined by the geometry of the partition and by the central charge of the field theory that describes the critical wave function.

Figure 1

The logarithmic contribution to entanglement entropy for three partitions of a disk. Partitions (a) and (c) do not change the total Euler characteristic, while partition (b) does. Note that it is the partition length scale, which may be finite even if $A$ or $B$ is infinite, that sets $R$ in the logarithm.