Entanglement entropy of 2D conformal quantum critical points: hearing the shape of a quantum drum

TL;DR

The entanglement entropy of a pure quantum state of a bipartite system A union or logical sumB is defined as the von Neumann entropy of the reduced density matrix obtained by tracing over one of the two parts.

Abstract

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. Critical ground states of local Hamiltonians in one dimension have entanglement that 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 the entanglement entropy for a standard class of $z=2$ quantum critical points in two spatial dimensions with scale invariant ground state wave functions: in addition to a nonuniversal ``area law'' contribution proportional to the size of the $AB$ boundary, there is generically a universal logarithmically divergent correction. This logarithmic term is completely determined by the geometry of the partition into subsystems and the central charge of the field theory that describes the equal-time correlations of the critical wavefunction.

Figures (1)

• 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.