We consider the bipartite entanglement entropy of ground states of extended quantum systems with a large degeneracy. Often, as when there is a spontaneously broken global Lie group symmetry, basis elements of the lowest-energy space form a natural geometrical structure. For instance, the spins of a spin-$1/2$ representation, pointing in various directions, form a sphere. We show that for subsystems with a large number $m$ of local degrees of freedom, the entanglement entropy diverges as $\frac{d}{2}\log m$, where $d$ is the fractal dimension of the subset of basis elements with nonzero coefficients. We interpret this result by seeing $d$ as the (not necessarily integer) number of zero-energy Goldstone bosons describing the ground state. We suggest that this result holds quite generally for largely degenerate ground states, with potential applications to spin glasses and quenched disorder.

• Received 3 April 2011

DOI:https://doi.org/10.1103/PhysRevLett.108.120401