title: Entanglement entropy of highly degenerate States and fractal dimensions.
abstract: 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 d/2 logm, 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.
authors.affiliation: Centre for Mathematical Science, City University London, Northampton Square EC1V 0HB, United Kingdom.
authors.forename: Olalla A
authors.initials: OA
authors.lastname: Castro-Alvaredo
authors.forename: Benjamin
authors.initials: B
authors.lastname: Doyon