Entanglement Entropy in the O(N) Model

Abstract

It is generally believed that in spatial dimension (d) > 1 the leading contribution to the entanglement entropy (S = - tr\rho_A log \rho_A) scales as the area of the boundary of subsystem (A). The coefficient of this "area law" is non-universal. However, in the neighbourhood of a quantum critical point (S) is believed to possess subleading universal corrections. In the present work, we study the entanglement entropy in the quantum (O(N)) model in 1 < (d) < 3. We use an expansion in (\epsilon = 3-d) to evaluate i) the universal geometric correction to (S) for an infinite cylinder divided along a circular boundary; ii) the universal correction to (S) due to a finite correlation length. Both corrections are different at the Wilson-Fisher and Gaussian fixed points, and the (\epsilon \to 0) limit of the Wilson-Fisher fixed point is distinct from the Gaussian fixed point. In addition, we compute the correlation length correction to the Renyi entropy (S_n = 1/1-n log tr {\rho_A}^n) in (\epsilon) and large-(N) expansions. For (N \to \infty), this correction generally scales as (N^2) rather than the naively expected (N). Moreover, the Renyi entropy has a phase transition as a function of (n) for (d) close to 3.