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 $S=\ensuremath{-}\text{tr}\text{ }{\ensuremath{\rho}}_{A}\text{ }\text{log}\text{ }{\ensuremath{\rho}}_{A}$ scales as the area of the boundary of subsystem $A$. The coefficient of this ``area law'' is nonuniversal. However, in the neighborhood 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 $ϵ=3\ensuremath{-}d$ 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 $ϵ\ensuremath{\rightarrow}0$ 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 ${S}_{n}=\frac{1}{1\ensuremath{-}n}\text{log}\text{ }\text{tr}\text{ }{\ensuremath{\rho}}_{A}^{n}$ in $ϵ$ and large-$N$ expansions. For $N\ensuremath{\rightarrow}\ensuremath{\infty}$, 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.