Entanglement Entropy in the O(N) model

TL;DR

The entanglement entropy in the quantum O(N) model in 1 0 limit of the Wilson-Fisher fixed point is distinct from the Gaussian fixed point and the correlation length correction to the Renyi entropy S_n = (log tr rho^n_A)/(1-n) in epsilon and large-N expansions is computed.

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 -> 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 = (log tr rho^n_A)/(1-n) in epsilon and large-N expansions. For N -> infinity, 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.

Figures (10)

• Figure 1: The cylindrical geometry considered in calculation of finite size correction to the entanglement entropy.
• Figure 2: Leading correction to the propagator $\delta {\cal G}^{1,0}$ due to the boundary perturbation. Here and below, a cross denotes an interaction vertex of $c$.
• Figure 3: Corrections to the propagator, a) $\delta {\cal G}^{0,1}$ and b) $\delta {\cal G}^{2,0}$. Here and below, a dot denotes an interaction vertex of $u$.
• Figure 4: Corrections to the propagator $\delta {\cal G}^{1,1}$.
• Figure 5: $\beta$-function of the boundary coupling $c_r$ for a) Non-interacting theory ($u$ = 0), b) Interacting theory, $n = 1$, c) Interacting theory, $n < n_c$, d) Interacting theory, $n > n_c$.