Entanglement entropy in the $O\left(N\right)$ model

It is generally believed that in spatial dimension $d>1$, the leading contribution to the entanglement entropy $S=-\text{tr}{\rho }_A\text{log}{\rho }_A$ 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 $O\left(N\right)$ model in $1<d<3$. We use an expansion in $ϵ=3-d$ 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 $ϵ\to 0$ 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 $S_n=\frac{1}{1-n}\text{log}\text{tr}{\rho }_A^n$ in $ϵ$ 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.

Figure 1

(Color online) The cylindrical geometry considered in calculation of finite-size correction to the entanglement entropy.

Figure 2

Leading correction to the propagator $\delta {\mathcal{G}}^{1,0}$ due to the boundary perturbation. Here and below, a cross denotes an interaction vertex of $c$.

Figure 3

Corrections to the propagator; (a) $\delta {\mathcal{G}}^{0,1}$ and (b) $\delta {\mathcal{G}}^{2,0}$. Here and below, a dot denotes an interaction vertex of $u$.

Figure 4

Corrections to the propagator $\delta {\mathcal{G}}^{1,1}$.

Figure 5

(Color online) $\beta$ function of the boundary coupling $c_r$ for (a) noninteracting theory $\left(u=0\right)$, (b) interacting theory $n=1$, (c) interacting theory $n<n_c$, and (d) interacting theory $n>n_c$.

Figure 6

Leading contributions to ${⟨{\varphi }^2\left(x\right)⟩}_n$ (denoted by a black square here and below): (a) mean-field result; (b) correction due to the boundary perturbation.

Figure 7

Contributions to ${⟨{\varphi }^2\left(x\right)⟩}_n-{⟨{\varphi }^2\left(x\right)⟩}_1$ at order $u$. The counterterm ${\delta }^{1}c$ is denoted by a circled cross here and below.

Figure 8

Contributions to ${⟨{\varphi }^2\left(x\right)⟩}_n-{⟨{\varphi }^2\left(x\right)⟩}_1$ at order $u^2$ (diagrams involving insertions of $c_r$ are not shown). The counterterm for the coupling $u$ is shown as a circled dot.

Figure 9

Contributions to the partition function at order $u$.

Figure 10

The function $G^{D=2}\left(\phi ,\phi \right)$ determining the dependence of $\gamma$ on the twist $\phi$ [Eq. (5.31)].