Entanglement Entropy beyond the Free Case

We present a perturbative method to compute the ground state entanglement entropy for interacting systems. We apply it to a collective model of mutually interacting spins in a magnetic field. At the quantum critical point, the entanglement entropy scales logarithmically with the subsystem size, the system size, and the anisotropy parameter. We determine the corresponding scaling prefactors and evaluate the leading finite-size correction to the entropy. Our analytical predictions are in perfect agreement with numerical results.

Figure 1

Entanglement entropy as a function of the magnetic field, with fixed $\gamma =1/4$, $\tau =1/4$ and $N=32$, 64, 128, 256 (from numerics) and $\infty$ (${\mathcal{E}}^{(0)}$). Arrows indicate the behavior of the finite-size correction in various regions. For $h<1$, we have plotted ${\mathcal{E}}^{(0)}+\ln 2$ for the reason given in the text. The inset is a zoom around $h=\sqrt{\gamma }$ where ${\mathcal{E}}^{(0)}=0$, for $N=64$ (black line) and $\infty$ (red line).

Figure 2

Exponents ${\chi }_{\gamma }$ and ${\chi }_{\tau }$ as a function of $1/N$ obtained from numerical diagonalization of $H$. For clarity, we plotted $2{\chi }_{\tau }$ and $6{\chi }_{\gamma }$ which are expected to be equal to 1 in the thermodynamic limit (dotted lines are guides for the eyes). Inset: entropy as a function of $\ln N$ at fixed $\gamma$ and $\tau$. The dotted line has a slope ${\chi }_N=1/6$.

Figure 3

Behavior of $N[{\mathcal{E}}_{\mathrm{num}}-{\mathcal{E}}^{(0)}]$ as a function of $h$ for fixed $\tau =1/4$ and $N=32$, 64, 128, 256 [from numerics (black lines)] and $\infty$ (${\mathcal{E}}^{(1)}$ obtained from the perturbative expansion (red lines). Inset: comparison between ${\mathcal{E}}^{(1)}$ (○) and its Taylor expansion (20) (solid line).