Universal Thermal Corrections to Entanglement Entropy for Conformal Field Theories on Spheres

TL;DR

This work studies universal thermal corrections to entanglement entropy for conformal field theories on ${\mathbb R} \times S^{d-1}$ with a cap-like region, in the presence of a finite-volume mass gap. Using a low-temperature expansion and a mapping to hyperbolic space via the modular Hamiltonian, it derives a universal correction $\delta S_A(T) = g \Delta I_d(\theta_0) e^{-\beta \Delta / R} + o(e^{-\beta \Delta / R})$ that depends only on the mass gap $\Delta$, degeneracy $g$, and cap angle $\theta_0$, plus a recurrence for $I_d(\theta)$. Numerical checks for a conformally coupled scalar show agreement only after replacing $I_d(\theta_0)$ with $I_{d-2}(\theta_0)$, explained by a boundary-term effect that is reconciled via a counter-term; mutual information calculations corroborate the numeric methods. The results reveal a universal, subleading thermal correction to entanglement entropy that is independent of the central charge and offer insights into holographic interpretations and potential extensions to Rényi entropies.

Abstract

We consider entanglement entropy of a cap-like region for a conformal field theory living on a sphere times a circle in d space-time dimensions. Assuming that the finite size of the system introduces a unique ground state with a nonzero mass gap, we calculate the leading correction to the entanglement entropy in a low temperature expansion. The correction has a universal form for any conformal field theory that depends only on the size of the mass gap, its degeneracy, and the angular size of the cap. We confirm our result by calculating the entanglement entropy of a conformally coupled scalar numerically. We argue that an apparent discrepancy for the scalar can be explained away through a careful treatment of boundary terms. In an appendix, to confirm the accuracy of the numerics, we study the mutual information of two cap-like regions at zero temperature.

Figures (3)

• Figure 1: The entanglement entropy difference $\delta S = S_0(T) - S_0(0)$ for a caplike region with angular size $\theta$. The plot demonstrates the cross over between small $T$ and small $\pi - \theta$ behavior. Left: Three dimensional case. From top to bottom, the data points correspond to $RT = 0.025$, 0.05, 0.075, and 0.1. Right: Four dimensional case. From top to bottom, the data points correspond to $RT = 0.05$, 0.075, and 0.1. The curves are the prediction (\ref{['mainresult']}) with $I_d(\theta)$ replaced by $I_{d-2}(\theta)$, as discussed in the text. The big dots mark the low temperature thermal entropy correction $1 + \Delta/RT$. The lattice used had 200 grid points.
• Figure 2: The entanglement entropy difference $\delta S_A = S_A(T) - S_A(0)$ for a cap like region $A$ with angular size $\theta$ in the limit where $T$ is sent to zero first: (left) $d=3$; (right) $d=4$. The curves are the prediction (\ref{['mainresult']}) with $I_d(\theta)$ replaced by $I_{d-2}(\theta)$, as discussed in the text. The points were numerically determined. The lattice used had 200 grid points.
• Figure 3: Mutual information $M_d$ for a conformally coupled scalar with two cap like regions centered around the north and south poles on $S^{d-1}$. The cross ratio $x$ is defined in eq. (\ref{['crangles']}). From top to bottom on the left hand side: $d=3$, 4, 5, and 6. The order is reversed on the right hand side. The straight lines on the right hand side are the analytic predictions by Huerta and Casini Casini:2009sr. The straight lines on the left hand side are the analytic predictions by Cardy Cardy:2013nua ($d=3$, 4) or using his method ($d=5$, 6). Zooming in on the plot reveals that the four curves intersect at six points rather than one point.