Renyi Entropy and Geometry

TL;DR

The paper addresses the problem of universal terms in $(3+1)$-dimensional Rényi entropies for general entangling geometries by proposing a geometric, $q$-dependent structure involving functions $f_a(q)$, $f_b(q)$, and $f_c(q)$. The authors test the conjecture through direct numerical computations in free scalar and fermion theories and by relating 4D massless Rényi entropies across cylindrical entangling surfaces to 3D massive Rényi entropies across circular surfaces, including a mapping to thermal free energy on $S^1\times\mathbb{H}^2$ for massless cases. They find evidence that $f_b(q)=f_c(q)$ for these theories and relate the 4D universal term to the large-$mR$ expansion in 3D via $(C_{-1}^q)^{\text{scalar}}= -\pi f_b(q)/240$ and $(C_{-1}^q)^{\text{fermion}}= -\pi f_b(q)/480$, while also computing massless Renyi entropies and perimeter-law coefficients $\beta^q$ that agree with analytic expectations. Together, these results advance the understanding of higher-dimensional Renyi entropies, their connection to Weyl data, and their behavior under massive deformations and geometric variations of the entangling surface.

Abstract

Entanglement entropy in even dimensional conformal field theories (CFTs) contains well-known universal terms arising from the conformal anomaly. Renyi entropies are natural generalizations of the entanglement entropy that are much less understood. Above two spacetime dimensions, the universal terms in the Renyi entropies are unknown for general entangling geometries. We conjecture a new structure in the dependence of the four-dimensional Renyi entropies on the intrinsic and extrinsic geometry of the entangling surface. We provide evidence for this conjecture by direct numerical computations in the free scalar and fermion field theories. The computation involves relating the four-dimensional free massless Renyi entropies across cylindrical entangling surfaces to corresponding three-dimensional massive Renyi entropies across circular entangling surfaces. Our numerical technique also allows us to directly probe other interesting aspects of three-dimensional Renyi entropy, including the massless renormalized Renyi entropy and calculable contributions to the perimeter law.

Figures (3)

• Figure 1: The coefficients $C^q_{-1}$ of the $1/(mR)$ term in the large $mR$ expansion of the Rényi entropy (see \ref{['Sqm']}) for the complex scalar (left) and Dirac fermion (right) theories. These coefficients are related to the $f_b(q)$ coefficients appearing in \ref{['Rd4']} in the $(3+1)$-dimensional Rényi entropy for the massless CFTs through \ref{['c1']}. The orange curves show the predictions from our conjecture that $f_b(q) = f_c(q)$, with the $f_c(q)$ given in \ref{['fac']}. The black points are the results of the numerical calculations. The numerical results agree with the analytic prediction to within 3% for the scalar and fermion theories for all $q$.
• Figure 2: The massless Rényi entropies $S^q$ in the free complex scalar (left) and Dirac fermion (right) theories as functions of the Rényi parameter $q$. The orange curves are the analytic predictions coming from the mapping to $S^1 \times \mathbb{H}^2$ (see \ref{['RenyiResult']}). The black points are the results of the numerical computation. We find that the numerical results agree with the analytic predictions to within 2% for the scalar and fermion theories across all $q$.
• Figure 3: The coefficients $- \beta^q$ in the large-$mR$ expansion of the free-field Rényi entropy \ref{['Sqm']}. An explicit computation of the $\beta^q$ in the wave-guide geometry Lewkowycz:2012qr combined with the assumption that the $\beta^q$-term obeys the perimeter law, as in \ref{['SqmSig']}, leads to the analytic prediction \ref{['betaqFS']} (solid orange) for both the real scalar and Dirac fermion theories. Our numerical results for $\beta^1$ agree with \ref{['betaqFS']} to better than $0.1$%, while the $\beta^q$ at large $q$ agree with the analytic expression to within $\sim$2%.