Entanglement entropy for a Dirac fermion in three dimensions: vertex contribution

TL;DR

This work analyzes the universal logarithmic contribution to the entanglement entropy in $(2+1)$-dimensional free Dirac theories arising from boundary vertices of a region. By mapping the 3D problem to a 2D spectral problem on a sphere with a cut and relating the Dirac Green function to a scalar counterpart via a mass relation, the authors derive an integral expression for the vertex coefficient $s_D(x)$ and obtain its analytic expansion near equal-angled vertices. The results are compared to the scalar case and to holographic predictions, showing that free fermions and scalars have distinct angular dependences while holography closely approximates the fermionic case within a few percent. The study also confirms strong subadditivity constraints for the vertex function and establishes the mass-independence of the logarithmic term, providing benchmarks for CFT classifications and AdS/CFT comparisons in three dimensions.

Abstract

In three dimensions there is a logarithmically divergent contribution to the entanglement entropy which is due to the vertices located at the boundary of the region considered. In this work we find the corresponding universal coefficient for a free Dirac field, and extend a previous work in which the scalar case was treated. The problem is equivalent to find the conformal anomaly in three dimensional space where multiplicative boundary conditions for the field are imposed on a plane angular sector. As an intermediate step of the calculation we compute the trace of the Green function of a massive Dirac field in a two dimensional sphere with boundary conditions imposed on a segment of a great circle.

Figures (2)

• Figure 1: The coefficient $s_D(x)$ of the logarithmic term in the entropy for a Dirac fermion computed by its Taylor series expansion around $x=\pi$ up to order $14$ (solid curve). The dashed lines are the Taylor expansions of lower order, from order $2$ to $12$. Also shown are three points of the curve at angles $\pi/4$, $\pi/2$, and $3/4 \pi$, obtained by direct numerical evaluation of $s_D(x)$ in a lattice (see the text for details).
• Figure 2: Comparison between the coefficients of the logarithmic term for different theories. The picture shows, from top to bottom, the functions $s_S$, $s_D$ and $s_H$ normalized to $s_D$. The differences increase toward the origin where these ratios get $s_S/s_D=1.099$ and $s_H/s_D=.975$ (not shown).