Modular invariance and entanglement entropy

TL;DR

This paper investigates the behavior of entanglement and Renyi entropies in 2D free conformal field theories at finite temperature and size, focusing on their modular properties and the impact of fermion spin structure summation.

Contribution

It introduces a detailed analysis of modular covariance of entanglement entropy in free 2D CFTs and clarifies the summation over fermion spin structures in different interval limits.

Findings

Entanglement entropy exhibits modular covariance, not invariance.

Two different summation methods over spin structures are identified for small and large intervals.

Results support Bose-Fermi duality in entanglement properties.

Abstract

We study the Renyi and entanglement entropies for free 2d CFT's at finite temperature and finite size, with emphasis on their properties under modular transformations of the torus. We address the issue of summing over fermion spin structures in the replica trick, and show that the relation between entanglement and thermal entropy determines two different ways to perform this sum in the limits of small and large interval. Both answers are modular covariant, rather than invariant. Our results are compared with those for a free boson at unit radius in the two limits and complete agreement is found, supporting the view that entanglement respects Bose-Fermi duality. We extend our computations to multiple free Dirac fermions having correlated spin structures, dual to free bosons on the Spin(2d) weight lattice.