Conformal perturbation theory and higher spin entanglement entropy on the torus

TL;DR

This paper investigates how chemical potential deformations affect entanglement entropy in a 1+1D free fermion theory on a torus, revealing finite corrections involving modular forms and proposing a consistent integral prescription.

Contribution

It provides the first detailed calculation of order ^2 corrections to entanglement entropy with spin-three chemical potential, including a novel integral prescription and modular form analysis.

Findings

Order ^2 corrections are finite and involve quasi-modular forms.

A new prescription for contact term integrals is proposed and validated.

Winding modes do not contribute to the perturbative expansion, indicating different entanglement structures.

Abstract

We study the free fermion theory in 1+1 dimensions deformed by chemical potentials for holomorphic, conserved currents at finite temperature and on a spatial circle. For a spin-three chemical potential \mu, the deformation is related at high temperatures to a higher spin black hole in hs[0] theory on AdS_3 spacetime. We calculate the order \mu^2 corrections to the single interval Renyi and entanglement entropies on the torus using the bosonized formulation. A consistent result, satisfying all checks, emerges upon carefully accounting for both perturbative and winding mode contributions in the bosonized language. The order \mu^2 corrections involve integrals that are finite but potentially sensitive to contact term singularities. We propose and apply a prescription for defining such integrals which matches the Hamiltonian picture and passes several non-trivial checks for both thermal corrections and the Renyi entropies at this order. The thermal corrections are given by a weight six quasi-modular form, whilst the Renyi entropies are controlled by quasi-elliptic functions of the interval length with modular weight six. We also point out the well known connection between the perturbative expansion of the partition function in powers of the spin-three chemical potential and the Gross-Taylor genus expansion of large-N Yang-Mills theory on the torus. We note the absence of winding mode contributions in this connection, which suggests qualitatively different entanglement entropies for the two systems.