Free Fermion Cyclic/Symmetric Orbifold CFTs and Entanglement Entropy

TL;DR

This paper analyzes two-dimensional free fermion orbifold CFTs, deriving partition functions and entanglement entropy expressions, revealing complex replica contributions, and exploring implications for symmetric and holographic CFTs.

Contribution

It constructs twist operators for cyclic and symmetric orbifolds, relates partition functions via Hecke operators, and computes entanglement entropy in finite and quenched states.

Findings

Partition function of Z_2 orbifold on a torus derived

Explicit relation between cyclic and symmetric orbifold partition functions

Entanglement entropy computed for finite temperature and quenched states

Abstract

In this paper we study the properties of two-dimensional CFTs defined by cyclic and symmetric orbifolds of free Dirac fermions, especially by focusing on the partition function and entanglement entropy. Via the bosonization, we construct the twist operators which glue two complex planes to calculate the partition function of Z_2 orbifold CFT on a torus. We also find an expression of Z_N cyclic orbifold in terms of Hecke operators, which provides an explicit relation between the partition functions of cyclic orbifolds and those of symmetric ones. We compute the entanglement entropy and Renyi entropy in cyclic orbifolds on a circle both for finite temperature states and for time-dependent states under quantum quenches. We find that the replica method calculation is highly non-trivial and new because of the contributions from replicas with different boundary conditions. We find the full expression for the Z_2 orbifold and show that the periodicity gets doubled. Finally, we discuss extensions of our results on entanglement entropy to symmetric orbifold CFTs and make a heuristic argument towards holographic CFTs.