On the M\"obius transformation in the entanglement entropy of fermionic chains

TL;DR

This paper explores the role of M"obius transformations on Riemann surfaces in understanding the symmetries and dualities of entanglement entropy in fermionic chains, supporting its use in analyzing phase transitions.

Contribution

It provides a comprehensive analysis of how M"obius transformations affect the Riemann surface associated with fermionic entanglement entropy, revealing the origin of observed symmetries and dualities.

Findings

Identifies the action of M"obius transformations on the Riemann surface.

Uncovers the origin of symmetries and dualities in entanglement entropy.

Supports entanglement entropy as a tool for phase transition analysis.

Abstract

There is an intimate relation between entanglement entropy and Riemann surfaces. This fact is explicitly noticed for the case of quadratic fermionic Hamiltonians with finite range couplings. After recollecting this fact, we make a comprehensive analysis of the action of the M\"obius transformations on the Riemann surface. We are then able to uncover the origin of some symmetries and dualities of the entanglement entropy already noticed recently in the literature. These results give further support for the use of entanglement entropy to analyse phase transition.