Timelike Entanglement Entropy and Phase Transitions in non-Conformal Theories

TL;DR

This paper introduces a holographic formalism for timelike entanglement entropy in non-conformal theories, revealing its complex nature and its role as a probe for confinement and phase transitions.

Contribution

It develops a new holographic approach to timelike entanglement entropy in non-conformal theories, including a method to merge surfaces and analyze phase transition behavior.

Findings

Timelike entanglement entropy is complex-valued with real and imaginary parts from spacelike and timelike surfaces.

Existence of a critical length for connected surfaces in confining theories.

Imaginary part of entropy signals confinement and phase transitions.

Abstract

We propose a holographic formalism for a timelike entanglement entropy in non-conformal theories. This pseudoentropy is a complex-valued measure of information, which, in holographic non-conformal theories, receives contributions from a set of spacelike surfaces and a finite timelike bulk surface with mirror symmetry. We suggest a method of merging the surfaces so that the boundary length of the subregion is exclusively specified by holography. We show that in confining theories, the surfaces can be merged in the bulk at the infrared tip of the geometry and are homologous to the boundary region. The timelike entanglement entropy receives its imaginary and real contributions from the timelike and the spacelike surfaces, respectively. Additionally, we demonstrate that in confining theories, there exists a critical length within which a connected non-trivial surface can exist, and the imaginary part of the timelike entanglement entropy is non-zero. Therefore, the timelike entanglement entropy exhibits unique behavior in confining theories, making it a probe of confinement and phase transitions. Finally, we discuss the entanglement entropy in Euclidean spacetime in confining theories and the effect of a simple analytical continuation from a spacelike subsystem to a timelike one.