An emergent geometric description for a topological phase transition in the Kitaev superconductor model

TL;DR

This paper develops an emergent geometric framework using RG transformations to describe topological phase transitions in the Kitaev superconductor model, linking field theory, geometry, and entanglement entropy.

Contribution

It introduces a novel geometric description of topological phase transitions via an emergent bulk action with an extra dimension representing RG scale.

Findings

Derived a bulk action with an extra dimension for the Kitaev model

Established a metric structure from the Callan-Symanzik equation

Linked entanglement entropy in the geometric and quantum field theory descriptions

Abstract

Resorting to Wilsonian renormalization group (RG) transformations, we propose an emergent geometric description for a topological phase transition in the Kitaev superconductor model. An effective field theory consists of an emergent bulk action with an extra dimension, an ultraviolet (UV) boundary condition for an initial value of a coupling function, and an infrared (IR) effective action with a fully renormalized coupling function. The bulk action describes the evolution of the coupling function along the direction of the extra dimension, where the extra dimension is identified with an RG scale and the resulting equation of motion is nothing but a $\beta-$function. In particular, the IR effective field theory turns out to be consistent with a Callan-Symanzik equation which takes into account both the bulk and IR boundary contributions. This derived Callan-Symanzik equation gives rise to a metric structure. Based on this emergent metric tensor, we uncover the equivalence of the entanglement entropy between the emergent geometric description and the quantum field theory in the vicinity of the quantum critical point.