Scaling Theory of Topological Phase Transitions

TL;DR

This paper introduces a scaling theory for topological phase transitions, using a curvature function and renormalization group approach to identify critical points and universal behavior in topologically ordered systems.

Contribution

It proposes a novel scaling procedure for the curvature function in inversion symmetric systems to analyze topological phase transitions.

Findings

The RG equations for energy parameters are derived from the scaling procedure.

A length scale from the curvature function characterizes critical behavior.

Universal critical behavior is observed across various examined systems.

Abstract

Topologically ordered systems are characterized by topological invariants that are often calculated from the momentum space integration of a certain function that represents the curvature of the many-body state. The curvature function may be Berry curvature, Berry connection, or other quantities depending on the system. Akin to stretching a messy string to reveal the number of knots it contains, a scaling procedure is proposed for the curvature function in inversion symmetric systems, from which the topological phase transition can be identified from the flow of the driving energy parameters that control the topology (hopping, chemical potential, etc.) under scaling. At an infinitesimal operation, one obtains the renormalization group (RG) equations for the driving energy parameters. A length scale defined from the curvature function near the gap-closing momentum is suggested to characterize the scale invariance at critical points and fixed points, and displays a universal critical behavior in a variety of systems examined.