Scaling Theory of ${\mathbb Z}_{2}$ Topological Invariants

TL;DR

This paper introduces two scaling schemes for ${ m Z}_2$ topological invariants in inversion-symmetric topological insulators and superconductors, enabling the detection of topological phase transitions via renormalization group flow analysis.

Contribution

It proposes novel scaling methods based on the Pfaffian's phase gradient and second derivative to identify topological phase transitions in ${ m Z}_2$ systems.

Findings

Pfaffian exhibits universal critical behavior near time-reversal invariant momenta

Scaling schemes successfully track topological phase transitions

Renormalization group flow reveals critical points in models

Abstract

For inversion-symmetric topological insulators and superconductors characterized by ${\mathbb Z}_{2}$ topological invariants, two scaling schemes are proposed to judge topological phase transitions driven by an energy parameter. The scaling schemes renormalize either the phase gradient or the second derivative of the Pfaffian of the time-reversal operator, through which the renormalization group flow of the driving energy parameter can be obtained. The Pfaffian near the time-reversal invariant momentum is revealed to display a universal critical behavior for a great variety of models examined.