Scaling behavior in a multicritical one-dimensional topological insulator

TL;DR

This paper investigates the scaling behavior at multicritical points in a one-dimensional topological insulator model, revealing universal critical exponents and gap-closing phenomena through scaling and renormalization group analyses.

Contribution

It introduces a detailed scaling analysis of topological phase transitions at multicritical points, including critical exponents and a basis-independent correlation function.

Findings

Critical exponents are identical for different transition orders.

Both second- and fourth-order transitions exhibit linear and parabolic gap closures.

A basis-independent correlation function characterizes the topological transition.

Abstract

A class of Aubry-Andr\'e-Harper models of spin-orbit coupled electrons exhibits a topological phase diagram where two regions belonging to the same phase are split up by a multicritical point. The critical lines which meet at this point each defines a topological quantum phase transition with a second-order nonanalyticity of the ground-state energy, accompanied by a linear closing of the spectral gap with respect to the control parameter; except at the multicritical point which supports fourth-order transitions with parabolic gap-closing. Here both types of criticality are characterized through a scaling analysis of the curvature function defined from the topological invariant of the model. We extract the critical exponents of the diverging curvature function at the non-high symmetry points in the Brillouin zone where the gap closes, and also apply a renormalization group approach to the flattening curvature function at high symmetry points. We also derive a basis-independent correlation function between Wannier states to characterize the transition. Intriguingly, we find that the critical exponents and scaling law defined with respect to the spectral gap remain the same regardless of the order of the transition.