Weakly Interacting Topological Insulators: Quantum Criticality and Renormalization Group Approach

TL;DR

This paper introduces a renormalization group method to efficiently identify topological phase transitions in weakly interacting topological insulators, revealing critical behavior and the impact of interactions on phase transition properties.

Contribution

It develops a numerically efficient RG approach based on Green's functions to analyze topological phase transitions and criticality in weakly interacting insulators, simplifying calculations and uncovering new statistical insights.

Findings

RG method reduces complex integrations to Green's function calculations

Interactions can alter critical exponents of topological phase transitions

Method applied to 1D and 2D models demonstrating its effectiveness

Abstract

For D-dimensional weakly interacting topological insulators in certain symmetry classes, the topological invariant can be calculated from a D- or (D+1)-dimensional integration over a certain curvature function that is expressed in terms of single-particle Green's functions. Based on the divergence of curvature function at the topological phase transition, we demonstrate how a renormalization group approach circumvents these integrations and reduces the necessary calculation to that for the Green's function alone, rendering a numerically efficient tool to identify topological phase transitions in a large parameter space. The method further unveils a number of statistical aspects related to the quantum criticality in weakly interacting topological insulators, including correlation function, critical exponents, and scaling laws, that can be used to characterize the topological phase transitions driven by either interacting or noninteracting parameters. We use 1D class BDI and 2D class A Dirac models with electron-electron and electron-phonon interactions to demonstrate these principles, and find that interactions may change the critical exponents of the topological insulators.