Correlation Length, Universality Classes, and Scaling Laws Associated with Topological Phase Transitions

TL;DR

This paper investigates the behavior of correlation functions near topological phase transitions in lattice models, revealing universal critical exponents and scaling laws that define universality classes in different dimensions.

Contribution

It introduces the concept of universality classes for topological phase transitions based on correlation length and symmetry, extending the understanding of critical phenomena in topological systems.

Findings

Correlation length diverges at topological transitions.

Universal critical exponents characterize different universality classes.

Scaling laws are valid even in interacting topological systems.

Abstract

The correlation functions related to topological phase transitions in inversion-symmetric lattice models described by $2\times 2$ Dirac Hamiltonians are discussed. In one dimension, the correlation function measures the charge-polarization correlation between Wannier states at different positions, while in two dimensions it measures the itinerant-circulation correlation between Wannier states. The correlation function is nonzero in both the topologically trivial and nontrivial states, and allows to extract a correlation length that diverges at topological phase transitions. The correlation length and the curvature function that defines the topological invariants are shown to have universal critical exponents, allowing the notion of universality classes to be introduced. Particularly in two dimensions, the universality class is determined by the orbital symmetry of the Dirac model. The scaling laws that constrain the critical exponents are revealed, and are predicted to be satisfied even in interacting systems, as demonstrated in an interacting topological Kondo Insulator.