Topological transition as a percolation of the Berry curvature

TL;DR

This paper investigates how the sign of the Berry curvature influences the topological phase transition in two-dimensional materials, revealing a percolation phenomenon of the Berry curvature sign during the transition.

Contribution

It introduces the concept of topological transition as a percolation of Berry curvature sign, supported by analysis of multiple models including Haldane and QWZ models.

Findings

Oppositely signed Berry curvature regions are localized in non-trivial topology.

These regions become delocalized and percolate during the transition to trivial topology.

The percolation of Berry curvature sign characterizes the topological phase transition.

Abstract

We first study the importance of the sign of the Berry curvature in the Euler characteristic of the two-dimensional topological material with two bands. Then we report an observation of a character of the topological transition as a percolation of the sign of the Berry curvature. The Berry curvature F has peaks at the Dirac points, enabling us to divide the Brillouin zone into two regions depending on the sign of the F: one with the same sign with a peak and the other with the opposite sign. We observed that when the Chern number is non-zero, the oppositely signed regions are localized. In contrast, in the case of a trivial topology, the oppositely signed regions are delocalized dominantly. Therefore, the oppositely signed region will percolate under the topological phase transition from non-trivial to trivial. We checked this for several models including the Haldane model, the extended Haldane model, and the QWZ model. Our observation may serve as a novel feature of the topological phase transition.