The insulator/Chern-insulator transition in the Haldane model

TL;DR

This paper investigates the transition between normal insulator and Chern insulator in the Haldane model, revealing how physical properties like the density matrix and Wannier functions behave and break down at the phase boundary.

Contribution

It provides a detailed analysis of the behavior of the density matrix and Wannier functions across the insulator-Chern insulator transition, highlighting the divergence of certain localization measures.

Findings

Density matrix decays exponentially in both phases

Power-law decay at the phase boundary

Wannier function spread diverges in the Chern-insulator phase

Abstract

We study the behavior of several physical properties of the Haldane model as the system undergoes its transition from the normal-insulator to the Chern-insulator phase. We find that the density matrix has exponential decay in both insulating phases, while having a power-law decay, more characteristic of a metallic system, precisely at the phase boundary. The total spread of the maximally-localized Wannier functions is found to diverge in the Chern-insulator phase. However, its gauge-invariant part, related to the localization length of Resta and Sorella, is finite in both insulating phases and diverges as the phase boundary is approached. We also clarify how the usual algorithms for constructing Wannier functions break down as one crosses into the Chern-insulator region of the phase diagram.