Short-ranged interaction effects on $Z_2$ topological phase transitions: The perturbative mean-field method

TL;DR

This paper introduces an analytical method combining perturbative and self-consistent mean-field approaches to accurately predict interaction-driven topological phase transitions in $Z_2$ topological insulators, surpassing traditional methods.

Contribution

The authors develop a novel combined perturbative and mean-field method to analyze interaction effects on topological phase transitions, improving upon existing approaches.

Findings

Method accurately predicts topological phase transitions beyond traditional mean-field and perturbative methods.

Results agree with quantum Monte Carlo simulations on the Kane-Mele model.

Interactions can either stabilize or destabilize the topological phase depending on Hamiltonian symmetries.

Abstract

Time-reversal symmetric topological insulator is a novel state of matter that a bulk-insulating state carries dissipationless spin transport along the surfaces, embedded by the $Z_2$ topological invariant. In the noninteracting limit, this exotic state has been intensively studied and explored with realistic systems, such as HgTe/(Hg,Cd)Te quantum wells. Yet an interacting topological insulator is still an elusive subject, and most related analyses rely on the mean-field approximation and numerical simulations. Among the approaches, the mean-field approximation fails to predict the topological phase transition, in particular at intermediate interaction strength without spontaneously breaking symmetry. In this review, we develop an analytical approach based on a combined perturbative and self-consistent mean-field treatment of interactions that is capable of capturing topological phase transitions beyond either method when used independently. As an illustration of the method, we study the effects of short-ranged interactions on the $Z_2$ topological insulator phase, also known as the quantum spin Hall phase, in three generalized versions of the Kane-Mele model at half-filling on the honeycomb lattice. The results are in excellent agreement with quantum Monte Carlo calculations on the same model, and cannot be reproduced by either a perturbative treatment or a self-consistent mean-field treatment of the interactions. Our analytical approach helps to clarify how the symmetries of the one-body terms of the Hamiltonian determine whether interactions tend to stabilize or destabilize a topological phase. Moreover, our method should be applicable to a wide class of models where topological transitions due to interactions are in principle possible, but are not correctly predicted by either perturbative or self-consistent treatments.