Topological invariants for interacting topological insulators: I. Efficient numerical evaluation scheme and implementations

TL;DR

This paper introduces an efficient numerical scheme for calculating topological invariants in interacting topological insulators using Green's functions, addressing finite-size effects and leveraging symmetry to improve accuracy and reduce computational effort.

Contribution

It presents a novel periodization method and demonstrates its effectiveness in computing Z2 and spin Chern numbers in interacting systems, advancing numerical techniques in topological matter.

Findings

Periodization overcomes finite-size effects in calculations.

Symmetry properties reduce computational complexity.

Scheme successfully applied to two 2D models of interacting TIs.

Abstract

The aim of this series of two papers is to discuss topological invariants for interacting topological insulators (TIs). In the first paper (I), we provide a paradigm of efficient numerical evaluation scheme for topological invariants, in which we demystify the procedures and techniques employed in calculating Z2 invariant and spin Chern number via zero-frequency single-particle Green's function in quantum Monte Carlo (QMC) simulations. Here we introduce a periodization process to overcome the ubiquitous finite-size effect, so that the calculated spin Chern number shows ideally quantized values. We also show that making use of symmetry properties of the underlying systems can greatly reduce the computational effort. To demonstrate the effectiveness of our numerical evaluation scheme, especially the periodization process, of topological invariants, we apply it on two independent two-dimensional models of interacting topological insulators. In the subsequent paper (II), we apply the scheme developed here to wider classes of models of interacting topological insulators, for which certain limitation of constructing topological invariant via single-particle Green's functions will be presented.