Characterizing the weak topological properties: Berry phase point of view

TL;DR

This paper develops classification schemes for two-dimensional topological phases with trivial Chern number but nontrivial edge states, using Berry phase analysis and Wilson loop methods, with applications to Wilson-Dirac models.

Contribution

It introduces new classification methods for weak topological phases based on Berry phase and Wilson loop analysis, extending previous strong topological phase classifications.

Findings

Weak topological phases can be characterized by quantized Berry phase pi.

Graphical methods successfully classify weak properties in anisotropic and next-nearest neighbor models.

Weak properties are linked to specific boundary-dependent edge states.

Abstract

We propose classification schemes for characterizing two-dimensional topological phases with nontrivial weak indices. Here, "weak" implies that the Chern number in the corresponding phase is trivial, while the system shows edge states along specific boundaries. As concrete examples, we analyze different versions of the so-called Wilson-Dirac model with (i) anisotropic Wilson terms, (ii) next nearest neighbor hopping terms, and (iii) a superlattice generalization of the model, here in the tight-binding implementation. For types (i) and (ii) a graphic classification of strong properties is successfully generalized for classifying weak properties. As for type (iii), weak properties are attributed to quantized Berry phase pi along a Wilson loop.