Universal phase diagrams for the quantum spin Hall systems

TL;DR

This paper explains how 3D quantum spin Hall phases emerge from insulators via topological monopoles in momentum space, describing the creation, movement, and annihilation of monopole-antimonopole pairs as external parameters vary.

Contribution

It introduces a topological framework based on monopoles in k space to describe phase transitions between insulators and quantum spin Hall phases in three dimensions.

Findings

Gapless phase arises between insulator and quantum spin Hall phases.

Monopole-antimonopole pairs are created and annihilated during phase transitions.

Topological monopoles govern the emergence of quantum spin Hall phases.

Abstract

We describe how the three-dimensional quantum spin Hall phase arises from the insulator phase by changing an external parameter. In 3D systems without inversion symmetry, a gapless phase should appear between the two phases with a bulk gap. The gapless points are monopoles and antimonopoles (in k space), whose topological nature is the source of this gapless phase. In general, when the external parameter is changed from the ordinary insulator phase, two monopole-antimonopole pairs are created and the system becomes gapless. The gap-closing points (monopoles and antimonopoles) then move in the k space as the parameter is changed further. They eventually annihilate in pairs, with changing partners from the pair creations, and the system opens a gap again, entering into the quantum spin Hall phase.