Topological phase transitions of three-dimensional topological insulator without energy gap closing

Abstract

In this paper, we demonstrate an anomalous topological phase transition without closing of bulk energy gap. We find such an effect in a model of three-dimensional (3D) topological insulator (TI) subjected to the in-plane exchange field. The energy spectrum, spin spectrum and momentum-dependent spin Chern numbers are calculated. It is shown that our system realizes both the 3D TI phase and the integer quantum Hall (QH) phase. By varying the strength of exchange field, a series of topological phase transitions takes place and in the mean time the energy gap remains open. However, the spin spectrum is closed at the transition and various topological phases are characterized with different number of nodes in spin spectrum. In a tight-binding form, the surface modes are discussed to confirm with the phase diagram. Particularly for a strong field, we find the flat band edge modes which may provide an opportunity for realizing the two-dimensional (2D) fractional QH effect on the boundary of our 3D system.