Topological Phase Transitions and Evolution of Boundary States Induced by Zeeman Fields in Second-Order Topological Insulators

Zheng-Yang Zhuang,Zhongbo Yan

doi:10.3389/fphy.2022.866347

Abstract

Second-order topological insulators (SOTIs) are a class of materials hosting gapless bound states at boundaries with dimension lower than the bulk by two. In this work, we investigate the effect of Zeeman field on two- and three-dimensional time-reversal invariant SOTIs. We find that a diversity of topological phase transitions can be driven by the Zeeman field, including both boundary and bulk types. For boundary topological phase transitions, we find that the Zeeman field can change the time-reversal invariant SOTIs to time-reversal symmetry breaking SOTIs, accompanying with the change of the number of robust corner or hinge states. Relying on the direction of Zeeman field, the number of bound states per corner or chiral states per hinge can be either one or two in the resulting time-reversal symmetry breaking SOTIs. Remarkably, for bulk topological phase transitions, we find that the transitions can result in Chern insulator phases with chiral edge states and topological semimetal phases with sharply-localized corner states in two dimensions, and hybrid-order Weyl semimetal phases with the coexistence of surface Fermi arcs and gapless hinge states in three dimensions. Our study reveals that the Zeeman field can induce very rich physics in higher-order topological materials.

Highlights

In the last few years, higher-order topological phases have attracted tremendous interest owing to the emergence of novel boundary physics beyond the description of conventional bulk-boundary correspondence [1–27]
In two dimensions, when the concerned time-reversal invariant second-order topological insulator (SOTI) with each corner harboring two zero-energy bound states is subjected to the Zeeman field, we find that the fate of the zeroenergy corner states depends on the direction, the strength, and the type of Zeeman field
With the increase of field strength, we find that the system can first undergo a boundary topological phase transition and enter a new SOTI phase with the number and the locations of zero-energy bound states depend on the direction of Zeeman field

Summary

INTRODUCTION

In the last few years, higher-order topological phases have attracted tremendous interest owing to the emergence of novel boundary physics beyond the description of conventional bulk-boundary correspondence [1–27]. When Bx < Mη and By < Mη, each corner of the square-lattice sample harbors two zero-energy bound states as the Zeeman field is not strong enough to induce a boundary topological phase transition. This result can be understood by noting that when the two Dirac masses with different origins have the same ratio on two near-neighboring edges, we can redefine the Pauli matrices while maintaining their anticommutative relationship. With the increase of Zeeman field, we have shown that in two dimensions a Chern insulator with chiral edge states will be realized after the bulk energy gap undergoes a closing-andreopening transition. Fermi arcs will appear on the top and bottom z-normal surfaces, indicating a different pattern of the distribution of the boundary states compared to the case with a z-direction Zeeman field

DISCUSSIONS AND CONCLUSION

DATA AVAILABILITY STATEMENT