Topological Insulators in Three Dimensions

Topological Insulators in Three Dimensions

11 Jul 2007 | Liang Fu, C.L. Kane and E.J. Mele
This paper presents a study of three-dimensional (3D) generalizations of the quantum spin Hall (QSH) effect. In two dimensions, the QSH effect is characterized by a single $ Z_2 $ topological invariant, but in three dimensions, there are four such invariants, leading to 16 distinct "topological insulator" phases. These phases are divided into two classes: weak topological insulators (WTIs) and strong topological insulators (STIs). WTI states are fragile and can be connected to band insulators by disorder, while STI states are robust and have surface states that realize a $ 2+1 $ dimensional parity anomaly without fermion doubling, giving rise to a novel "topological metal" surface phase. The authors introduce a tight-binding model on a distorted diamond lattice that realizes both WTI and STI phases. This model allows for the explicit study of surface states. The paper discusses the relevance of this model to real 3D materials, including bismuth. The QSH effect in 3D is generalized by considering the topological invariants and their physical meaning. The $ Z_2 $ invariant for the bulk QSH phase is connected to spin-filtered edge states. The paper also discusses the surface states and their properties, including the surface Fermi arcs and the topological protection of the surface states. The paper concludes by noting that the 4-band diamond lattice model, while simple, may provide insights into the behavior of real crystals. It also suggests that searching for materials in the STI phase, which occur on the "other side of diamond" in the sequence of crystal structures, could be an interesting direction for future research. The authors hope that the exotic surface properties predicted for the STI phase will stimulate further experimental and theoretical efforts.This paper presents a study of three-dimensional (3D) generalizations of the quantum spin Hall (QSH) effect. In two dimensions, the QSH effect is characterized by a single $ Z_2 $ topological invariant, but in three dimensions, there are four such invariants, leading to 16 distinct "topological insulator" phases. These phases are divided into two classes: weak topological insulators (WTIs) and strong topological insulators (STIs). WTI states are fragile and can be connected to band insulators by disorder, while STI states are robust and have surface states that realize a $ 2+1 $ dimensional parity anomaly without fermion doubling, giving rise to a novel "topological metal" surface phase. The authors introduce a tight-binding model on a distorted diamond lattice that realizes both WTI and STI phases. This model allows for the explicit study of surface states. The paper discusses the relevance of this model to real 3D materials, including bismuth. The QSH effect in 3D is generalized by considering the topological invariants and their physical meaning. The $ Z_2 $ invariant for the bulk QSH phase is connected to spin-filtered edge states. The paper also discusses the surface states and their properties, including the surface Fermi arcs and the topological protection of the surface states. The paper concludes by noting that the 4-band diamond lattice model, while simple, may provide insights into the behavior of real crystals. It also suggests that searching for materials in the STI phase, which occur on the "other side of diamond" in the sequence of crystal structures, could be an interesting direction for future research. The authors hope that the exotic surface properties predicted for the STI phase will stimulate further experimental and theoretical efforts.
Reach us at info@study.space
[slides] Topological insulators in three dimensions. | StudySpace