Topological Surface States in Three-Dimensional Magnetic Insulators (original) (raw)

2008, Physical Review Letters

An electron moving in a magnetically ordered background feels an effective magnetic field that can be both stronger and more rapidly varying than typical externally applied fields. One consequence is that insulating magnetic materials in three dimensions can have topologically nontrivial properties of the effective band structure. For the simplest case of two bands, these "Hopf insulators" are characterized by a topological invariant as in quantum Hall states and Z2 topological insulators, but instead of a Chern number or parity, the underlying invariant is the Hopf invariant that classifies maps from the 3-sphere to the 2-sphere. This paper gives an efficient algorithm to compute whether a given magnetic band structure has nontrivial Hopf invariant, a double-exchange-like tight-binding model that realizes the nontrivial case, and a numerical study of the surface states of this model. PACS numbers: 73.20.At, 03.65.Vf Recent theoretical and experimental work has shown that there exist nonmagnetic band insulators in which spin-orbit coupling plays a role similar to that of the magnetic field in the integer quantum Hall effect (IQHE). In two dimensions [1], these "topological insulators" have robust edge states, observed in HgTe/(Hg,Cd)Te heterostructures , and are predicted to show a spin quantum Hall effect. The existence of a genuinely threedimensional topological insulator phase [3, 4, 5] with protected surface states, recently observed in Bi 0.9 Sb 0.1 [6], is rather surprising because the IQHE does not have a fully three-dimensional version, but only layered versions of the 2D case. Both 2D and 3D topological insulators are nonmagnetic, and in fact unbroken time-reversal invariance is required for the edge state to remain gapless. The edge or surface states of topological insulators and IQHE states exist because there are topological invariants that distinguish these insulating states from ordinary insulators, and across a boundary between one of these states and an ordinary insulator, the energy gap must close.