Hopf insulators and their topologically protected surface states

Abstract

Three-dimensional (3D) topological insulators in general need to be protected by certain kinds of symmetries other than the presumed U (1) charge conservation. A peculiar exception is the Hopf insulators which are 3D topological insulators characterized by an integer Hopf index. To demonstrate the existence and physical relevance of the Hopf insulators, we construct a class of tight-binding model Hamiltonians which realize all kinds of Hopf insulators with arbitrary integer Hopf index. These Hopf insulator phases have topologically protected surface states and we numerically demonstrate the robustness of these topologically protected states under general random perturbations without any symmetry other than the U (1) charge conservation that is implicit in all kinds of topological insulators.Topological phases of matter may be divided into two classes: the intrinsic ones and the symmetry protected ones 1 . Symmetry protected topological (SPT) phases are gapped quantum phases that are protected by symmetries of the Hamiltonian and cannot be smoothly connected to the trivial phases under perturbations that respect the same kind of symmetries. Intrinsic topological (IT) phases, on the other hand, do not require symmetry protection and are topologically stable under arbitrary perturbations. Unlike SPT phases, IT phases may have exotic excitations bearing fractional or even non-Abelian statistics in the bulk 2 . Fractional 3 quantum Hall states and spin liquids 4 belong to these IT phases. Remarkable examples of the SPT phases include the well known 2D and 3D topological insulators and superconductors protected by time reversal symmetry 5-7 , and the Haldane phase of the spin-1 chain protected by the SO(3) spin rotational symmetry 8 . For interacting bosonic systems with on-site symmetry G, distinct SPT phases can be systematically classified by group cohomology of G 1 , while for free fermions, the SPT phases can be systematically described by K-theory or homotopy group theory 9 , which leads to the well known periodic table for topological insulators and superconductors 10,11 .Most 3D topological insulators have to be protected by some other symmetries 10,11 , such as time reversal, particle hole or chrial symmetry, and the U (1) charge conservation symmetry 12 . A peculiar exception occurs when the Hamiltonian has just two effective bands. In this case, interesting topological phases, the so-called Hopf insulators 13 , may exist. These Hopf insulator phases have no symmetry other than the prerequisite U (1) charge conservation. To elucidate why this happens, let us consider a generic band Hamiltonian in 3D with m filled bands and n empty bands. Without symmetry constraint, the space of such Hamiltonians is topologically equivalent to the Grassmannian manifold G m,m+n and can be classified by the homotopy group of this Grassmannian 11 . Since the homotopy group π 3 (G m,m+n ) = {0} for all (m, n) = (1, 1), there exists no nontrivial topological phase in general. However, when m = n = 1, G 1,2 is topologically equival...

New p- and n-type ferromagnetic semiconductors: Cr-doped BaZn2As2

“…According to first-principles calculations, the reason for this is the same as that for Li(Zn,Mn)As. 6 Recently, a Cr-doped p-type I−II−V DMS, Li(Zn,Cr)As with T c about 218 K has been reported in experiment, 7 which is higher than that for (Ga,Mn)As.…”

“…The introduction of holes was presumably because of the excess Li + in substitutional Zn 2+ sites. 5 Later, another I−II−V DMS, i.e., Li(Zn,Mn)P was reported in an experiment with T c = 34 K. 6 Li(Zn,Mn)P with excess Li was determined to be of p-type as well in the experiment. According to first-principles calculations, the reason for this is the same as that for Li(Zn,Mn)As.…”

Symmetry-protected topological Hopf insulator and its generalizations

“…4,5, but the key symmetry that protects the Hopf insulator was not identified. Without a proper symmetry, the Hopf insulator is actually trivial, which we will explain later in this paper.…”

“…4,5, but one has to study it with caution. The Brillouin zone of a 3d lattice model is a three-torus T 3 .…”

“…Without a proper symmetry, the Hopf insulator is actually trivial, which we will explain later in this paper. We demonstrate that the topological nature of the 3d Hopf insulator state, previously described only by the homotopy group π 3 [S 2 ] = Z in the simplest two band model [4,5], is more generally given by π 3 [Sp(2N )/U(N )] = Z 2 in a multi-band model [6], which is the key to stabilizing this state under physical conditions. As long as the system has the translation and a C particle-hole symmetry (to be defined later), this Hopf insulator has a Z 2 classification.…”