Classification of topological insulators and superconductors in three spatial dimensions

Abstract

We systematically study topological phases of insulators and superconductors (SCs) in 3D. We find that there exist 3D topologically non-trivial insulators or SCs in 5 out of 10 symmetry classes introduced by Altland and Zirnbauer within the context of random matrix theory. One of these is the recently introduced Z_2 topological insulator in the symplectic symmetry class. We show there exist precisely 4 more topological insulators. For these systems, all of which are time-reversal (TR) invariant in 3D, the space of insulating ground states satisfying certain discrete symmetry properties is partitioned into topological sectors that are separated by quantum phase transitions. 3 of the above 5 topologically non-trivial phases can be realized as TR invariant SCs, and in these the different topological sectors are characterized by an integer winding number defined in momentum space. When such 3D topological insulators are terminated by a 2D surface, they support a number (which may be an arbitrary non-vanishing even number for singlet pairing) of Dirac fermion (Majorana fermion when spin rotation symmetry is completely broken) surface modes which remain gapless under arbitrary perturbations that preserve the characteristic discrete symmetries. In particular, these surface modes completely evade Anderson localization. These topological phases can be thought of as 3D analogues of well known paired topological phases in 2D such as the chiral p-wave SC. In the corresponding topologically non-trivial and topologically trivial 3D phases, the wavefunctions exhibit markedly distinct behavior. When an electromagnetic U(1) gauge field and fluctuations of the gap functions are included in the dynamics, the SC phases with non-vanishing winding number possess non-trivial topological ground state degeneracies.

20 pages. Changed title, added two tables

References in corpus (11)

• Quantum Spin Hall Insulator State in HgTe Quantum Wells
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• The Helical Liquid and the Edge of Quantum Spin Hall Systems
• Electron transport in disordered graphene
• Resonantly-paired fermionic superfluids
• Topological delocalization of two-dimensional massless Dirac fermions
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• Z2 topological term, the global anomaly, and the two-dimensional symplectic symmetry class of Anderson localization
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• Two-dimensional spin-filtered chiral network model for the Z_2 quantum spin-Hall effect
• Many-body generalization of the Z2 topological invariant for the quantum spin Hall effect

Cited by in corpus (7)

• Topological Phases of Noncentrosymmetric Superconductors: Edge States, Majorana Fermions, and the Non-Abelian Statistics
• Fractional Quantum Hall Effect via Holography: Chern-Simons, Edge States, and Hierarchy
• Topological properties of spin-triplet superconductors and the Fermi surface topology in the normal state
• Ballistic transport in disordered graphene
• Lattice model of three-dimensional topological singlet superconductor with time-reversal symmetry
• Fractional vortex lattice structures in spin triplet superconductors
• First observation of spin-helical Dirac fermions and topological phases in undoped and doped Bi2Te3 demonstrated by spin-ARPES spectroscopy