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We shall denote the nontrivial topological phases in all five symmetry classes by the term "topological insulator," although in the context of superconducting systems (or superfluids) this might be considered a misnomer. In the case of the BdG symmetry classes DIII and CI, as well as for class AIII, when interpreted as a superconductor (Ref.), the term "insulator" refers to the fact that the BCS quasiparticles are fully gapped in the bulk by the mean-field pairing gap. That is, it refers to an insulating behavior as manifested, e.g., by thermal transport properties or by spin transport in superconductors when spin is a good quantum number. Alternatively, a superconductor or a superfluid with nontrivial topological character in BdG fermionic excitations can be called topological superconductor or topological superfluid, respectively.
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see also arXiv:cond-mat/0006360 (unpublished).
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This is not to say that the 3D IQHE on a lattice, discussed by M. Kohmoto, B. I. Halperin, and Y. S. Wu, Phys. Rev. B 10.1103/PhysRevB.45.13488 45, 13488 (1992), is not possible: in the above work by Kohmoto, quantum ground states constructed from filled 3D Bloch states are characterized by a triplet of Chern numbers, each describing the winding of a map from the 2D torus, which is a subspace of 3D BZ, onto Gm,m+n (C). Hence, the 3D IQHE is essentially a layered version of the 2D IQHE. Similarly, in symmetry class AII, there exists a 3D topological state, which consists of layered 2D Z2 topological quantum states, and which has been termed "weak topological insulator" (see Ref.). By analogy, we argue that also in symmetry classes D, C, and DIII (see Table 1) there exists a layered version of the 2D topological quantum states.
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Topological insulators in TR invariant BdG classes were discussed in terms of the Z2 number in Refs., thereby emphasizing the fermion number parity in the ground state (see also Sec. 7). Here, however, we find that an additional discrete symmetry (i.e., PHS) allows one to define an integral winding number ν, which protects an arbitrary number (an arbitrary even number for class CI) of surface Dirac (Majorana for class DIII) fermion states against the opening of a gap.
-
Topological insulators in TR invariant BdG classes were discussed in terms of the Z2 number in Refs., thereby emphasizing the fermion number parity in the ground state (see also Sec. 7). Here, however, we find that an additional discrete symmetry (i.e., PHS) allows one to define an integral winding number ν, which protects an arbitrary number (an arbitrary even number for class CI) of surface Dirac (Majorana for class DIII) fermion states against the opening of a gap.
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In B. Andrei Bernevig and Han-Dong Chen, arXiv:cond-mat/0611766 (unpublished), a topological integer was introduced in the context of the Z2 topological insulator in three dimensions, for a subset of 4×4 Hamiltonians in the symplectic symmetry class. The condition that selects this subset turns out to be PHS. Hence, the relevant symmetry classes studied in the above work can be identified as class DIII or class AIII, rather than the symplectic symmetry class.
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While the topological term of Pruisken type is possible for classes A, C, and D in two dimensions, such a term need not necessarily be realized at a surface of a 3D topological insulator. What distinguishes the Pruisken terms from Z2 topological terms or the WZW terms is their tunability: the Pruisken term depends on one parameter (topological angle), which can be tuned by changing microscopic details.
-
While the topological term of Pruisken type is possible for classes A, C, and D in two dimensions, such a term need not necessarily be realized at a surface of a 3D topological insulator. What distinguishes the Pruisken terms from Z2 topological terms or the WZW terms is their tunability: the Pruisken term depends on one parameter (topological angle), which can be tuned by changing microscopic details.
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Note that the three classes AIII, BDI, and CII possess sublattice symmetry (SLS) and can thus be realized as nearest-neighbor hopping models on a 1D lattice (which is always bipartite). There are two dimerized states, and for a finite lattice one of them has a zero-mode "edge state" at each of the two boundaries. The topological integer specifies the occupation number of such a zero-mode edge state.
-
Note that the three classes AIII, BDI, and CII possess sublattice symmetry (SLS) and can thus be realized as nearest-neighbor hopping models on a 1D lattice (which is always bipartite). There are two dimerized states, and for a finite lattice one of them has a zero-mode "edge state" at each of the two boundaries. The topological integer specifies the occupation number of such a zero-mode edge state.
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It is possible to design an interacting spin model for which a fermionic many-body state constructed from a slave-particle mean-field Hamiltonian (projective construction) is an exact ground state (Refs.). Indeed, in Ref., Kitaev discussed a spin-1/2 model on the honeycomb lattice, whose ground state is constructed from noninteracting Majorana fermions and which lies in the universality class of the Moore-Read Pfaffian state. Following the spirit of Ref. we can construct an exactly solvable spin-3/2 model of Kitaev type on the diamond lattice whose ground state can be obtained from a fermionic ground state of a class DIII topological insulator.
-
It is possible to design an interacting spin model for which a fermionic many-body state constructed from a slave-particle mean-field Hamiltonian (projective construction) is an exact ground state (Refs.). Indeed, in Ref., Kitaev discussed a spin-1/2 model on the honeycomb lattice, whose ground state is constructed from noninteracting Majorana fermions and which lies in the universality class of the Moore-Read Pfaffian state. Following the spirit of Ref. we can construct an exactly solvable spin-3/2 model of Kitaev type on the diamond lattice whose ground state can be obtained from a fermionic ground state of a class DIII topological insulator.
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135
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When using the supersymmetry method for disorder averaging, the resulting theories are, in general, WZW models on GL(1 1) and Osp(2 2) supergroup manifolds, possessing Kac-Moody current algebra symmetry.
-
When using the supersymmetry method for disorder averaging, the resulting theories are, in general, WZW models on GL(1 1) and Osp(2 2) supergroup manifolds, possessing Kac-Moody current algebra symmetry.
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