It has been known that an anti-unitary symmetry such as time-reversal or charge conjugation is needed to realize Z2 topological phases in non-interacting systems. Topological insulators and superconducting nanowires are representative examples of suc
h Z2 topological matters. Here we report the first-known Z2 topological phase protected by only unitary symmetries. We show that the presence of a nonsymmorphic space group symmetry opens a possibility to realize Z2 topological phases without assuming any anti-unitary symmetry. The Z2 topological phases are constructed in various dimensions, which are closely related to each other by Hamiltonian mapping. In two and three dimensions, the Z2 phases have a surface consistent with the nonsymmorphic space group symmetry, and thus they support topological gapless surface states. Remarkably, the surface states have a unique energy dispersion with the Mobius twist, which identifies the Z2 phases experimentally. We also provide the relevant structure in the K-theory.
In a recent paper by Schmitt et al. (arXiv:1402.1091), signs of two exchange interactions of one-dimensional alternating chains in distorted honeycomb systems of Na3Cu2SbO6 and Na2Cu2TeO6 are argued by theoretical calculations. Although the authors r
eport that they have clarified that the interaction J1a is ferromagnetic between the spins within a structural dimer, and J1b is antiferromagnetic between these dimers, these results have been known for a long time by our two papers, which are cited by Schmitt et al.
We construct a minimal generating set of the level 2 mapping class group of a nonorientable surface of genus $g$, and determine its abelianization for $gge4$.
It has been widely believed that half quantum vortices are indispensable to realize topological stable Majorana zero modes and non-Abelian anyons in spinful superconductors/superfluids. Contrary to this wisdom, we here demonstrate that integer quantu
m vortices in spinful superconductors can host topologically stable Majorana zero modes because of the mirror symmetry. The symmetry protected Majorana fermions may exhibit non-Abelian anyon braiding.
Crystal point group symmetry is shown to protect Majorana fermions (MFs) in spinfull superconductors (SCs). We elucidate the condition necessary to obtain MFs protected by the point group symmetry. We argue that superconductivity in Sr2RuO4 hosts a t
opological phase transition to a topological crystalline SC, which accompanies a d-vector rotation under a magnetic field along the c-axis. Taking all three bands and spin-orbit interactions into account, symmetry-protected MFs in the topological crystalline SC are identified. Detection of such MFs provides evidence of the d-vector rotation in Sr2RuO4 expected from Knight shift measurements but not yet verified.
We give a new upper bound on the stable commutator length of Dehn twists in hyperelliptic mapping class groups, and determine the stable commutator length of some elements. We also calculate values and the defects of homogeneous quasimorphisms derive
d from omega-signatures, and show that they are linearly independent in the mapping class groups of pointed 2-spheres when the number of points is small.
Topological stability of the edge states is investigated for non-Hermitian systems. We examine two classes of non-Hermitian Hamiltonians supporting real bulk eigenenergies in weak non-Hermiticity: SU(1,1) and SO(3,2) Hamiltonians. As an SU(1,1) Hamil
tonian, the tight-binding model on the honeycomb lattice with imaginary on-site potentials is examined. Edge states with ReE=0 and their topological stability are discussed by the winding number and the index theorem, based on the pseudo-anti-Hermiticity of the system. As a higher symmetric generalization of SU(1,1) Hamiltonians, we also consider SO(3,2) models. We investigate non-Hermitian generalization of the Luttinger Hamiltonian on the square lattice, and that of the Kane-Mele model on the honeycomb lattice, respectively. Using the generalized Kramers theorem for the time-reversal operator Theta with Theta^2=+1 [M. Sato et al., arXiv:1106.1806], we introduce a time-reversal invariant Chern number from which topological stability of gapless edge modes is argued.
For ordinary hermitian Hamiltonians, the states show the Kramers degeneracy when the system has a half-odd-integer spin and the time reversal operator obeys Theta^2=-1, but no such a degeneracy exists when Theta^2=+1. Here we point out that for non-h
ermitian systems, there exists a degeneracy similar to Kramers even when Theta^2=+1. It is found that the new degeneracy follows from the mathematical structure of split-quaternion, instead of quaternion from which the Kramers degeneracy follows in the usual hermitian cases. Furthermore, we also show that particle/hole symmetry gives rise to a pair of states with opposite energies on the basis of the split quaternion in a class of non-hermitian Hamiltonians. As concrete examples, we examine in detail NxN Hamiltonians with N=2 and 4 which are non-hermitian generalizations of spin 1/2 Hamiltonian and quadrupole Hamiltonian of spin 3/2, respectively.
We propose a topological field theory for a spin-less two-dimensional chiral superconductor that contains fundamental Majorana fields. Due to a fermionic gauge symmetry, the Majorana modes survive as dynamical degrees of freedom only at magnetic vort
ex cores, and on edges. We argue that these modes have the topological properties pertinent to a p-wave superconductor including the non-abelian braiding statistics, and support this claim by calculating the ground state degeneracy on a torus. We also briefly discuss the connection to the Moore-Read Pfaffian quantum Hall state, and extensions to the spinful case and to three-dimensonal topological superconductors.
Superconductivity is a phenomenon where the macroscopic quantum coherence appears due to the pairing of electrons. This offers a fascinating arena to study the physics of broken gauge symmetry. However, the important symmetries in superconductors are
not only the gauge invariance. Especially, the symmetry properties of the pairing, i.e., the parity and spin-singlet/spin-triplet, determine the physical properties of the superconducting state. Recently it has been recognized that there is the important third symmetry of the pair amplitude, i.e., even or odd parity with respect to the frequency. The conventional uniform superconducting states correspond to the even-frequency pairing, but the recent finding is that the odd-frequency pair amplitude arises in the spatially non-uniform situation quite ubiquitously. Especially, this is the case in the Andreev bound state (ABS) appearing at the surface/interface of the sample. The other important recent development is on the nontrivial topological aspects of superconductors. As the band insulators are classified by topological indices into (i) conventional insulator, (ii) quantum Hall insulator, and (iii) topological insulator, also are the gapped superconductors. The influence of the nontrivial topology of the bulk states appears as the edge or surface of the sample. In the superconductors, this leads to the formation of zero energy ABS (ZEABS). Therefore, the ABSs of the superconductors are the place where the symmetry and topology meet each other which offer the stage of rich physics. In this review, we discuss the physics of ABS from the viewpoint of the odd-frequency pairing, the topological bulk-edge correspondence, and the interplay of these two issues. It is described how the symmetry of the pairing and topological indices determines the absence/presence of the ZEABS, its energy dispersion, and properties as the Majorana fermions.
