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Register a new userThe classification of surface states of topological insulators and superconductors with magnetic point group symmetry
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Authors
Ken Shiozaki
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We present the exhaustive classification of surface states of topological insulators and superconductors protected by crystallographic magnetic point group symmetry in three spatial dimensions. Recently, Cornfeld and Chapman [Phys. Rev. B {bf 99}, 075105 (2019)] pointed out that the topological classification of mass terms of the Dirac Hamiltonian with point group symmetry is recast as the extension problem of the Clifford algebra, and we use their results extensively. Comparing two-types of Dirac Hamiltonians with and without the mass-hedgehog potential, we establish the irreducible character formula to read off which Hamiltonian in the whole $K$-group belongs to fourth-order topological phases, which are atomic insulators localized at the center of the point group.
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Topological classification in our previous paper [K. Shiozaki and M. Sato, Phys. Rev. B ${bf 90}$, 165114 (2014)] is extended to nonsymmorphic crystalline insulators and superconductors. Using the twisted equivariant $K$-theory, we complete the classification of topological crystalline insulators and superconductors in the presence of additional order-two nonsymmorphic space group symmetries. The order-two nonsymmorphic space groups include half lattice translation with $Z_2$ flip, glide, two-fold screw, and their magnetic space groups. We find that the topological periodic table shows modulo-2 periodicity in the number of flipped coordinates under the order-two nonsymmorphic space group. It is pointed out that the nonsymmorphic space groups allow $mathbb{Z}_2$ topological phases even in the absence of time-reversal and/or particle-hole symmetries. Furthermore, the coexistence of the nonsymmorphic space group with the time-reversal and/or particle-hole symmetries provides novel $mathbb{Z}_4$ topological phases, which have not been realized in ordinary topological insulators and superconductors. We present model Hamiltonians of these new topological phases and the analytic expression of the $mathbb{Z}_2$ and $mathbb{Z}_4$ topological invariants. The half lattice translation with $Z_2$ spin flip and glide symmetry are compatible with the existence of the boundary, leading to topological surface gapless modes protected by such order-two nonsymmorphic symmetries. We also discuss unique features of these gapless surface modes.
We identify four types of higher-order topological semimetals or nodal superconductors (HOTS), hosting (i) flat zero-energy Fermi arcs at crystal hinges, (ii) flat zero-energy hinge arcs coexisting with surface Dirac cones, (iii) chiral or helical hinge modes, or (iv) flat zero-energy hinge arcs connecting nodes only at finite momentum. Bulk-boundary correspondence relates the hinge states to the bulk topology protecting the nodal point or loop. We classify all HOTS for all tenfold-way classes with an order-two crystalline (anti-)symmetry, such as mirror, twofold rotation, or inversion.
Systems of free fermions are classified by symmetry, space dimensionality, and topological properties described by K-homology. Those systems belonging to different classes are inequivalent. In contrast, we show that by taking a many-body/Fock space viewpoint it becomes possible to establish equivalences of topological insulators and superconductors in terms of duality transformations. These mappings connect topologically inequivalent systems of fermions, jumping across entries in existent classification tables, because of the phenomenon of symmetry transmutation by which a symmetry and its dual partner have identical algebraic properties but very different physical interpretations. To constrain our study to established classification tables, we define and characterize mathematically Gaussian dualities as dualities mapping free fermions to free fermions (and interacting to interacting). By introducing a large, flexible class of Gaussian dualities we show that any insulator is dual to a superconductor, and that fermionic edge modes are dual to Majorana edge modes, that is, the Gaussian dualities of this paper preserve the bulk-boundary correspondence. Transmutation of relevant symmetries, particle number, translation, and time reversal is also investigated in detail. As illustrative examples, we show the duality equivalence of the dimerized Peierls chain and the Majorana chain of Kitaev, and a two-dimensional Kekule-type topological insulator, including graphene as a special instance in coupling space, dual to a p-wave superconductor. Since our analysis extends to interacting fermion systems we also briefly discuss some such applications.
We theoretically study the effect of magnetic moire superlattice on the topological surface states by introducing a continuum model of Dirac electrons with a single Dirac cone moving in the time-reversal symmetry breaking periodic pontential. The Zeeman-type moire potentials generically gap out the moire surface Dirac cones and give rise to isolated flat Chern minibands with Chern number $pm1$. This result provides a promising platform for realizing the time-reversal breaking correlated topological phases. In a $C_6$ periodic potential, when the scalar $U_0$ and Zeeman $Delta_1$ moire potential strengths are equal to each other, we find that energetically the first three bands of $Gamma$-valley moire surface electrons are non-degenerate and realize i) an $s$-orbital model on a honeycomb lattice, ii) a degenerate $p_x,p_y$-orbitals model on a honeycomb lattice, and iii) a hybridized $sd^2$-orbital model on a kagome lattice, where moire surface Dirac cones in these bands emerge. When $U_0 eqDelta_1$, the difference between the two moire potential serves as an effective spin-orbit coupling and opens a topological gap in the emergent moire surface Dirac cones.
The edge states of a two-dimensional quantum spin Hall (QSH) insulator form a one-dimensional helical metal which is responsible for the transport property of the QSH insulator. Conceptually, such a one-dimensional helical metal can be attached to any scattering region as the usual metallic leads. We study the analytical property of the scattering matrix for such a conceptual multiterminal scattering problem in the presence of time reversal invariance. As a result, several theorems on the connectivity property of helical edge states in two-dimensional QSH systems as well as surface states of three-dimensional topological insulators are obtained. Without addressing real model details, these theorems, which are phenomenologically obtained, emphasize the general connectivity property of topological edge/surface states from the mere time reversal symmetry restriction.
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