161 research outputs found
Atiyah-Hirzebruch Spectral Sequence in Band Topology: General Formalism and Topological Invariants for 230 Space Groups
We study the Atiyah-Hirzebruch spectral sequence (AHSS) for equivariant
K-theory in the context of band theory. Various notions in the band theory such
as irreducible representations at high-symmetric points, the compatibility
relation, topological gapless and singular points naturally fits into the AHSS.
As an application of the AHSS, we get the complete list of topological
invariants for 230 space groups without time-reversal or particle-hole
invariance. We find that a lot of torsion topological invariants appear even
for symmorphic space groups.Comment: 65 pages, many figures and table
Z2-topology in nonsymmorphic crystalline insulators: Mobius twist in surface states
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 such 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.Comment: 10 pages, 5 figure
The classification of surface states of topological insulators and superconductors with magnetic point group symmetry
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 -group belongs to fourth-order
topological phases, which are atomic insulators localized at the center of the
point group.Comment: 18+18 page
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