Kiyonori Gomi

Equivariant smooth Deligne cohomology

On the basis of Brylinski’s work, we introduce a notion of equ ivariant smooth Deligne cohomology group, which is a generalization of both ordinary smooth Deligne cohomology and ordinary equivariant cohomology. Using the cohomology group, we classify equivariant circle bundles with connection, and equivariant gerbes with connection.

Z2topology in nonsymmorphic crystalline insulators: Möbius 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.

Classification of “Real” Bloch-bundles: Topological quantum systems of type AI

Abstract We provide a classification of type AI topological quantum systems in dimension d = 1 , 2 , 3 , 4 which is based on the equivariant homotopy properties of “Real” vector bundles. This allows us to produce a fine classification able to take care also of the non stable regime which is usually not accessible via K -theoretic techniques. We prove the absence of non-trivial phases for one-band AI free or periodic quantum particle systems in each spatial dimension by inspecting the second equivariant cohomology group which classifies “Real” line bundles. We also show that the classification of “Real” line bundles suffices for the complete classification of AI topological quantum systems in dimension d ⩽ 3 . In dimension d = 4 the determination of different topological phases (for free or periodic systems) is fixed by the second “Real” Chern class which provides an even labeling identifiable with the degree of a suitable map. Finally, we provide explicit realizations of non trivial 4-dimensional free models for each given topological degree.

Topological crystalline materials: General formulation, module structure, and wallpaper groups

We formulate topological crystalline materials on the basis of the twisted equivariant

Topology of nonsymmorphic crystalline insulators and superconductors

K

CONNECTIONS AND CURVINGS ON LIFTING BUNDLE GERBES

-theory. Basic ideas of the twisted equivariant

A Variant of K-Theory and Topological T-Duality for Real Circle Bundles

K

Twisted K-Theory and Finite-Dimensional Approximation

-theory is explained with application to topological phases protected by crystalline symmetries in mind, and systematic methods of topological classification for crystalline materials are presented. Our formulation is applicable to bulk gapful topological crystalline insulators/superconductors and their gapless boundary and defect states, as well as bulk gapless topological materials such as Weyl and Dirac semimetals, and nodal superconductors. As an application of our formulation, we present a complete classification of topological crystalline surface states, in the absence of time-reversal invariance. The classification works for gapless surface states of three-dimensional insulators, as well as full gapped two-dimensional insulators. Such surface states and two-dimensional insulators are classified in a unified way by 17 wallpaper groups, together with the presence or the absence of (sublattice) chiral symmetry. We identify the topological numbers and their representations under the wallpaper group operation. We also exemplify the usefulness of our formulation in the classification of bulk gapless phases. We present a new class of Weyl semimetals and Weyl superconductors that are topologically protected by inversion symmetry.

Twists on the Torus Equivariant under the 2-Dimensional Crystallographic Point Groups

glide, twofold screw, and their magnetic space groups. We nd that the topological periodic table shows modulo-2 periodicity in the number of ipped coordinates under the order-two nonsymmorphic space group. It is pointed out that the nonsymmorphic space groups allow Z2 topological phases even in the absence of time-reversal and/or particle-hole symmetries. Furthermore, the coexistence of the nonsymmorphic space group with time-reversal and/or particle-hole symmetries provides novel Z4 topological phases, which have not been realized in ordinary topological insulators and superconductors. We present model Hamiltonians of these new topological phases and analytic expressions of the Z2 and Z4 topological invariants. The half lattice translation with Z2 spin ip and glide symmetry are compatible with the existence of boundaries, leading to topological surface gapless modes protected by the order-two nonsymmorphic symmetries. We also discuss unique features of these gapless surface modes.

Central extensions of gauge transformation groups of higher abelian gerbes

We construct a connection and a curving on a bundle gerbe associated with lifting a structure group of a principal bundle to a central extension. The construction is based on certain structures on the bundle, i.e. connections and splittings. The Deligne cohomology class of the lifting bundle gerbe with the connection and with the curving coincides with the obstruction class of the lifting problem with these structures.