掲載種別：研究論文（学術雑誌）   出版者・発行元：Forum of Mathematics, Sigma  

We introduce an index for symmetry-protected topological (SPT) phases of infinite fermionic chains with an on-site symmetry given by a finite group G. This index takes values in Z2 × H1(G, Z2) × H2(G,U(1)p) with a generalised Wall group law under stacking. We show that this index is an invariant of the classification of SPT phases. When the ground state is translation invariant and has reduced density matrices with uniformly bounded rank on finite intervals, we derive a fermionic matrix product representative of this state with on-site symmetry.

DOI： 10.1017/fms.2021.19

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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：Reviews in Mathematical Physics  

For parity-conserving fermionic chains, we review how to associate Z2-indices to ground states in finite systems with quadratic and higher-order interactions as well as to quasifree ground states on the infinite CAR algebra. It is shown that the Z2-valued spectral flow provides a topological obstruction for two systems to have the same Z2-index. A rudimentary definition of a Z2-phase label for a class of parity-invariant and pure ground states of the one-dimensional infinite CAR algebra is also provided. Ground states with differing phase labels cannot be connected without a closing of the spectral gap of the infinite GNS Hamiltonian.

DOI： 10.1142/S0129055X20500282

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掲載種別：研究論文（学術雑誌）   出版者・発行元：International Journal of Mathematics  

We use the Cayley transform to provide an explicit isomorphism at the level of cycles from van Daele K-theory to KK-theory for graded C-algebras with a real structure. Isomorphisms between KK-theory and complex or real K-theory for ungraded C-algebras are a special case of this map. In all cases, our map is compatible with the computational techniques required in physical and geometrical applications, in particular, index pairings and Kasparov products. We provide applications to real K-theory and topological phases of matter.

DOI： 10.1142/S0129167X20500743

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掲載種別：研究論文（学術雑誌）   出版者・発行元：Annales Henri Poincare  

We examine the non-commutative index theory associated with the dynamics of a Delone set and the corresponding transversal groupoid. Our main motivation comes from the application to topological phases of aperiodic lattices and materials and applies to invariants from tilings as well. Our discussion concerns semifinite index pairings, factorisation properties of Kasparov modules and the construction of unbounded Fredholm modules for lattices with finite local complexity.

DOI： 10.1007/s00023-019-00764-9

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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：Mathematical Physics Analysis and Geometry  

In order to study continuous models of disordered topological phases, we construct an unbounded Kasparov module and a semifinite spectral triple for the crossed product of a separable C∗-algebra by a twisted ℝd-action. The spectral triple allows us to employ the non-unital local index formula to obtain the higher Chern numbers in the continuous setting with complex observable algebra. In the case of the crossed product of a compact disorder space, the pairing can be extended to a larger algebra closely related to dynamical localisation, as in the tight-binding approximation. The Kasparov module allows us to exploit the Wiener–Hopf extension and the Kasparov product to obtain a bulk-boundary correspondence for continuous models of disordered topological phases.

DOI： 10.1007/s11040-018-9274-4

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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：Journal of Physics A: Mathematical and Theoretical  

The search for strong topological phases in generic aperiodic materials and meta-materials is now vigorously pursued by the condensed matter physics community. In this work, we first introduce the concept of patterned resonators as a unifying theoretical framework for topological electronic, photonic, phononic etc (aperiodic) systems. We then discuss, in physical terms, the philosophy behind an operator theoretic analysis used to systematize such systems. A model calculation of the Hall conductance of a 2-dimensional amorphous lattice is given, where we present numerical evidence of its quantization in the mobility gap regime. Motivated by such facts, we then present the main result of our work, which is the extension of the Chern number formulas to Hamiltonians associated to lattices without a canonical labeling of the sites, together with index theorems that assure the quantization and stability of these Chern numbers in the mobility gap regime. Our results cover a broad range of applications, in particular, those involving quasi-crystalline, amorphous as well as synthetic (i.e. algorithmically generated) lattices.

DOI： 10.1088/1751-8121/aac093

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出版者・発行元：Springer International Publishing  

DOI： 10.1007/978-3-319-72299-3_10

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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：Springer Nature  

We study the application of Kasparov theory to topological insulator systems and the bulk–edge correspondence. We consider observable algebras as modelled by crossed products, where bulk and edge systems may be linked by a short exact sequence. We construct unbounded Kasparov modules encoding the dynamics of the crossed product. We then link bulk and edge Kasparov modules using the Kasparov product. Because of the anti-linear symmetries that occur in topological insulator models, real C∗-algebras and KKO-theory must be used.

DOI： 10.1007/s00023-016-0541-2

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記述言語：英語  

DOI： 10.1017/S000497271600037X

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記述言語：英語   出版者・発行元：Reviews in Mathematical Physics  

We study topological insulators, regarded as physical systems giving rise to topological invariants determined by symmetries both linear and anti-linear. Our perspective is that of non-commutative index theory of operator algebras. In particular, we formulate the index problems using Kasparov theory, both complex and real. We show that the periodic table of topological insulators and superconductors can be realized as a real or complex index pairing of a Kasparov module capturing internal symmetries of the Hamiltonian with a spectral triple encoding the geometry of the sample's (possibly non-commutative) Brillouin zone.

DOI： 10.1142/S0129055X16500045

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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：Letters in Mathematical Physics  

We prove the bulk-edge correspondence in K-theory for the quantum Hall effect by constructing an unbounded Kasparov module from a short exact sequence that links the bulk and boundary algebras. This approach allows us to represent bulk topological invariants explicitly as a Kasparov product of boundary invariants with the extension class linking the algebras. This paper focuses on the example of the discrete integer quantum Hall effect, though our general method potentially has much wider applications.

DOI： 10.1007/s11005-015-0781-y

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開催年月日： 2023年11月

記述言語：日本語   会議種別：口頭発表（招待・特別）  

開催地：京都大学数理解析研究所  

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開催年月日： 2023年8月

記述言語：英語   会議種別：口頭発表（招待・特別）  

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開催年月日： 2023年7月

記述言語：英語   会議種別：口頭発表（招待・特別）  

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開催年月日： 2022年12月

記述言語：英語   会議種別：公開講演，セミナー，チュートリアル，講習，講義等  

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記述言語：英語   会議種別：公開講演，セミナー，チュートリアル，講習，講義等  

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記述言語：英語   会議種別：口頭発表（招待・特別）  

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記述言語：英語   会議種別：口頭発表（一般）  

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記述言語：英語   会議種別：口頭発表（招待・特別）  

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記述言語：英語   会議種別：口頭発表（招待・特別）  

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記述言語：英語  

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記述言語：英語  

開催地：Univeristy of Sydney  

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記述言語：英語   会議種別：口頭発表（一般）  

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会議種別：公開講演，セミナー，チュートリアル，講習，講義等  

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集中講義
テーマ：非可換幾何と物理学の応用の入門