Controlled Topological Phases and Bulk-edge Correspondence

Kubota, Yosuke

doi:10.1007/s00220-016-2699-3

Cited by 78 publications

(74 citation statements)

References 60 publications

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“…Some of our results show parallels with those of Kubota and of Ewert-Meyer, who study topological phases associated to Delone sets and the corresponding Roe algeba [31,54]. Briefly, the Roe algebra, by its universal nature, provides a means to compare topological phases from different lattice configurations (see [54,Lemma 2.19]). Conversely, the transversal groupoid algebra is used to determine the topological phase of Hamiltonians associated to a fixed lattice configuration.…”

Section: Introductionsupporting

confidence: 76%

Index Theory and Topological Phases of Aperiodic Lattices

Bourne

Mesland

2019

Ann. Henri Poincaré

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We examine the noncommutative index theory associated to 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.

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Section: Introductionsupporting

confidence: 76%

Index Theory and Topological Phases of Aperiodic Lattices

Bourne

Mesland

2019

Ann. Henri Poincaré

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“…This duality, with its origins in string theory, relates the K-theory of two torus bundles over the same base manifold. In the presence of lattice symmetries, Kubota [34] establishes bulk-boundary correspondence at the level of strong invariants, by using the tools of coarse geometry. Following ideas of Freed-Moore [19], he combines the "quantum symmetries" of the Tenfold Way and the lattice symmetries into a single group.…”

Section: Introductionmentioning

confidence: 99%

Bulk-Boundary Correspondence for Disordered Free-Fermion Topological Phases

Alldridge

Max

Zirnbauer

2019

Commun. Math. Phys.

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Guided by the many-particle quantum theory of interacting systems, we develop a uniform classification scheme for topological phases of disordered gapped free fermions, encompassing all symmetry classes of the Tenfold Way. We apply this scheme to give a mathematically rigorous proof of bulk-boundary correspondence. To that end, we construct real C$$^*$$ ∗ -algebras harbouring the bulk and boundary data of disordered free-fermion ground states. These we connect by a natural bulk-to-boundary short exact sequence, realising the bulk system as a quotient of the half-space theory modulo boundary contributions. To every ground state, we attach two classes in different pictures of real operator $$K$$ K -theory (or $$KR$$ KR -theory): a bulk class, using Van Daele’s picture, along with a boundary class, using Kasparov’s Fredholm picture. We then show that the connecting map for the bulk-to-boundary sequence maps these $$KR$$ KR -theory classes to each other. This implies bulk-boundary correspondence, in the presence of disorder, for both the “strong” and the “weak” invariants.

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“…Another important feature of them is that the topological feature of the QSH phase is determined only by the bulk states. This is so called 'bulk-edge correspondence' [43][44][45][46][47]. The value of Z 2 topological invariant is calculated by the bulk wave functions of materials under this concept.…”

Section: Introductionmentioning

confidence: 99%

Quantum spin Hall phase in honeycomb nanoribbons with two different atoms: edge shape effect to bulk-edge correspondence

Kondo

Ito

2019

J. Phys. Commun.

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In this study, we consider a quantum spin Hall (QSH) phase in both the zigzag and the armchair type of honeycomb nanoribbons with two different atoms from the viewpoint of bulk-edge correspondence. Generally, the QSH phase in honeycomb nanoribbons is determined by the topology of the bulk Hamiltonian. However, the armchair type of nanoribbons seems to become the QSH phase in a very different region compared with bulk materials. On the other hand, the zigzag type of nanoribbons seems to become the QSH phase in almost the same region as bulk materials. We study the reason why the QSH phase in nanoribbons seems to be different from that of bulk materials using the extended Kane-Mele Hamiltonian. As a result, there is a clear difference in the edge states in the QSH phase between the zigzag and the armchair type of nanoribbons. We find that the QSH phase region in nanoribbons is actually different from that of bulk materials. This is because the coherence lengths of edge wave functions of nanoribbons are extremely influenced by their edge-shapes. We can conclude that the bulk-edge correspondence does not hold for relatively narrow nanoribbons compared with their coherence lengths and that the edge shapes of nanoribbons make their coherence lengths of edge wavefunctions different, which largely influences the QSH phase.

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Controlled Topological Phases and Bulk-edge Correspondence

Cited by 78 publications

References 60 publications

Index Theory and Topological Phases of Aperiodic Lattices

Index Theory and Topological Phases of Aperiodic Lattices

Bulk-Boundary Correspondence for Disordered Free-Fermion Topological Phases

Quantum spin Hall phase in honeycomb nanoribbons with two different atoms: edge shape effect to bulk-edge correspondence

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