T-Duality Simplifies Bulk-Boundary Correspondence

Mathai, Varghese; Thiang, Guo Chuan

doi:10.1007/s00220-016-2619-6

Cited by 35 publications

(51 citation statements)

References 71 publications

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“…Results covering all symmetry classes in any dimension were obtained by Hannabuss, Mathai, and Thiang. In a series of papers [23,40,39], they apply the concept of T -duality to the bulk-boundary correspondence. This duality, with its origins in string theory, relates the K-theory of two torus bundles over the same base manifold.…”

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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Section: Introductionmentioning

confidence: 99%

Bulk-Boundary Correspondence for Disordered Free-Fermion Topological Phases

Alldridge

Max

Zirnbauer

2019

Commun. Math. Phys.

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“…On the other hand, Ktheory of another C * -algebra called the Roe algebra, which is originally introduced by Roe [Roe88,Roe93] for the purpose of studying index theory of noncompact manifolds, is easy to handle and widely studied. A useful tool is the (equivariant) coarse Baum-Connes map [HR95,Yu97b,Sha08], which gives an isomorphism between a certain K-homology groups and K-groups of Roe algebras (note that there is another use of the Baum-Connes isomorphism in connection with the T-duality studied in [MT15a,MT15b,MT15c,HMT15]). For example, when X is a lattice of E d , the groups K * (C * (X)) are isomorphic to K d− * (pt).…”

Section: Introductionmentioning

confidence: 99%

Controlled Topological Phases and Bulk-edge Correspondence

Kubota

2016

Commun. Math. Phys.

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Abstract. In this paper, we introduce a variation of the notion of topological phase reflecting metric structure of the position space. This framework contains not only periodic and non-periodic systems with symmetries in Kitaev's periodic table but also topological crystalline insulators. We also define the bulk and edge indices as invariants taking values in the twisted equivariant K-groups of Roe algebras as generalizations of existing invariants such as the Hall conductance or the Kane-Mele Z2-invariant. As a consequence, we obtain a new mathematical proof of the bulk-edge correspondence by using the coarse MayerVietoris exact sequence.

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“…Reversing the argument, this says that the extension defined by α X ⊕ ∂ X indeed correctly captures the geometry of the bulk-boundary relationship. Theorem 4.1 generalises to the hyperbolic plane geometry, the Euclidean space result [52] and the Nil and Solv geometry results [28,29]. 4.5.…”

Section: T-duality Simplifies the Bulk-boundary Correspondencementioning

confidence: 92%

“…Abstractly, such extensions may be studied using K-homology or Kasparov theoryindeed a KK-theoretic formulation was worked out in [12]. In [52,53], we showed that under appropriate T-duality transformations (a geometric Fourier transform closely related to the Baum-Connes assembly map), the Euclidean space bulk-boundary maps simplify into geometric restriction-to-boundary maps, and these results were extended to Nil and Solv geometries in [28,29].…”

Section: Introductionmentioning

confidence: 98%

Topological phases on the hyperbolic plane: fractional bulk-boundary correspondence

Mathai

Thiang

2019

Advances in Theoretical and Mathematical Physics

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We study topological phases in the hyperbolic plane using noncommutative geometry and T-duality, and show that fractional versions of the quantised indices for integer, spin and anomalous quantum Hall effects can result. Generalising models used in the Euclidean setting, a model for the bulk-boundary correspondence of fractional indices is proposed, guided by the geometry of hyperbolic boundaries. 1 arXiv:1712.02952v4 [cond-mat.str-el]

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T-Duality Simplifies Bulk-Boundary Correspondence

Cited by 35 publications

References 71 publications

Bulk-Boundary Correspondence for Disordered Free-Fermion Topological Phases

Bulk-Boundary Correspondence for Disordered Free-Fermion Topological Phases

Controlled Topological Phases and Bulk-edge Correspondence

Topological phases on the hyperbolic plane: fractional bulk-boundary correspondence

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