Magnetic twisted actions on general abelian $C^*$-algebras

Belmonte, Fabián; Lein, Max; Măntoiu, Marius

doi:10.7900/jot.2010jun30.1896

Cited by 12 publications

(15 citation statements)

References 21 publications

(37 reference statements)

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“…We encode a magnetic twist on our groupoid via a construction from [12], which considered twisted crossed products of commutative C * -algebras. We construct a magnetic field in d dimensions as a 2-form B ∈ 2 R d .…”

Section: Algebra and Representationsmentioning

confidence: 99%

Non-commutative Chern numbers for generic aperiodic discrete systems

Bourne¹,

Prodan²

2018

J. Phys. A: Math. Theor.

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The search for strong topological phases in generic aperiodic materials and metamaterials 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.

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Section: Algebra and Representationsmentioning

confidence: 99%

Non-commutative Chern numbers for generic aperiodic discrete systems

Bourne¹,

Prodan²

2018

J. Phys. A: Math. Theor.

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“…Definition 2. 7 We say that a * -algebra A is a pre-C * -algebra if it is (i) Fréchet, i.e. complete and metrizable such that the multiplication is jointly continuous; (ii) Isomorphic to a proper dense * -subalgebra ι(A) of a C * -algebra A, where ι :…”

Section: Definition 25mentioning

confidence: 99%

“…is a pre-Morita equivalence bimodule described by Eqs. ( 6) and (7). Because H is étale, the right-action in Eq.…”

Section: Frames Of Translates and Wannier Bases From Groupoid Equival...mentioning

confidence: 99%

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Localised Module Frames and Wannier Bases from Groupoid Morita Equivalences

Bourne

Mesland

2021

J Fourier Anal Appl

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Following the operator algebraic approach to Gabor analysis, we construct frames of translates for the Hilbert space localisation of the Morita equivalence bimodule arising from a groupoid equivalence between Hausdorff groupoids, where one of the groupoids is étale and with a compact unit space. For finitely generated and projective submodules, we show these frames are orthonormal bases if and only if the module is free. We then apply this result to the study of localised Wannier bases of spectral subspaces of Schrödinger operators with atomic potentials supported on (aperiodic) Delone sets. The noncommutative Chern numbers provide a topological obstruction to fast-decaying Wannier bases and we show this result is stable under deformations of the underlying Delone set.

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“…Example 2.2), though we will consider constant field strength for simplicity. We direct the reader to [10,58,62] for a more detailed study on magnetic fields and twisted crossed products. The Schrödinger operator is given by…”

Section: Applications To Disordered Quantum Systems and Topological Pmentioning

confidence: 99%

Chern Numbers, Localisation and the Bulk-edge Correspondence for Continuous Models of Topological Phases

Bourne

Rennie

2018

Math Phys Anal Geom

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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 R d -action. The spectral triple allows us to employ the nonunital 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.

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Magnetic twisted actions on general abelian $C^*$-algebras

Abstract: We introduce magnetic twisted actions of X = R n on general abelian C * -algebras and study the associated twisted crossed product and pseudodifferential algebras in the framework of strict deformation quantization.

Cited by 12 publications

References 21 publications

Non-commutative Chern numbers for generic aperiodic discrete systems

Non-commutative Chern numbers for generic aperiodic discrete systems

Localised Module Frames and Wannier Bases from Groupoid Morita Equivalences

Chern Numbers, Localisation and the Bulk-edge Correspondence for Continuous Models of Topological Phases

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