Fredholm Homotopies for Strongly-Disordered 2D Insulators

Bols, Alex; Schenker, Jeffrey; Shapiro, Jacob

doi:10.48550/arxiv.2110.07068

Cited by 3 publications

(4 citation statements)

References 23 publications

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“…The spectrum of Figure 1 corresponds to an H with a non-trivial Fu-Kane-Mele index. This characterisation of the Fu-Kane-Mele index parallels the characterisation for translation invariant free fermion systems through the bulk-boundary correspondence [27,23,12] where the bands associated to modes bound to the magnetic flux are replaced by bands formed by edge modes.…”

Section: Idea Behind the Constructionmentioning

confidence: 54%

A Many-Body Index for Quantum Charge Transport

Bachmann

Bols

Roeck

et al. 2019

Commun. Math. Phys.

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We propose an index for pairs of a unitary map and a clustering state on manybody quantum systems. We require the map to conserve an integer-valued charge and to leave the state, e.g. a gapped ground state, invariant. This index is integer-valued and stable under perturbations. In general, the index measures the charge transport across a fiducial line. We show that it reduces to (i) an index of projections in the case of non-interacting fermions, (ii) the charge density for translational invariant systems, and (iii) the quantum Hall conductance in the two-dimensional setting without any additional symmetry. Example (ii) recovers the Lieb-Schultz-Mattis theorem, and (iii) provides a new and short proof of quantization of Hall conductance in interacting many-body systems.

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Section: Idea Behind the Constructionmentioning

confidence: 54%

A Many-Body Index for Quantum Charge Transport

Bachmann

Bols

Roeck

et al. 2019

Commun. Math. Phys.

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“…The presence of a mobility gap, meaning that at the chemical potential there are only localized states, is still required, otherwise the system would lose its insulating nature. A different rigorous approach to strongdisordered systems is that of the so-called deterministic disorder, see for example [56] and section 7 of [57] with the definition of SULE, and the more recent works [54,58]. In the formulation of the Bott index a band gap has been assumed.…”

Section: Discussion and Perspectivesmentioning

confidence: 99%

“…( 72) and the Chern number Ch(P) taking advantage of an approach developed in [44] for bounded operators. The operators (72) have been already suggested in the context of the integer quantum Hall effect by Kitaev in appendix C of [52], and more recently, with a suitable modification, in the context of infinite-dimensional Hilbert spaces by the authors of [53,54]. Being the Bott index an integer this shows that the correction O(λ) in Eq.…”

Section: An Exact Approach: Homotopies On the Torus: The Right And Th...mentioning

confidence: 93%

On the Bott index of unitary matrices on a finite torus

Toniolo

2022

Lett Math Phys

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This article reviews the foundations of the theory of the Bott index of a pair of unitary matrices in the context of condensed matter theory, as developed by Hastings and Loring (J. Math. Phys. 51, 015214 (2010), Ann. Phys. 326, 1699 (2011), providing a novel proof of the equality with the Chern number. The Bott index is defined for a pair of unitary matrices, then extended to a pair of invertible matrices and homotopic invariance of the index is proven. An insulator defined on a lattice on a two-torus, that is a rectangular lattice with periodic boundary conditions, is considered and a pair of quasi-unitary matrices associated to this physical system are introduced. It is shown that their Bott index is well defined and the connection with the transverse conductance, the Chern number, is established proving the equality of the two quantities, in certain units.

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“…We appeal to the bulk-edge correspondence for time-reversal invariant topological insulators [15,13,2,6] which links the bulk index to an edge index associated to a time-reversal symmetric unitary acting on the edge modes. Inspired by [3], we prove a symmetric Wold decomposition for such unitaries which implies in particular that the absolutely continuous spectrum of this unitary covers the whole unit circle if the edge index is non-trivial.…”

Section: Introductionmentioning

confidence: 99%