Chambers’s Formula for the Graphene and the Hou Model with Kagome Periodicity and Applications

Helffer, Bernard; Kerdelhué, Philippe; Royo-Letelier, Jimena

doi:10.1007/s00023-015-0415-z

Cited by 14 publications

(20 citation statements)

References 24 publications

(69 reference statements)

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“…(8.49) Therefore bands of Q Λ (Φ) are in one-to-one correspondence with bands of Λ B , and thus also with bands of H B . That the bands of Q Λ (Φ) do not overlap is shown in Section 6 of[HKL16]. Thus, the unique correspondence among bands of Q Λ (Φ) and H B shows that the non-overlapping of bands holds true for H B as well.Remark 9.…”

mentioning

confidence: 86%

“…This operator has already been studied, in different contexts, for rational flux quanta in [KL14], [HKL16], and [AEG14].…”

Section: Quick Observations About H 2πp/qθmentioning

confidence: 99%

“…Thus, the unique correspondence among bands of Q Λ (Φ) and H B shows that the non-overlapping of bands holds true for H B as well.Remark 9. For Φ 2π = 1 2 the spectral bands of Q Λ (Φ) are touching and given by[HKL16] Lemma 8.15 the bands of H B on each Hill band are touching as well, seeFig. 6.…”

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confidence: 97%

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Cantor spectrum of graphene in magnetic fields

2019

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mentioning

confidence: 86%

“…This operator has already been studied, in different contexts, for rational flux quanta in [KL14], [HKL16], and [AEG14].…”

Section: Quick Observations About H 2πp/qθmentioning

confidence: 99%

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Cantor spectrum of graphene in magnetic fields

2019

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“…The Kagome lattice has attracted attention in the physics and mathematical physics community in connection with magnetic properties of certain crystal structures (see, e.g., [35,36]) and due to the emergence of butterfly spectra [37][38][39].…”

Section: Existence Of Finitely Supported Eigenfunctions On Planar Graphsmentioning

confidence: 99%

Unique continuation principles and their absence for Schrödinger eigenfunctions on combinatorial and quantum graphs and in continuum space

Peyerimhoff¹,

Täufer²,

Veselić³

2017

Nanosystems: Phys. Chem. Math.

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For the analysis of the Schrödinger and related equations it is of central importance whether a unique continuation principle (UCP) holds or not. In continuum Euclidean space, quantitative forms of unique continuation imply Wegner estimates and regularity properties of the integrated density of states (IDS) of Schrödinger operators with random potentials. For discrete Schrödinger equations on the lattice, only a weak analog of the UCP holds, but it is sufficient to guarantee the continuity of the IDS. For other combinatorial graphs, this is no longer true. Similarly, for quantum graphs the UCP does not hold in general and consequently, the IDS does not need to be continuous.

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“…An algebraic approach to this operator was put forward by Bellissard (for more details see [Be92], [Be94]). Note that there are results about the Hofstadter-type spectrum of the Harper model on other planar graphs (the triangular, hexagonal, Kagome lattices) (see [Hou09], [Ke92], [KeR14], [HKeR16], and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Invariants for Laplacians on periodic graphs

Saburova

2019

Math. Ann.

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We consider a magnetic Laplacian with periodic magnetic potentials on periodic discrete graphs. Its spectrum consists of a finite number of bands, where degenerate bands are eigenvalues of infinite multiplicity. We obtain a specific decomposition of the magnetic Laplacian into a direct integral in terms of minimal forms. A minimal form is a periodic function defined on edges of the periodic graph with a minimal support on the period. It is crucial that fiber magnetic Laplacians (matrices) have the minimal number of coefficients depending on the quasimomentum and the minimal number of coefficients depending on the magnetic potential. We show that these numbers are invariants for the magnetic Laplacians on periodic graphs. Using this decomposition, we estimate the position of each band, the Lebesgue measure of the magnetic Laplacian spectrum and a variation of the spectrum under a perturbation by a magnetic field in terms of these invariants and minimal forms. In addition, we consider an inverse problem: we determine necessary and sufficient conditions for matrices depending on the quasimomentum on a finite graph to be fiber magnetic Laplacians. Moreover, similar results for magnetic Schrödinger operators with periodic potentials are obtained.

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Chambers’s Formula for the Graphene and the Hou Model with Kagome Periodicity and Applications

Cited by 14 publications

References 24 publications

Cantor spectrum of graphene in magnetic fields

Cantor spectrum of graphene in magnetic fields

Unique continuation principles and their absence for Schrödinger eigenfunctions on combinatorial and quantum graphs and in continuum space

Invariants for Laplacians on periodic graphs

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