Tight-binding reduction and topological equivalence in strong magnetic fields

Shapiro, Jacob; Weinstein, Michael I.

doi:10.1016/j.aim.2022.108343

Cited by 9 publications

(4 citation statements)

References 57 publications

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“…A rigorous derivation of () from () was carried out by Shapiro, Fefferman, and Weinstein in a sequence of papers, 26,36,37 following earlier work of Lee‐Thorp, Fefferman, and Weinstein 38 as well as work of Helffer and Sjöstrand 39,40 . The basic idea is to replace V by

\breve{\lambda }^2 V

and then consider the limit of large

\breve{\lambda }

.…”

Section: Ssh Model and Main Resultsmentioning

confidence: 99%

On the Su–Schrieffer–Heeger model of electron transport: Low‐temperature optical conductivity by the Mellin transform

2023

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We describe the low-temperature optical conductivity as a function of frequency for a quantum-mechanical system of electrons that hop along a polymer chain. To this end, we invoke the Su-Schrieffer-Heeger tight-binding Hamiltonian for noninteracting spinless electrons on a one-dimensional (1D) lattice. Our goal is to show via asymptotics how the interband conductivity of this system behaves as the smallest energy bandgap tends to close. Our analytical approach includes: (i) the Kubotype formulation for the optical conductivity with a nonzero damping due to microscopic collisions, (ii) reduction of this formulation to a 1D momentum integral over the Brillouin zone, and (iii) evaluation of this integral in terms of elementary functions via the threedimensional Mellin transform with respect to key physical parameters and subsequent inversion in a region of the respective complex space. Our approach reveals an intimate connection of the behavior of the conductivity to particular singularities of its Mellin transform.The analytical results are found in good agreement with direct numerical computations.

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\breve{\lambda }^2 V

and then consider the limit of large

\breve{\lambda }

.…”

Section: Ssh Model and Main Resultsmentioning

confidence: 99%

On the Su–Schrieffer–Heeger model of electron transport: Low‐temperature optical conductivity by the Mellin transform

2023

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“…The Hamiltonian

H_\sharp

in (2.6) arises as the limit of the 1 − electron model in the strong binding regime; see, for example, [10, 19, 28]. The operator

H_\sharp

is bounded and self‐adjoint on

l^2(\mathbb {H}_\sharp )

; see Section 2.5.3 and relation ().…”

Section: Mathematical Frameworkmentioning

confidence: 99%

“…The relationship between the underlying continuum single electron Schroedinger equation for graphene and the tight-binding limit is investigated in detail in [11]. For a general discussion of tight binding models and their relationship to the underlying continuum PDEs see, for example, [10,19,28].…”

Section: Introductionmentioning

confidence: 99%

Discrete honeycombs, rational edges, and edge states

Fefferman,

Fliss,

Weinstein

2023

Comm Pure Appl Math

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Consider the tight binding model of graphene, sharply terminated along an edge l parallel to a direction of translational symmetry of the underlying period lattice. We classify such edges l into those of “zigzag type” and those of “armchair type”, generalizing the classical zigzag and armchair edges. We prove that zero energy / flat band edge states arise for edges of zigzag type, but never for those of armchair type. We exhibit explicit formulas for flat band edge states when they exist. We produce strong evidence for the existence of dispersive (non flat) edge state curves of nonzero energy for most l.

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“…Exponential upper and lower bounds on the hopping function have been proved in the context of a continuum model of graphene by Fefferman-Lee-Thorp-Weinstein [24]. These bounds were generalized to the setting of a strong magnetic field by Fefferman-Shapiro-Weinstein [27,40].…”

Section: 4mentioning

confidence: 99%

Bistritzer-MacDonald dynamics in twisted bilayer graphene

Watson¹,

Kong²,

MacDonald³

et al. 2022

Preprint

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The Bistritzer-MacDonald (BM) model, introduced in [10], attempts to capture the electronic properties of twisted bilayer graphene (TBG), even at incommensurate twist angles, by an effective periodic model over the bilayer moiré pattern. Starting from a tight-binding model, we identify a regime where the BM model emerges as the effective dynamics for electrons modeled as wave-packets spectrally concentrated at the monolayer Dirac points, up to error that can be rigorously estimated. Using measured values of relevant physical constants, we argue that this regime is realized in TBG at the first "magic" angle.

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Tight-binding reduction and topological equivalence in strong magnetic fields

Cited by 9 publications

References 57 publications

On the Su–Schrieffer–Heeger model of electron transport: Low‐temperature optical conductivity by the Mellin transform

On the Su–Schrieffer–Heeger model of electron transport: Low‐temperature optical conductivity by the Mellin transform

Discrete honeycombs, rational edges, and edge states

Bistritzer-MacDonald dynamics in twisted bilayer graphene

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