On the Construction of Wannier Functions in Topological Insulators: the 3D Case

Cornean, Horia D.; Monaco, Domenico

doi:10.1007/s00023-017-0621-y

Cited by 12 publications

(16 citation statements)

References 21 publications

(61 reference statements)

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“…When these vanish, as is the case for instance for systems with time-reversal symmetry (TRS), it is possible to construct exponentially localised Wannier functions. Different algorithms exist to construct them, either by optimising a smoothness criterion starting from a physical initial guess [18,22,24], or by building a smooth gauge from the ground up [11,5,6,3,14,10].…”

Section: Introductionmentioning

confidence: 99%

Localised Wannier Functions in Metallic Systems

Cornean¹,

Gontier²,

Levitt³

et al. 2019

Ann. Henri Poincaré

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The existence and construction of exponentially localised Wannier functions for insulators is a well-studied problem. In comparison, the case of metallic systems has been much less explored, even though localised Wannier functions constitute an important and widely used tool for the numerical band interpolation of metallic condensed matter systems. In this paper we prove that, under generic conditions, N energy bands of a metal can be exactly represented by N + 1 Wannier functions decaying faster than any polynomial. We also show that, in general, the lack of a spectral gap does not allow for exponential decay.Let H be a complex Hilbert space of dimension M ∈ N ∪ {+∞}. We consider a family of selfadjoint operators H(k) acting on H parametrised by k ∈ R d with d ≤ 3, and satisfying the following properties:

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Section: Introductionmentioning

confidence: 99%

Localised Wannier Functions in Metallic Systems

Cornean¹,

Gontier²,

Levitt³

et al. 2019

Ann. Henri Poincaré

Self Cite

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“…where C is the anti-unitary operator given by the complex conjugation. In particular, this implies that the Chern number of this family equals zero and that one can construct [13,23,9,6] a global smooth sectionφ0 : T * → F such that…”

Section: (Non)existence Of Localized Wannier Functionsmentioning

confidence: 99%

Peierls' substitution for low lying spectral energy windows

2019

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We consider a 2d magnetic Schrödinger operator perturbed by a weak magnetic field which slowly varies around a positive mean. In a previous paper we proved the appearance of a 'Landau type' structure of spectral islands at the bottom of the spectrum, under the hypothesis that the lowest Bloch eigenvalue of the unperturbed operator remained simple on the whole Brillouin zone, even though its range may overlap with the range of the second eigenvalue. We also assumed that the first Bloch spectral projection was smooth and had a zero Chern number.In this paper we extend our previous results to the only two remaining possibilities: either the first Bloch eigenvalue remains isolated while its corresponding spectral projection has a non-zero Chern number, or the first two Bloch eigenvalues cross each other.

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“…Following [9], we will explain how to produce an approximation α ′ (k 2 , k 3 ) of α δ (k 2 , k 3 ) with the following properties:…”

Section: Parseval Frames For Fermi-like Magnetic Projectionsmentioning

confidence: 99%

“…Finally let us sketch the main ideas borrowed from [9] which are behind the proof of the three properties of α ′ listed above.…”

Section: Parseval Frames For Fermi-like Magnetic Projectionsmentioning

confidence: 99%

Parseval Frames of Exponentially Localized Magnetic Wannier Functions

Cornean

Monaco

Moscolari

2019

Commun. Math. Phys.

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Motivated by the analysis of Hofstadter-like Hamiltonians for a 2-dimensional crystal in presence of a uniform transverse magnetic field, we study the possibility to construct spanning sets of exponentially localized (generalized) Wannier functions for the space of occupied states.When the magnetic flux per unit cell satisfies a certain rationality condition, by going to the momentum-space description one can model m occupied energy bands by a real-analytic and Z 2periodic family {P (k)} k∈R 2 of orthogonal projections of rank m. More generally, in dimension d ≤ 3, a moving orthonormal basis of Ran P (k) consisting of real-analytic and Z d -periodic Bloch vectors can be constructed if and only if the first Chern number(s) of P vanish(es). Here we are mainly interested in the topologically obstructed case.First, by dropping the generating condition, we show how to construct a collection of m − 1 orthonormal, real-analytic, and Z d -periodic Bloch vectors. Second, by dropping the orthonormality condition, we can construct a Parseval frame of m + 1 real-analytic and Z d -periodic Bloch vectors which generate Ran P (k). Both constructions are based on a two-step logarithm method which produces a moving orthonormal basis in the topologically trivial case.A moving Parseval frame of analytic, Z d -periodic Bloch vectors corresponds to a Parseval frame of exponentially localized composite Wannier functions. We extend this construction to the case of Hofstadter-like Hamiltonians with an irrational magnetic flux per unit cell and show how to produce Parseval frames of exponentially localized generalized Wannier functions also in this setting.

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On the Construction of Wannier Functions in Topological Insulators: the 3D Case

Cited by 12 publications

References 21 publications

Localised Wannier Functions in Metallic Systems

Localised Wannier Functions in Metallic Systems

Peierls' substitution for low lying spectral energy windows

Parseval Frames of Exponentially Localized Magnetic Wannier Functions

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