Construction of Real-Valued Localized Composite Wannier Functions for Insulators

Fiorenza, Domenico; Monaco, Domenico; Panati, Gianluca

doi:10.1007/s00023-015-0400-6

Cited by 33 publications

(47 citation statements)

References 48 publications

(16 reference statements)

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“…The vanishing of the latter allows to extend U obs on F , and therefore to cure Φ F . Notice that this construction was already indicated in [21] (see also [11]) for the case of the torus, that is, g = 1.…”

Section: B Extension Of Projections On General Surfacesmentioning

confidence: 72%

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: B Extension Of Projections On General Surfacesmentioning

confidence: 72%

Localised Wannier Functions in Metallic Systems

Cornean¹,

Gontier²,

Levitt³

et al. 2019

Ann. Henri Poincaré

Self Cite

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“…where α is a shorthand for either t or (t, s) in the Pauli, with s a spin index, and t the cell-periodic position coordinate that is defined with the equivalence t ∼ t + R (R being a Bravais-lattice vector). We define α =V unit cell dt, where V is the volume of the BZ, 132 and there is an additional sum over the spin index s for spinor wavefunctions. The inner product over one unit cell is then defined as…”

Section: Proof Of Equivalence Of µ ⊥ =3 and Nonzero Mirrormentioning

confidence: 99%

“…where V is the volume of the BZ. 132 For Wannier functions {|W αj j,R } j,αj ,R that are eigenfunctions of P rP with eigenvalues { j +R} j,R in the atomic limit (which exist for strong EBRs only), the non-Abelian Berry connection defined in Eq. (K56) reads as…”

Section: Using the Adiabatic Theoremmentioning

confidence: 99%

Topological Bloch oscillations

Höller

Alexandradinata

2018

Phys. Rev. B

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Bloch oscillations originate from the translational symmetry of crystals. These oscillations occur with a fundamental period that a semiclassical wavepacket takes to traverse a Brillouin-zone loop. We introduce a new type of Bloch oscillations whose periodicity is an integer (µ>1) multiple of the fundamental period. The period multiplier µ is a topological invariant protected by the space groups of crystals, which include more than just translational symmetries. For example, µ divides n for crystals with an n-fold rotational or screw symmetry; with a reflection, inversion or glide symmetry, µ equals two. We identify the commonality underlying all period-multiplied oscillations: the multi-band Berry-Zak phases, which encode the holonomy of adiabatic transport of Bloch functions in quasimomentum space, differ pairwise by integer multiples of 2π/µ. For a class of multi-band subspaces whose projected-position operators commute, period multiplication has a complementary explanation through the real space distribution of Wannier functions. This complementarity follows from a one-to-one correspondence between Berry-Zak phases and the centers of Wannier functions. A Wannier description of period multiplication does not always exist, as we exemplify with band subspaces with either a nonzero Chern number or Z2 Kane-Mele topological order. In the former case, we present general constraints between Berry-Zak phases and Chern numbers, as well as introduce a recipe to construct nontrivial Chern bands -by splitting elementary band representations. To help identify band subspaces with µ>1, a general theorem is presented that outputs Zak phases that are symmetry-protected to integer multiples of 2π/n, given the point-group symmetry representation of any gapped band subspace. A cold-atomic experiment that has observed period-multiplied Bloch oscillations is discussed, and directions are provided for future experiments.CONTENTS arXiv:1708.02943v4 [cond-mat.other]

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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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Construction of Real-Valued Localized Composite Wannier Functions for Insulators

Cited by 33 publications

References 48 publications

Localised Wannier Functions in Metallic Systems

Localised Wannier Functions in Metallic Systems

Topological Bloch oscillations

Peierls' substitution for low lying spectral energy windows

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