On the Construction of Composite Wannier Functions

Cornean, Horia D.; Herbst, Ira; Nenciu, G.

doi:10.1007/s00023-016-0489-2

Cited by 31 publications

(88 citation statements)

References 32 publications

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“…2 The reduction of the problem from periodic time-reversal symmetric Hamiltonians to families of projectors as in Assumption 2.1 is given e. g. in [CHN,Sec. 1.2].…”

Section: Statement Of the Problem And Main Resultsmentioning

confidence: 99%

“…Since the family of projectors {P (k)} k∈R d satisfies the analyticity assumption (P 1 ), one may ask whether real analytic Bloch frames can be constructed for {P (k)} k∈R d . Arguing as in [CHN,Lemma 2.3], one can indeed show that whenever a continuous Bloch frame exists, than it can be easily modified into a real analytic one; the procedure preserves the symmetries (periodicity and TRS) whenever they are required. The result of Theorem 2.5 can therefore be strengthened and gives the existence of real analytic Bloch frames with the prescribed symmetry properties.…”

Section: 2mentioning

confidence: 97%

“…It follows then that the family of fiber projectors {P Σ (k)} k∈R d is unitarily equivalent to a family of projectors {P (k)} k∈R d satisfying Assumption 2.1 (compare [CHN,Sec. 2.1]).…”

Section: 2mentioning

confidence: 99%

“…With respect to a possibly different increasing labelling, the arguments {φ j (k 2 )} 1≤j≤m of the eigenvalues of α gen (k 2 ) can be chosen in [0, 2π) for k 2 ∈ [1/2 − ǫ 1 , 1/2], and depend analytically on k 2 on the same interval: this can be achieved in view of the (local) real analyticity of α gen (k 2 ) around half-integer values of k 2 , which implies by the Analytic Rellich Theorem [CHN,Sec. 2.7.4] that also its eigenvalues can be chosen to be real analytic.…”

Section: 31mentioning

confidence: 99%

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Wannier functions and ℤ2 invariants in time-reversal symmetric topological insulators

2017

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Abstract. We provide a constructive proof of exponentially localized Wannier functions and related Bloch frames in 1-and 2-dimensional time-reversal symmetric (TRS) topological insulators. The construction is formulated in terms of periodic TRS families of projectors (corresponding, in applications, to the eigenprojectors on an arbitrary number of relevant energy bands), and is thus model-independent. The possibility to enforce also a TRS constraint on the frame is investigated. This leads to a topological obstruction in dimension 2, related to Z2 topological phases.We review several proposals for Z2 indices that distinguish these topological phases, including the ones by Fu-Kane [FK], Prodan [Pr2], Graf-Porta [GP] and Fiorenza-Monaco-Panati [FMP2]. We show that all these formulations are equivalent. In particular, this allows to prove a geometric formula for the the Z2 invariant of 2-dimensional TRS topological insulators, originally indicated in [FK], which expresses it in terms of the Berry connection and the Berry curvature.

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“…2 The reduction of the problem from periodic time-reversal symmetric Hamiltonians to families of projectors as in Assumption 2.1 is given e. g. in [CHN,Sec. 1.2].…”

Section: Statement Of the Problem And Main Resultsmentioning

confidence: 99%

Section: 2mentioning

confidence: 97%

Section: 2mentioning

confidence: 99%

Section: 31mentioning

confidence: 99%

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Wannier functions and ℤ2 invariants in time-reversal symmetric topological insulators

2017

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“…This question is crucial in the study of conduction/insulation properties in crystals via maximally localized Wannier functions (see e. g. [21,4]). There are by now several techniques that are able to construct analytic frames out of continuous ones preserving moreover all the symmetries, for example by convolution with suitable kernels [6,7]. These are all incarnations of the more general Oka's principle, which states that in fair generality the obstruction to the triviality of a vector bundle in the continuous category can be lifted to the analytic one [23].…”

Section: Remark 2 (Analytic Bloch Frames)mentioning

confidence: 99%

Advances in Quantum Mechanics

Michelangeli¹,

Dell’Antonio²

2017

Springer INdAM Series

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The use of topological invariants to describe geometric phases of quantum matter has become an essential tool in modern solid state physics. The first instance of this paradigmatic trend can be traced to the study of the quantum Hall effect, in which the Chern number underlies the quantization of the transverse Hall conductivity. More recently, in the framework of time-reversal symmetric topological insulators and quantum spin Hall systems, a new topological classification has been proposed by Fu, Kane and Mele, where the label takes value in Z 2 . We illustrate how both the Chern number c ∈ Z and the Fu-Kane-Mele invariant δ ∈ Z 2 of 2-dimensional topological insulators can be characterized as topological obstructions. Indeed, c quantifies the obstruction to the existence of a frame of Bloch states for the crystal which is both continuous and periodic with respect to the crystal momentum. Instead, δ measures the possibility to impose a further time-reversal symmetry constraint on the Bloch frame.

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The Localization Dichotomy for Gapped Periodic Systems and Its Relevance for Macroscopic Transport

Panati

2018

Springer Proceedings in Mathematics &Amp; Statistics

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On the Construction of Composite Wannier Functions

Cited by 31 publications

References 32 publications

Wannier functions and ℤ2 invariants in time-reversal symmetric topological insulators

Wannier functions and ℤ2 invariants in time-reversal symmetric topological insulators

Advances in Quantum Mechanics

The Localization Dichotomy for Gapped Periodic Systems and Its Relevance for Macroscopic Transport

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