Parseval frames of exponentially localized magnetic Wannier functions

Abstract

Motivated by the analysis of gapped periodic quantum systems in presence of a uniform magnetic field in dimension $d\le 3$, 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 ${\mathbb{Z}}^d$-periodic family $\{P(\mathbf{k}){\}}_{\mathbf{k}\in {\mathbb{R}}^d}$ of orthogonal projections of rank $m$. A moving orthonormal basis of $\mathrm{Ran}P(\mathbf{k})$ consisting of real-analytic and ${\mathbb{Z}}^d$-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 algorithmically construct a collection of $m-1$ orthonormal, real-analytic, and periodic Bloch vectors. Second, by dropping the linear independence condition, we construct a Parseval frame of $m+1$ real-analytic and periodic Bloch vectors which generate $\mathrm{Ran}P(\mathbf{k})$. Both algorithms 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, periodic Bloch vectors corresponds to a Parseval frame of exponentially localized composite Wannier functions. We extend this construction to the case of magnetic 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. Our results are illustrated in crystalline insulators modelled by $2d$ discrete Hofstadter-like Hamiltonians, but apply to certain continuous models of magnetic Schrödinger operators as well.

40 pages. Improved exposition and minor corrections. Final version matches published paper on Commun. Math. Phys

References in corpus (10)

• Spectral and Fermi surface properties from Wannier interpolation
• Optimal decay of Wannier functions in Chern and Quantum Hall insulators
• The Faraday effect revisited: Thermodynamic limit
• Robust determination of maximally-localized Wannier functions
• Wannier functions and Z_2 invariants in time-reversal symmetric topological insulators
• Tight frames of exponentially decaying Wannier functions
• Gauge-theoretic invariants for topological insulators: A bridge between Berry, Wess-Zumino, and Fu-Kane-Mele
• On the construction of Wannier functions in topological insulators: the 3D case
• Good Wannier bases in Hilbert modules associated to topological insulators
• Numerical construction of Wannier functions through homotopy

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• Good Wannier bases in Hilbert modules associated to topological insulators
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• Localised Wannier functions in metallic systems
• Matrix representation of Magnetic pseudo-differential operators via tight Gabor frames
• Topology vs localization in synthetic dimensions
• Purely linear response of the quantum Hall current to space-adiabatic perturbations
• Exploiting $\vartheta -$functions for the identification of topological materials