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.

arXiv comment40 pages. Improved exposition and minor corrections. Final version matches published paper on Commun. Math. Phys

References in corpus (20)

• Maximally localized Wannier functions: Theory and applications
• A beginner's guide to the modern theory of polarization
• Spectral and Fermi surface properties from Wannier interpolation
• Triviality of Bloch and Bloch-Dirac bundles
• Tensor network trial states for chiral topological phases in two dimensions and a no-go theorem in any dimension
• Optimal decay of Wannier functions in Chern and Quantum Hall insulators
• ${\mathbb{Z}}_2$ invariants of topological insulators as geometric obstructions
• The Faraday effect revisited: Thermodynamic limit
• Robust determination of maximally-localized Wannier functions
• Construction of real-valued localized composite Wannier functions for insulators
• Symmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry
• Wannier functions and Z_2 invariants in time-reversal symmetric topological insulators
• On the construction of composite Wannier functions
• Peierls substitution for magnetic Bloch bands
• On the Lipschitz continuity of spectral bands of Harper-like and magnetic Schroedinger operators
• 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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