Optimal Decay of Wannier functions in Chern and Quantum Hall Insulators

Abstract

We investigate the localization properties of independent electrons in a periodic background, possibly including a periodic magnetic field, as e. g. in Chern insulators and in Quantum Hall systems. Since, generically, the spectrum of the Hamiltonian is absolutely continuous, localization is characterized by the decay, as |x| → ∞, of the composite (magnetic) Wannier functions associated to the Bloch bands below the Fermi energy, which is supposed to be in a spectral gap. We prove the validity of a localization dichotomy, in the following sense: either there exist exponentially localized composite Wannier functions, and correspondingly the system is in a trivial topological phase with vanishing Hall conductivity, or the decay of any composite Wannier function is such that the expectation value of the squared position operator, or equivalently of the Marzari-Vanderbilt localization functional, is +∞. In the latter case, the Bloch bundle is topologically non-trivial, and one expects a non-zero Hall conductivity.(2) Throughout this Section, we use Hartree atomic units, and moreover we reabsorb the reciprocal of the speed of light 1/c in the definition of the function A Γ .

A New Approach to Transport Coefficients in the Quantum Spin Hall Effect

“…The analysis of periodic operators is best performed in the so-called (magnetic) Bloch-Floquet-Zak representation (see e.g. [40,45,50] and references therein). The (magnetic) Bloch-Floquet-Zak transform is initially defined on compactly supported functions ψ ∈…”

“…For the sake of simplicity, we consider only d ≤ 3 and we ignore the "spin space" C N , but similar results holds true if, for example, V Γ is matrix-valued and acts non-trivially on these degrees of freedom. With the help of Kato's theory [35], and arguing as in [45] on the basis [11], it is not difficult to prove that, if A = A Γ is Γ -periodic, and C 1 denotes the fundamental cell of the lattice, then for the validity of (H 1 ) it is sufficient to assume either of the following two sets of hypotheses:…”

Robust determination of maximally localized Wannier functions

“…As already noticed [30], the existence of a smooth Bloch frame is not trivial in view of the competition between the smoothness of u and the property (6), which encodes a global topological constraint. For instance, it is known that in some models with broken time-reversal symmetry (e. g. in the presence of a magnetic field or in a Chern insulator) there cannot exist any such continuous frame, due to a topological obstruction [11,16,2,24].…”

Localised Module Frames and Wannier Bases from Groupoid Morita Equivalences

“…Because the operator H has a periodic potential, such bases exist by the Bloch-Floquet transform. The regularity of Wannier bases changes drastically depending on the topological properties of the spectral band of the Schrödinger operator, where a delocalised Wannier basis can be used as an indicator that the system has a non-trival topological phase, see [15,16,24,29,32] for example. Wannier bases with exponential decay can be constructed for periodic and aperiodic Hamiltonians such that the compression of a position operator by the Fermi projection has uniform spectral gaps [40,41].…”