Low lying spectral gaps induced by slowly varying magnetic fields

Abstract

We consider a periodic Schr\"odinger operator in two dimensions perturbed by a weak magnetic field whose intensity slowly varies around a positive mean. We show in great generality that the bottom of the spectrum of the corresponding magnetic Schr\"odinger operator develops spectral islands separated by gaps, reminding of a Landau-level structure. First, we construct an effective Hofstadter-like magnetic matrix which accurately describes the low lying spectrum of the full operator. The construction of this effective magnetic matrix does not require a gap in the spectrum of the non-magnetic operator, only that the first and the second Bloch eigenvalues do not cross but their ranges might overlap. The crossing case is more difficult and will be considered elsewhere. Second, we perform a detailed spectral analysis of the effective matrix using a gauge-covariant magnetic pseudo-differential calculus adapted to slowly varying magnetic fields. As an application, we prove in the overlapping case the appearance of spectral islands separated by gaps.Comment: 55 pages, to appear in J. Funct. Ana

Magnetic pseudodifferential operators represented as generalized Hofstadter-like matrices

“…Continuity of the spectrum can be proved under quite general conditions on the Hamiltonians [1,3,4], while more refined properties like the Lipschitz behaviour of spectral edges were first proved by Bellissard [5] for discrete Hofstadter-like models [17]. Cornean, Purice and Helffer [7,8,9,11,12]…”

Peierls' substitution for low lying spectral energy windows

“…We continue by noticing that Proposition 6.4 and the properties of the magnetic composition of symbols (see Proposition B.12 in [3]) imply that for any seminorm ν on S −∞ (Ξ) there exists Cν > 0 such that…”

“…We consider a more abstract setting and give the details of the arguments sketched in the third item of Subsection 2.3 that gives a procedure to apply the Feshbah-Schur method in a more complicate situation when the commutator H, π is no longer controlled by a small parameter (as was the situation in our previous paper [3]).…”

“…The magnetic quasi-band projections Following step by step the arguments in Section 3 of [3] we can associate with the quasi-band orthogonal projection π ∈ L(H) some magnetic versions of it: π ǫ and π ǫ,κ . Starting from the principal quasi Wannier function ψ0 introduced in Definition 3.2 and the magnetic field introduced in (1.5) we define the following objects:…”

“…We can also construct the Γ-periodic symbol pǫ ∈ S −∞ (Ξ) such that π ǫ = Op ǫ (pǫ) and the symbol pǫ,κ ∈ S −∞ (Ξ) such that π ǫ,κ = Op ǫ,κ (pǫ,κ) and the same proof as in [3] gives the following result: Note that in this case the commutator H ǫ,κ , π ǫ,κ is no longer small (due to the arbitrary deformation which was made in constructing the quasi Wannier function).…”