We consider a model of an electron in a crystal moving under the influence of an external electric field: Schrödinger's equation in one spatial dimension with a potential which is the sum of a periodic function V and a smooth function W . We assume that the period of V is much shorter than the scale of variation of W and denote the ratio of these scales by . We consider the dynamics of semiclassical wavepacket asymptotic (in the limit ↓ 0) solutions which are spectrally localized near to a crossing of two Bloch band dispersion functions of the periodic operator − 1 2 ∂ 2 z + V (z). We show that the dynamics is qualitatively different from the case where bands are well-separated: at the time the wavepacket is incident on the band crossing, a second wavepacket is 'excited' which has opposite group velocity to the incident wavepacket. We then show that our result is consistent with the solution of a 'Landau-Zener'-type model. arXiv:1709.05575v2 [math-ph] 1 Jun 2018

Abstract. We consider a model of an electron in a crystal moving under the influence of an external electric field: Schrödinger's equation with a potential which is the sum of a periodic function and a general smooth function. We identify two dimensionless parameters: (re-scaled) Planck's constant and the ratio of the lattice spacing to the scale of variation of the external potential. We consider the special case where both parameters are equal and denote this parameter ǫ. In the limit ǫ ↓ 0, we prove the existence of solutions known as semiclassical wavepackets which are asymptotic up to 'Ehrenfest time' t ∼ ln 1/ǫ. To leading order, the center of mass and average quasi-momentum of these solutions evolve along trajectories generated by the classical Hamiltonian given by the sum of the Bloch band energy and the external potential. We then derive all corrections to the evolution of these observables proportional to ǫ. The corrections depend on the gauge-invariant Berry curvature of the Bloch band, and a coupling to the evolution of the wave-packet envelope which satisfies Schrödinger's equation with a time-dependent harmonic oscillator Hamiltonian. This infinite dimensional coupled 'particle-field' system may be derived from an 'extended' ǫ-dependent Hamiltonian. It is known that such coupling of observables (discrete particle-like degrees of freedom) to the wave-envelope (continuum field-like degrees of freedom) can have a significant impact on the overall dynamics.

We consider the chiral model of twisted bilayer graphene introduced by Tarnopolsky, Kruchkov, and Vishwanath (TKV). TKV proved that for inverse twist angles α such that the effective Fermi velocity at the moiré K point vanishes, the chiral model has a perfectly flat band at zero energy over the whole Brillouin zone. By a formal expansion, TKV found that the Fermi velocity vanishes at α ≈ 0.586. In this work, we give a proof that the Fermi velocity vanishes for at least one α between 0.57 and 0.61 by rigorously justifying TKV’s formal expansion of the Fermi velocity over a sufficiently large interval of α values. The idea of the proof is to project the TKV Hamiltonian onto a finite-dimensional subspace and then expand the Fermi velocity in terms of explicitly computable linear combinations of modes in the subspace while controlling the error. The proof relies on two propositions whose proofs are computer-assisted, i.e., numerical computation together with worst-case estimates on the accumulation of round-off error, which show that round-off error cannot possibly change the conclusion of computation. The propositions give a bound below on the spectral gap of the projected Hamiltonian, an Hermitian 80 × 80 matrix whose spectrum is symmetric about 0, and verify that two real 18-th order polynomials, which approximate the numerator of the Fermi velocity, take values with a definite sign when evaluated at specific values of α. Together with TKV’s work, our result proves the existence of at least one perfectly flat band of the chiral model.

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We study the eigenvalue problem for a one-dimensional Dirac operator with a spatially varying "mass" term. It is well-known that when the mass function has the form of a kink, or domain wall, transitioning between strictly positive and strictly negative asymptotic mass, ±κ∞, at ±∞, the Dirac operator has a simple eigenvalue of zero energy (geometric multiplicity equal to one) within a gap in the continuous spectrum, with corresponding exponentially localized zero mode.We consider the eigenvalue problem for the one-dimensional Dirac operator with mass function defined by "glue-ing" together n domain wall-type transitions, assuming that the distance between transitions, 2δ, is sufficiently large, focusing on the illustrative cases n = 2 and 3. When n = 2 we prove that the Dirac operator has two real simple eigenvalues of opposite sign and of order e −2|κ∞|δ . The associated eigenfunctions are, up to L 2 error of order e −2|κ∞|δ , linear combinations of shifted copies of the single domain wall zero mode. For the case n = 3, we prove the Dirac operator has two non-zero simple eigenvalues as in the two domain wall case and a simple eigenvalue at energy zero. The associated eigenfunctions of these eigenvalues can again, up to small error, be expressed as linear combinations of shifted copies of the single domain wall zero mode. When n > 3 no new technical difficulty arises and the result is similar. Our methods are based on a Lyapunov-Schmidt reduction/Schur complement strategy, which maps the Dirac operator eigenvalue problem for eigenstates with near-zero energies to the problem of determining the kernel of an n × n matrix reduction, which depends nonlinearly on the eigenvalue parameter.The class of Dirac operators we consider controls the bifurcation of topologically protected "edge states" from Dirac points (linear band crossings) for classes of Schrödinger operators with domain-wall modulated periodic potentials in one and two space dimensions. The present results may be used to construct a rich class of defect modes in periodic structures modulated by multiple domain walls.

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