Wavepackets in inhomogeneous periodic media: Effective particle-field dynamics and Berry curvature

Abstract

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.

Wavepackets in Inhomogeneous Periodic Media: Propagation Through a One-Dimensional Band Crossing

“…The bound (3.6) follows immediately from (3.2). When we consider the dynamics of wavepackets associated with E n (p) or E n+1 (p) and spectrally localized close to p * , the gap condition (3.2) and Corollary 3.3 will allow us to bound contributions to the solution from all bands other than E n (p) and E n+1 (p) uniformly through the crossing time, see Appendix D of [47] for details.…”

“…asymptotics is known to be the general limiting time-scale of validity of wavepacket, or coherent state, approximations [41]. These results generalize to d-dimensional analogs of (1.1) [47,6], see also [24,25,39].…”

“…In this section we recall the simple Lemma which we use in the proofs of Theorem 3.12 and Theorem 3.20 to estimate the corrector to a wave-packet approximate solution. A similar strategy was followed in [6,47] Lemma 4.1. For 0 < T ≤ ∞, let ψ ∈ C 0 ([t 0 , T ); L 2 (R)) denote the unique solution of the initial value problem (1.1) with initial data ψ 0 (x) given at t = t 0 :…”

Multiband effects in equations of motion of observables beyond the semiclassical approach

“…For multiband wave packets, especially when there are crossing points in the band structure, this issue is currently under investigation. Recent studies have shown that the collapse time of a packet initially localized in a single band whose evolution allows interband transitions after encountering a crossing point is also relatively long and it is determined by the 1/h scale [23,24]. Also, it has been shown that a multiband description of a wave packet on a generic honeycomb potential with Dirac points accurately approximates the full solution to the Schroedinger equation for large times even when the packet is centered at the crossing point [25].…”

“…Also, in a transport calculation, which includes scattering (not considered here), the time scale for spreading of the packet in real space must be larger than the scattering time. The collapse of the packet in reciprocal space is currently under investigation as noted earlier, although it has been shown that even for multiband wave packets with crossing points in the band structure, the collapse times are relatively long [22][23][24][25]32]. Although this issue needs further separate investigations, the EOM for the pair of linear bands can be obtained as…”