Noncommutative geometry and nonabelian Berry phase in the wave-packet dynamics of Bloch electrons

TL;DR

This paper develops a semiclassical framework describing Bloch electron wave-packet dynamics using nonabelian gauge fields linked to Berry phases, with applications to Hall and polarization currents in spintronics.

Contribution

It introduces a covariant, gauge-invariant set of equations of motion for wave packets in noncommutative geometry, generalizing Ehrenfest's theorem for systems with band degeneracy.

Findings

Derived exact semiclassical equations of motion incorporating nonabelian Berry connections.

Classified gauge fields into band projection and basis variation origins.

Demonstrated applications to various Hall and polarization currents, including parity polarization.

Abstract

Motivated by a recent proposal on the possibility of observing a monopole in the band structure, and by an increasing interest on the role of Berry phase in spintronics, we studied the adiabatic motion of a wave packet of Bloch functions, under a perturbation varying slowly and incommensurately to the lattice structure. We show using only the fundamental principles of quantum mechanics that its effective wave-packet dynamics is conveniently described by a set of equations of motion (EOM) for a semiclassical particle coupled to a nonabelian gauge field associated with a geometric Berry phase. Our EOM can be viewed as a generalization of the standard Ehrenfest's theorem, and their derivation was asymptotically exact in the framework of linear response theory. Our analysis is entirely based on the concept of local Bloch bands, a good starting point for describing the adiabatic motion of a wave packet. One of the advantages of our approach is that the various types of gauge fields were classified into two categories by their different physical origin: (i) projection onto specific bands, (ii) time-dependent local Bloch basis. Using those gauge fields, we write our EOM in a covariant form, whereas the gauge-invariant field strength stems from the noncommutativity of covariant derivatives along different axes of the reciprocal parameter space. The degeneracy of Bloch bands makes the gauge fields nonabelian. We applied our formalism to the analyses on various types of Hall and polarization currents. We highlighted their behavior under time reversal (T) and space inversion (I). The concept of parity polarization current was also introduced. Together with charge/spin Hall/polarization currents, this type of orbital current is expected to be a potential probe for detecting and controling Berry phase.