Orbital Wigner functions and quantum transport in multiband systems

Traditional theories of electron transport in crystals are based on the Boltzmann equation and do not capture physics arising from quantum coherence. We introduce a transport formalism based on ''orbital Wigner functions'', which accurately captures quantum coherent physics in multiband fermionic systems. We illustrate the power of this approach compared to traditional semiclassical transport theory by testing it numerically against microscopic simulations of one-dimensional, non-interacting, two-band systems -- the simplest systems capable of exhibiting inter-orbital coherence. We show that orbital Wigner functions accurately capture strongly non-equilibrium features of electron dynamics that lie beyond conventional Boltzmann theory, such as the ballistic transport of a relative phase between microscopic orbitals and topological Thouless pumping of charge both at non-zero temperature and away from the adiabatic limit. Our approach is motivated in part by modern ultracold atom experiments that can prepare and measure far-from-equilibrium charge transport and phase coherence in multiband fermionic systems, calling for correspondingly precise theories of transport. The quantitative accuracy exhibited by our approach, together with its capacity to capture nontrivial physics even at the ballistic scale, establishes orbital Wigner functions as an ideal starting point for developing a fully systematic theory of transport in crystals.