Abstract adiabatic charge pumping

TL;DR

This paper analyzes a general formula for quantum adiabatic charge pumping in closed systems with time-dependent Hamiltonians, deriving leading terms and relating pumped charge to dynamical and geometric phases.

Contribution

It provides a rigorous analysis of the adiabatic charge pumping formula, including explicit leading order terms and their relation to phases in time-periodic Hamiltonians.

Findings

Leading order charge expressed via dynamical and geometric phases.

Charge over a period related to derivatives of phases.

Analysis under the gap hypothesis for non-degenerate spectra.

Abstract

This paper is devoted to the analysis of an abstract formula describing quantum adiabatic charge pumping in a general context. We consider closed systems characterized by a slowly varying time-dependent Hamiltonian depending on an external parameter $\alpha$. The current operator, defined as the derivative of the Hamiltonian with respect to $\alpha$, once integrated over some time interval, gives rise to a charge pumped through the system over that time span. We determine the first two leading terms in the adiabatic parameter of this pumped charge under the usual gap hypothesis. In particular, in case the Hamiltonian is time periodic and has discrete non-degenerate spectrum, the charge pumped over a period is given to leading order by the derivative with respect to $\alpha$ of the corresponding dynamical and geometric phases.