A systematic derivative expansion of the adiabatic phase

Abstract

We present several methods for obtaining a derivative expansion for the adiabatic phase in quantum systems. We consider Hamiltonians of the form H( t) = H 0 + R i ( t) T i , where R i ( t) are external parameters, T i generate a simple three parameter Lie algebra and H 0 is an element of the center of the algebra. We identify the first correction to the classical adiabatic theorem as Berry's phase and compute the next two orders. In the course of the calculation we also reanalyze an example first studied by Berry and show that due to topological triviality Berry's phase for this system can be removed by a canonical transformation.