Effective Action in Density Functional Theory and the Berry Phase

Abstract

This paper addresses two important questions of principle: (1) the contradiction in the traditional time-dependent (td) density functional theory (tdDFT) which seems to require that causality be taken into account explicitly in the variational principle, as pointed out in a critical review article by Gross et al [1]; and (2) to obtain the Berry phase that should be present in the time evolution of any given initial state when the system Hamiltonian has time varying parameters such as ion coordinates, magnetic field, etc. The Berry phase within the density functional theory is not expected to be explicitly found at first sight, since phases cancel out in forming the density, and thus are of no consequence in practice. We here show that the resolution of these two issues comes about by reformulating the traditional Dirac-Frenkel td action principle to take explicit account of the time-path in the form suggested by Jackiw and Kerman [2] instead of that used in the development of the tdDFT [1]. (Dr. Robert van Leeuwen has recently informed me that he too has a time-path version of the action principle which I have not seen.) This reformulation leads to an effective action functional which generates appropriate td Green’s functions and contains also the Berry phase as an identifiable piece of the action functional. It should be mentioned that the Berry phase is also identifiable in the Dirac-Frenkel action functional [3].