2 First order adiabatic theory

Jianlan Chen, Meisheng Zhao, Jianda Wu +1 more

In this paper we present an invariant perturbation theory to adiabatic process according to the concepts of adiabatic orbits, adiabatic evolution orbit and U(1)-invariant adiabatic orbit. The probabilities of keeping the adiabatic orbit in the first-order and the second-order approximation are calculated, respectively. We also give a convenient sufficient condition.

G. V. Dubrovskiǐ, O.M. Pokrovskii

Gustavo Rigolin, Gerardo Ortíz, V. H. Ponce

We introduce a perturbative approach to solving the time-dependent Schr\"odinger equation, named adiabatic perturbation theory (APT), whose zeroth-order term is the quantum adiabatic approximation. The small parameter in the power series expansion of the time-dependent wave function is the inverse of the time it takes to drive the system's Hamiltonian from the initial to its final form. We review other standard perturbative and nonperturbative ways of going beyond the adiabatic approximation, extending and finding exact relations among them, and also compare the efficiency of those methods against the APT. Most importantly, we determine APT corrections to the Berry phase by use of the Aharonov-Anandan geometric phase. We then solve several time-dependent problems, allowing us to illustrate that the APT is the only perturbative method that gives the right corrections to the adiabatic approximation. Finally, we propose an experiment to measure the APT corrections to the Berry phase and show, for a particular spin-$1∕2$ problem, that to first order in APT the geometric phase should be two and a half times the (adiabatic) Berry phase.

Jonas Larson, Erik Sjöqvist, Patrik Öhberg

Rachel Berkowitz

Fan Zhang, Kieron Burke

In density functional theory, the adiabatic coupling may be treated perturbatively. This G\"orling-Levy perturbation theory can be applied to both nondegenerate and degenerate states. Formulas are given to second order. Some excitations are shown to be poorly described to first order (i.e., in exact exchange), no matter how weakly correlated the system is. Results are demonstrated on a simple model system, two fermions in a one-dimensional parabolic well with a contact interaction.