The geometry of Floquet states

Abstract

This paper studies the moduli space of "good" periodic Hamiltonians, i.e., periodic Hamiltonians with nondegenerate Floquet spectrum. The corresponding Floquet states determine a geometric model for the moduli space of such Hamiltonians. The model is expressed in terms of configuration spaces, flag manifolds, and principal bundles. A corollary of the main result is the complete determination of the rational cohomology ring of the space of good periodic Hamiltonians. The paper also considers the adiabatic driving of a 1-parameter family of good periodic Hamiltonians. We formulate two conjectures describing the resulting asymptotic quantum evolution. These are Floquet analogues of the adiabatic theorem in quantum mechanics.