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.

The paper studies the global geometry of smooth, period-one Hamiltonians on an $n$-dimensional complex Hilbert space $W$ whose monodromy operator has simple spectrum. It asks how their Floquet states encode the space $\mathcal H_F(W)$ of such Hamiltonians, and formulates geometric descriptions of evolution under slow variation of a periodic drive. The main theorem gives a diffeomorphism, natural under unitary isomorphisms, $\mathcal H_F(W)\cong\widetilde C_n(S^1)\times\Omega SU(W)\times\Omega\mathbb R\times F(W)$, where $F(W)$ is the complete flag manifold and $\widetilde C_n(S^1)$ is the universal cover of the unordered configuration space. The proof represents Hamiltonians by quasi-periodic orthonormal frames, quotients their phase and spectral-ordering freedoms, and uses determinant winding to select a unique representative. Since two factors are contractible, this gives $\mathcal H_F(W)\simeq\Omega SU(W)\times F(W)$ and determines its rational cohomology ring: a polynomial algebra with generator degrees $2,4,\ldots,2n-2$, tensored with the flag manifold’s coinvariant algebra. A related principal bundle is classified using eigenline Chern classes. For a fixed smooth path of these Hamiltonians traversed over time $O(N)$, an adiabatic conjecture predicts uniform $O(N^{-1})$ tracking of the frozen Floquet eigenline. A refinement specifies the endpoint state, including dynamical phases and parallel transport under a constructed Floquet connection, to the same accuracy. These dynamical statements remain conjectural: only a formal derivation is reported, with rigorous evolution and phase estimates deferred to future work. The setting is finite-dimensional and excludes degenerate monodromy spectra.