  We introduce the concept of a Floquet gauge pump whereby a dynamically
engineered Floquet Hamiltonian is employed to reveal the inherent degeneracy of
the ground state in interacting systems. We demonstrate this concept in a
one-dimensional XY model with periodically driven couplings and transverse
field. In the high frequency limit, we obtain the Floquet Hamiltonian
consisting of the static XY and dynamically generated Dzyaloshinsky-Moriya
interaction (DMI) terms. The dynamically generated magnetization current
depends on the phases of complex coupling terms, with the XY interaction as the
real and DMI as the imaginary part. As these phases are cycled, it reveals the
ground state degeneracies that distinguish the ordered and disordered phases.
We discuss experimental requirements needed to realize the Floquet gauge pump
in a synthetic quantum spin system of interacting trapped ions.
