  Besides exploring novel transition patterns, acquiring a full understanding
of the transition nature is an ultimate pursuit in studies of phase
transitions. The fundamental models of light-matter interactions manifest
single-qubit topological phase transitions, which is calling for an analytical
demonstration apart from numerical studies. We present a rigorous study for
topological transitions in Jaynes-Cummings Model generally with Stark
non-linear Coupling. In terms of the properties of Hermite polynomials, we show
that the topological structure of the eigen wave function has an exact
correspondence to the spin winding by nodes, which yields a full spin winding
without anti-winding nodes. The spurious fractional contribution to the winding
number of the winding angle at infinity is found to be actually integer. Thus,
the phase transitions in the model have a nature of topological phase
transitions and the excitation number is endowed as a topological quantum
number. The principal transition establishes a paradigmatic case that a
transition is both symmetry-breaking Landau class of transition and
symmetry-protected topological class of transition simultaneously, while
conventionally these two classes of transitions are incompatible due to the
contrary symmetry requirements. We also give an understanding for the origin of
unconventional topological transitions in the presence of counter-rotating
terms. Our results may provide a deeper insight for the few-body phase
transitions in light-matter interactions.
