Spin winding and topological nature of transitions in the Jaynes-Cummings model with Stark nonlinear coupling

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 (TPTs), which call for an analytical demonstration apart from numerical studies. We present a rigorous study of TPTs in the Jaynes-Cummings model generally with Stark nonlinear coupling. In terms of the properties of Hermite polynomials, we show that the topological structure of the eigenfunction has an exact correspondence to the spin winding by nodes, which yields a full spin winding without antiwinding nodes. We find it is the superposition between neighboring Fock states that leads to such a nontrivial spin winding. The spurious fractional contribution to the winding number of the winding angle at infinity is found to be actually an integer. Thus, the phase transitions in the model have the nature of TPTs and the excitation number is endowed as a topological quantum number. The principal transition establishes a paradigmatic case in which a transition is of both the symmetry-breaking Landau class of transition and the symmetry-protected topological class of transition, while conventionally these two classes of transitions are incompatible due to the contrary symmetry requirements. Such a transition-class reconciliation is realized by a preserved higher symmetry (here the parity) which protects the TPTs, while the symmetry breaking involves the subsymmetries. We also explain the origin of unconventional TPTs in the presence of counterrotating terms. Our results may provide deeper insight into the few-body phase transitions in light-matter interactions.