Absence of floating phase in superconductors with time-reversal symmetry breaking on any lattice

Due to the interplay of multicomponent order parameters [e.g., a twisted bilayer superconductor with interlayer Josephson coupling or a frustrated (n≥3)-band superconductor], a superconductor can possess a U(1)×Z₂ symmetry, corresponding to the superconducting Tc and time-reversal symmetry breaking transition TTRSB, respectively. It was then conjectured that in this class of Hamiltonians, there exists a vast parameter regime O such that the system exhibits vestigial TRSB, i.e., TTRSB>Tc, while at the boundary ∂O, the system possesses a single phase transition TTRSB=Tc. In this paper, we provide evidence towards this conjecture by mathematically eliminating the possibility of a floating phase, i.e., TTRSB<Tc, for the strong coupling regime. More specifically, we prove that the correlation functions of $U(1)$ spins are bounded above by that of Z₂ spins for all temperatures and lattice structures (e.g., Zᵈ for all d). In particular, this guarantees the existence of high-Tc TRSB (and consequently topological) superconductivity in a large class of Hamiltonians. Note that the same property can also be proven for a certain parameter regime (Δ≥4/5) of the generalized XY model on any lattice structure, despite belonging to an entirely distinct class of U(1)×Z₂ Hamiltonians.