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Due to the interplay of multi-component order parameters (e.g., a bilayersuperconductor with vanishing 1st order Josephson tunneling [1] or a frustrated$n\ge 3$-band superconductor [2]), a superconductor can possess a $U(1)\times\mathbb{Z}_2$ symmetry, corresponding to the superconducting $T_c$ andtime-reversal symmetry breaking transition $T_\text{TRSB}$, respectively. Itwas then conjectured that in vast parameter regimes (schematically denoted by$J_2$) and spatial dimensions $d$, there is only a single phase transition,i.e., $T_c = T_\text{TRSB}$. Mathematically, the conjecture is equivalent tothe following two properties being true simultaneously: (1) the absence of afloating phase $T_\text{TRSB} \ge T_c$ and (2) the absence of vestigial TRSB$T_\text{TRSB} \le T_c$. In this paper, we study the system in the strongcoupling limit $J_2 \to \infty$. By using exact dual graphical representations,we provide 2 independent proofs showing that the correlation functions of$U(1)$ spins is bounded above by that of $\mathbb{Z}_2$ spins for alltemperatures and lattice structures (e.g., $\mathbb{Z}^d$ for all $d$). Inparticular, this implies that property (1) must hold true on any latticestructure and thus guarantees the existence of high $T_c$ TRSBsuperconductivity. Further implications of this correlation inequality (such aswhether the conjectured single transition is first/second order) are alsodiscussed.  [1] O. Can, T. Tummuru, R. P. Day, I. Elfimov, A. Damascelli, and M. Franz,High-temperature topological superconductivity in twisted double-layer copperoxides, Nature Physics 17, 519 (2021)  [2] T. A. Bojesen, E. Babaev, and A. Sudb{\o}, Phase transitions andanomalous normal state in superconductors with broken time-reversal symmetry,Physical Review B 89, 104509 (2014).

Exact Inequality and the Absence of Floating Phase in a Superconductor with Time-reversal Symmetry Breaking on any Lattice