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

Abstract

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.

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