Absence of vestigial time-reversal symmetry breaking above $T_c$ in the strong-coupling limit of a twisted bilayer superconductor

Phase ordering in a 2D superconductor occurs via a BKT transition. However, in bilayer systems with vanishing first-order interlayer Josephson coupling, the superconducting phase can also exhibit spontaneous time-reversal symmetry breaking (TRSB) due to the locking of the phase difference $phi$ across the junction via a second-order Josephson coupling $J_2 cos (2phi)$ with a sign that favors $phi = pm pi/2$. In particular, in unconventional superconductors (SC), the first-order term $-J_1( heta) cos phi$ can be tuned to vanish via twisting the relative orientation (e.g., by $ heta=pi/4$ in a $d$-wave SC), so that the second order coupling $J_2 cos (2phi)$ becomes the dominant interlayer interaction. However, it remains unsettled whether TRSB occurs above the BKT transition in the normal state ( extit{vestigial order}). Here we show that in the strong-coupling (large $J_2$) limit,

the TRSB onsets precisely at the superconducting transition when $J_1= 0$, and at a lower critical temperature when $J_1>0$.