Tuning topological superconductivity within the $t$-$J$-$U$ model of twisted bilayer cuprates

We carry out a theoretical study of unconventional superconductivity in twisted bilayer cuprates (TBC) as a function of electron density and layer twist angle. The bilayer $t$-$J$-$U$ model is employed and analyzed within the framework of a generalized variational wave function approach in the statistically-consistent Gutzwiller formulation. The constructed phase diagram encompasses both gapless $d$-wave state (reflecting the pairing symmetry of untwisted copper-oxides) and gapped $d+{\mathrm{e}}^{i\phi }d$ phase that breaks spontaneously time-reversal-symmetry (TRS) and is characterized by nontrivial Chern number. We find that $d+{\mathrm{e}}^{i\phi }d$ state occupies a non-convex butterfly-shaped region in the doping vs. twist-angle plane, and demonstrate the presence of previously unreported reentrant TRS-breaking phase on the underdoped side of the phase diagram. This circumstance supports the emergence of topological superconductivity for fine-tuned twist angles in TBC away from ${45}^{\circ }$. Our analysis of the microscopically derived Landau free energy functional points toward sensitivity of the superconducting order parameter to small perturbations close to the topological state boundary.