Incoherent tunneling and topological superconductivity in twisted cuprate bilayers

Twisting two monolayers of a high-$T_c$ cuprate superconductor can engender a chiral topological state with spontaneously broken time reversal symmetry $\mathcal{T}$. A crucial ingredient required for the emergence of a gapped topological phase is electron tunneling between the CuO$_2$ planes, whose explicit form (in an ideal clean sample) is dictated by the symmetry of the atomic orbitals. However, a large body of work on the interlayer transport in cuprates indicates importance of disorder-mediated incoherent tunneling which evades the symmetry constraints present in an idealized crystal. The latter arises even in the cleanest single-crystal samples through oxygen vacancies in layers separating the CuO$_2$ planes, introduced to achieve the hole doping necessary for superconductivity. Here we assess the influence of incoherent tunneling on the phase diagram of a twisted bilayer. We show that the model continues to support a fully gapped topological phase with broken $\mathcal{T}$, even in the limit of disorder-mediated interlayer coupling. Compared to the model with a constant, momentum conserving interlayer coupling, the extent of the topological phase around the 45$^\circ$ twist decreases with increasing incoherence, but remains robustly present for parameters likely relevant to Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$.