Edge currents as probe of topology in twisted cuprate bilayers

Bilayers made of high-$T_c$ cuprate superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ assembled with a twist angle close to $45^\circ$ have been recently shown to spontaneously break time reversal symmetry $\mathcal{T}$, consistent with theoretical predictions for emergent chiral topological $d_{x^2-y^2}+id_{xy}$ phase in such twisted $d$-wave superconductors. Here we use a minimal microscopic model to estimate the size of spontaneous chiral edge currents expected to occur in the $\mathcal{T}$-broken phase. In accord with previous theoretical studies of chiral $d$-wave superconductors we find small but non-vanishing edge currents which we nevertheless predict to be above the detection threshold of the state-of-the-art magnetic scanning probe microscopy. In addition, by deriving a simple relation between the edge current and the electron spectral function we help elucidate the longstanding disparity between the size of edge currents in chiral $d$-wave and $p$-wave superconductors.