Inter-layer bias effect on time-reversal symmetry breaking in twisted bilayer cuprates

We study a one-band Hubbard model of twisted bilayer cuprates with a twist angle of ${53.13}^{\circ }$. By introducing an inter-layer bias, we simulate heterobilayers of different dopings. Using the variational cluster approximation (VCA) we probe the effect of this bias on the time-reversal-symmetry (TRS) breaking phase. Doping differences between layers affect the region where TRS-breaking occurs; we construct a phase diagram mapping out the TRS-breaking phase in the $n_1$-$n_2$ plane, $n_{\mathrm{\ell }}$ being the electron density on layer $\mathrm{\ell }$. We also map the spontaneous supercurrent on the same plane.