Interlayer 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 interlayer bias, we simulate heterobilayers of different dopings. Using the variational cluster approximation, we probe the effect of this bias on the time-reversal symmetry-breaking (TRSB) phase. Doping differences between layers affect the region where TRSB occurs; we construct a phase diagram mapping out the TRSB phase in the $n_1-n_2$ plane, with $n_{\ell }$ being the electron density on layer $\ell$. We also map the spontaneous supercurrent on the same plane.