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We study the twisted Hubbard model of a cuprate bilayer at a fixed twistangle $\theta=53.13^{\circ}$ using the variational cluster approximation, amethod that treats short-range dynamical correlations exactly. At intermediateinterlayer tunneling, the phase difference $\phi$ between the $d$-wave orderparameters of two layers is $\pi$ in the overdoped regime, while it is zero inthe underdoped regime, close to the Mott phase. At strong interlayer tunneling,we observe a clear time-reversal symmetry breaking phase near optimal doping,in which the phase difference $\phi$ changes continuously from 0 to $\pi$.However, this phase has trivial topology. We also apply a cluster extension ofdynamical mean field theory to the same problem, but fail to detect atime-reversal breaking phase with that method.

Doping phase diagram of a Hubbard model for twisted bilayer cuprates