We study the twisted Hubbard model of a cuprate bilayer at a fixed twist angle $\theta ={53.13}^{\circ }$ and as a function of doping, using the variational cluster approximation, a method that treats short-range dynamical correlations exactly. At intermediate inter-layer tunneling, we observe a sudden change of the relative sign of the $d$-wave order parameters of two layers between the underdoped and overdoped regimes. At strong inter-layer tunneling, we observe a clear time-reversal symmetry breaking phase near optimal doping, in which the relative phase of the two layers changes continuously from 0 to $\pi$. However, this phase has trivial topology. We also apply a cluster extension of dynamical mean field theory to the same problem, but fail to detect a time-reversal breaking phase with that method, probably owing to the very small energy difference between the different states involved.

中文翻译：

---

扭曲双层铜酸盐的 Hubbard 模型的掺杂相图

我们研究了固定扭曲角的铜酸盐双层的扭曲 Hubbard 模型$\theta ={53.13}^{\circ }$并且作为掺杂的函数，使用变分簇近似，一种精确处理短程动态相关性的方法。在中间层间隧穿中，我们观察到相对符号的突然变化$d$- 在欠掺杂和过掺杂状态之间的两层的波阶参数。在强层间隧道效应下，我们观察到在最佳掺杂附近出现明显的时间反演对称破缺相，其中两层的相对相位从 0 到$\pi$. 然而，这个阶段的拓扑结构很简单。我们还将动态平均场理论的集群扩展应用于同一问题，但未能使用该方法检测到时间反转中断阶段，这可能是由于所涉及的不同状态之间的能量差异非常小。