Competing orders in a three-orbital model of cuprates: Is there a loop-current order?

Apart from the prominent d  -wave superconductivity in cuprates, there are competing phases that are believed to be responsible for the pseudogap normal state. Proposals range from spin and charge density waves (SDW and CDW) at finite momentum to the more exotic loop-current order (LCO) at zero momentum. The latter is however beyond the scope of the effective single-orbital Zhang–Rice model. Aiming at treating all possible orderings in a unified framework, here we use a three-orbital model in terms of the copper dx2-y2dx2-y2 orbital and the oxygen pxpx and pypy orbitals, and perform functional renormalization group calculations for the effect of electron correlations. We find that in the parameter regime relevant to experiments, antiferromagnetic SDW and d  -wave superconductivity are the main ordering tendencies, with dominant contents from the copperdx2-y2dx2-y2-orbital, suggesting that the Zhang–Rice model is sufficient to capture the essential physics. If the Coulomb interaction between nearest copper and oxygen is hypothetically strong, an intra-unitcell CDW describing charge imbalance between copper and oxygen sites is realized. The long-sort LCO is too weak even in the three-orbital model, and should be irrelevant in cuprates.