Emergence of charge order in a staggered loop-current phase of cuprate high-temperature superconductors

We study the emergence of charge-ordered phases within a pi-loop-current (pi LC) model for the pseudogap based on a three-band model for underdoped cuprate superconductors. Loop currents and charge ordering are driven by distinct components of the short-range Coulomb interactions: loop currents result from the repulsion between nearest-neighbor copper and oxygen orbitals, while charge order results from repulsion between neighboring oxygen orbitals. We find that the leading pi LC phase has an antiferromagnetic pattern similar to previously discovered staggered flux phases, and that it emerges abruptly at hole dopings p below the Van Hove filling. Subsequent charge-ordering tendencies in the pi LC phase reveal that diagonal d-charge density waves (dCDWs) are suppressed by the loop currents while axial order competes more weakly. In some cases we find a wide temperature range below the loop-current transition, over which the susceptibility towards an axial dCDW is large. In these cases, short-range axial charge order may be induced by doping-related disorder. A unique feature of the coexisting dCDW and pi LC phases is the emergence of an incommensurate modulation of the loop currents. If the dCDW is biaxial (checkerboard) then the resulting incommensurate current pattern breaks all mirror and time-reversal symmetries, thereby allowing for a polar Kerr effect.