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 $\ensuremath{\pi}$-loop-current $(\ensuremath{\pi}\mathrm{LC})$ 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 $\ensuremath{\pi}\mathrm{LC}$ 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 $\ensuremath{\pi}\mathrm{LC}$ 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 $\ensuremath{\pi}\mathrm{LC}$ 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.