Relationship between Magnetic Anisotropy Below Pseudogap Temperature and Short-Range Antiferromagnetic Order in High-Temperature Cuprate Superconductor

The central issue in high-temperature cuprate superconductors is the pseudogap state appearing below the pseudogap temperature $T^*$, which is well above the superconducting transition temperature. In this study, we theoretically investigate the rapid increase of the magnetic anisotropy below the pseudogap temperature detected by the recent torque-magnetometry measurements on YBa$_2$Cu$_3$O$_y$ [Y. Sato et al., Nat. Phys., 13, 1074 (2017)]. Applying the spin Green's function formalism including the Dzyaloshinskii--Moriya interaction arising from the buckling of the CuO$_2$ plane, we obtain results that are in good agreement with the experiment and find a scaling relationship. Our analysis suggests that the characteristic temperature associated with the magnetic anisotropy, which coincides with $T^*$, is not a phase transition temperature but a crossover temperature associated with the short-range antiferromagnetic order.