In the replica symmetric approximation and the static limit in Matsubara ``imaginary time,'' we have investigated a quantum XY spin-glass system with the ferromagnetic coupling and the uniaxial anisotropy numerically. We found that, for
S=1, under the uniaxial anisotropy, the spin-glass phase breaks into two phases: a longitudinal spin-glass phase, and a spin-glass phase; the mixed phase of the spin glass and the ferromagnet breaks into two phases: a mixed phase of the longitudinal spin glass and the longitudinal ferromagnet, a mixed phase of the spin glass and the ferromagnet; the peak in the curve of the specific heat versus temperature is split into two peaks: the peak of uniaxial anisotropy and the peak of the ferromagnetic coupling. The system will be dominated by the random exchange interaction if the probability of the random exchange interaction taking negative value is greater than 15.87%. In the absence of the uniaxial anisotropy, there is a mean-interaction translational invariance in the spin-glass phase and the paramagnetic phase. In the presence of the uniaxial anisotropy, there is a mean-interaction translational invariance in the spin-glass phase, the longitudinal spin-glass phase and the paramagnetic phase. In these phases, the entropy, the specific heat and the susceptibility do not depend on the mean-interaction.
In the generalized quantum replica symmetric approximation, a quantum XY spin-glass model with ferromagnetic coupling and planar Dzyaloshinski–Mariya (DM) interactions is investigated theoretically. Entropy and specific heat are calculated numerically as well as spin self-interactions and spin-glass order parameters for spin . It is found that the double-cusp of the specific heat, which corresponds to the collinear ferromagnetic and the paramagnetic ferromagnetic transitions respectively, coincides completely due to DM-anisotropy interactions. Additionally, the local susceptibility and the spontaneous magnetization are evaluated numerically.