We investigate the magnetic properties of spin-$1/2$ charged Fermi gases with ferromagnetic coupling via mean-field theory, and find the interplay among the paramagnetism, diamagnetism and ferromagnetism. Paramagnetism and diamagnetism compete with e
ach other. When increasing the ferromagnetic coupling the spontaneous magnetization occurs in a weak magnetic field. The critical ferromagnetic coupling constant of the paramagnetic phase to ferromagnetic phase transition increases linearly with the temperature. Both the paramagnetism and diamagnetism increase when the magnetic field increases. It reveals the magnetization density $bar M$ increases firstly as the temperature increases, and then reaches a maximum. Finally the magnetization density $bar M$ decreases smoothly in the high temperature region. The domed shape of the magnetization density $bar M$ variation is different from the behavior of Bose gas with ferromagnetic coupling. We also find the curve of susceptibility follows the Curie-Weiss law, and for a given temperature the susceptibility is directly proportional to the Land{e} factor.
Within the mean-field theory, we investigate the magnetic properties of a charged spin-1 Bose gas in two dimension. In this system the diamagnetism competes with paramagnetism, where Lande-factor $g$ is introduced to describe the strength of the para
magnetic effect. The system presents a crossover from diamagnetism to paramagnetism with the increasing of Lande-factor. The critical value of the Lande-factor, $g_{c}$, is discussed as a function of the temperature and magnetic field. We get the same value of $g_{c}$ both in the low temperature and strong magnetic field limit. Our results also show that in very weak magnetic field no condensation happens in the two dimensional charged spin-1 Bose gas.
Magnetic properties of a charged spin-1 Bose gas with ferromagnetic interactions is investigated within mean-field theory. It is shown that a competition between paramagnetism, diamagnetism and ferromagnetism exists in this system. It is shown that d
iamagnetism, being concerned with spontaneous magnetization, cannot exceed ferromagnetism in very weak magnetic field. The critical value of reduced ferromagnetic coupling of paramagnetic phase to ferromagnetic phase transition $bar I_{c}$ increases with increasing temperature. The Lande-factor $g$ is introduced to describe the strength of paramagnetic effect which comes from the spin degree of freedom. The magnetization density $bar M$ increases monotonically with $g$ for fixed reduced ferromagnetic coupling $bar I$ as $bar I>bar I_{c}$. In a weak magnetic field, ferromagnetism makes immense contribution to the magnetization density. While at a high magnetic field, the diamagnetism inclines to saturate. Evidence for condensation can be seen in the magnetization density at weak magnetic field.
Within the two-leg $t$-J ladder, the spin dynamics of the pressure-induced two-leg ladder cuprate superconductor Sr$_{14-x}$Ca$_{x}$Cu$_{24}$O$_{41}$ is studied based on the kinetic energy driven superconducting mechanism. It is shown that in the pre
ssure-induced superconducting state, the incommensurate spin correlation appears in the underpressure regime, while the commensurate spin fluctuation emerges in the optimal pressure and overpressure regimes. In particular, the spin-lattice relaxation time is dominated by a temperature linear dependence term at low temperature followed by a peak developed below the superconducting transition temperature, in qualitative agreement with the experimental observation on Sr$_{14-x}$Ca$_{x}$Cu$_{24}$O$_{41}$.
