Incommensurable state of a spin density wave and superconductivity in quasi-two-dimensional systems with an anisotropic energy spectrum

We investigate phase transitions in quasi-two-dimensional systems with an anisotropic energy spectrum and a deviation from the half-filling of the energy band $(\mu\ne0)$. We demonstrate the possibility of the transition of an insulator into a half-metallic state when the nesting condition is violated because the parameter $\mu\ne0$ and of taking the umklapp processes into account. We obtain the basic equations for the parameters of the superconducting $(\Delta)$ and magnetic $(M)$ orders and determine the conditions for the emergence of superconductivity on the background of a spin-density-wave state and also for the coexistence of superconductivity and magnetism. We show that the transition of a magnetic system into a superconducting state as the parameter $\mu$ increases can be a first-order phase transition at low temperatures. We also obtain an expression for the heat capacity jump $(C_{\text{S}}-C_{\text{N}})$ at $T=T_{\text{c}}$, which depends on $M$ and $\mu$ and differs essentially from the case of the Bardeen–Cooper–Schrieffer theory. We also consider the transformations related to the density of electron states of the relevant anisotropic system, which can undergo essential changes under pressure or doping.