Exact solution on the magnetism and ferroelectricity in the Ising spin chain

In this paper, we explore the magnetoelectric coupling in Ca3CoMnO6-type compound with the consideration of interaction between spins. Under the linear approximation of nearest-neighbor spin interaction with respect to the ion displacement, both the Hamiltonian and the partition function of the system can be simplified as the summation of two independent items, one is linear harmonic oscillator relevant to the lattice vibration, and the other is only relevant to the spin. We obtain the magnetic and ferroelectric quantities of Ca3CoMnO6-type rigorously on the basis of the transfer-matrix method, qualitatively exhibiting the corresponding curves taken in the presence of temperature for zero magnetic field and various magnetic fields, respectively, and our calculation results are basically consistent with the behaviors in the experiment. We find that the magnetic susceptibility in the absence of magnetic field takes on the features of ferromagnetic Ising-like behavior. Moreover, the influence of different next-nearest-neighbor exchange interaction on the magnetic susceptibility and relative dielectric constant is given as well, exhibiting the corresponding complex response of the magnetic susceptibility based on the up–up–down–down spin structure