Study Of Villain Transformation For Ferrimagnetic Spin Chain

A variety of exotic physical phenomena in the low dimensional magnetic systems have been attracting much interest in recent years. In these systems, the physical pictures obtained from the classical approach are often greatly modified or even in contradiction with the result of the strong quantum fluctuation and topological effect. For the Heisenberg ferromagnetic spin chain, Haldane conjectured that the one-dimensional integer-spin chain with the nearest neighbor coupling has an energy gap in the spin excitation spectrum and the spin correlation decays exponentially with distance, whereas that of the half-odd-integer spin chain is gapless and the spin correlation decays algebraically with distance. For the Heisenberg ferrimagnetic spin chain, Wu study its ground state and thermodynamic properties by using the Schwinger-Boson Mean Field Theory. Brehemer et al. show that the absolute ground state of this model has a ferrimagnetic long-range order and obtained the low-lying excitation, by using the spin-wave theory and the Quantum Monte Carlo method. The above methods are good example of calculating ferrimagnetic spin chain.In this article, we will use the Villain transformation calculate the ferrimagnetic spin system, and there are five chapters in this article. Chapter 1, we introduce the developments of the low dimensional magnetic system. Chapter 2, we calculate the ferromagnetic and anti-ferromagnetic spin chain by using the Villain transformation, and obtain the dispersion relation and the correlation function. Chapter 3, Firstly, we use the Villain transformation calculate the ferrimagnetic spin chain, the thermodynamic properties as well as longitudinal and transverse correlation function are obtained; then, we calculate the ferrimagnetic spin chain under the external magnetic field, the magnetic susceptibility is obtained. Chapter 4, we study the ferrimagnetic spin chain with anisotropy under the magnetic field, the effect of the anisotropy on the thermodynamic variables are discussed. Chapter 5, the concludes are made.