Models of Magnetic System in the High Magnetic Field

From the historical point of view, the invention of the magnetic model is to describe the microscopic phase transition phenomena or the critical behavior of the variety of physical systems. The simplest magnetic model is called Ising model where each site can have two values (+1 and -1) and interactions only occur at neighboring sites. The model was rstly proposed by Wihelm Lenz (1920) in order to explain ferromagnetism from microscopic rst principles. Ernst Ising revisited the model in 1925 and solved the one-dimensional case. He found that there was no phase transition (or say Tc=0), and then roughly argued the same result in the higher dimensional case. However, Dramers and Wannier (1941) gave the finite Tc result qualitatively, and Lars Onsager (1944) derived the analytic free energy and gave the nite Tc result quantitatively. Both of them studied the two-dimensional case, and disproved the Ising's prediction. Ising model, as a harbinger of the microscopic model, has been becoming one of the pillars of statistical mechanics. There are lots of models after Ising. Heisenberg model (1928), for instance, is the most essential and representative model. There are three quantum spin components in it, and the non-commutability of the components leads to the quantum nature of the model. Throughout the thesis I divide into two topics about magnetic systems. One is classical and the other is quantum mechanical. Before start- ing I must explain what are the differences between classical and quantum spins. Generally we should denote the spin as an operator whose components adopt the commutation relations, Si;Sj =ihei jkSk. “Classical” means that one can take h!0 and thus all the components are com- mutable. This can be done when the temperature is high or the spin value is huge. Another possibility is due to some kinds of anisotropic limit (Jx,Jy << Jz) such that only z-component exists in the model (Ising), and we regard it as “classical” rather than “quantum”. Firstly I discuss the classical antiferromagnetic Ising chain with single-ion anisotropy term. I will use the transfer matrix method to find the analytic solvability of the one-dimensional Ising case even when the extra nonlinear term was added. In addition, I also propose a new method to deal with such a problem without solving the eigenvalue problems. In the second part I will start from a recently discovered magnetic material, called Han purple with the structure of quantum magnetic dimmer. Under the high magnetic eld this material will present BEC phenomena which has been discovered by the experiment and analyzed by numerical works recently. However, in these works they made some artifial transformation to study the model. In this thesis I will use a new mean field method to theoretically understand the physical insight of this material, and compare the analytical result with the experiment and the QMC (Quantum Monte Carlo) simulation.1 Exact ground-state phase diagram of antiferromagnetic Ising spin-S (S 1) chain in external magnetic eld and with single-ion anisotropy 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 S=1 Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 S= 3 2 and S=2 Cases . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Rules and the General Case S=s, s= 1 2 ;1; 3 2 ;2; 5 2 ; : : : . . . . . . . . 6 1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.7 Appendix 1. Details about the transition lines of the general case S=s, s= 1 2 ;1; 3 2 ;2; 5 2 ; : : : in h-D diagram. . . . . . . . . . . . . . . . . . 8 1.8 Appendix 2. Proof of the spin con guration which is only to be bipartite. 9 2 Magnetic Quantum Phase Transition of Han Purple 19 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Mapping from Spin to Bond Operator . . . . . . . . . . . . . . . . . 20 2.3 Mean Field of the Hamiltonian . . . . . . . . . . . . . . . . . . . . . 23 2.4 Excitations of the Mean Field Hamiltonian . . . . . . . . . . . . . . . 27 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.6 Appendix 1 De nitions of the notations in (2.52) . . . . . . . . . . . 32 2.7 Appendix 2 De nitions of the notations in (2.74) . . . . . . . . . . . 3