Exact Solution for Three-Dimensional Ising Model

摘要: Three-dimensional Ising model in zero external field is exactly solved by operator algebras, similar to the Onsager's approach in two dimensions. The partition function of the simple cubic crystal imposed by the periodic boundary condition along both $(0 1 0)$ and $(0 0 1)$ directions and the screw boundary condition along the $(1 0 0)$ direction is calculated rigorously. In the thermodynamic limit an integral replaces a sum in the formula of the partition function. A order-disorder transition in the infinite crystal occurs at a temperature $T=T_c$ determined by the condition: $\sinh\frac{2J}{k_BT_c}\sinh\frac{2(J_1+J_2)}{k_BT_c}=1$, where $(J_1 J_2 J)$ are the interaction energies in three directions, respectively. The analytical expressions for the internal energy and the specific heat are also given. It is also shown that the thermodynamic properties of 3D Ising model with $J_1=J_2$ are connected to those in 2D Ising model with the interaction energies $(J_1 J_{2D})$ by the relation $(\frac{J_{2D}}{k_BT})^*=(\frac{J}{k_BT})^*-\frac{J_1}{k_BT}$, where $x^*=\frac{1}{2}{\rm ln coth} x={\rm tanh}^{-1}(e^{-2x})$.