For the mean-field version of an important lattice-spin model due to Blume and Capel, we prove unexpected connections among the asymptotic behavior of the magnetization, the structure of the phase transitions, and a class of polynomials that we call the Ginzburg-Landau polynomials. The model depends on the parameters n, beta, and K, which represent, respectively, the number of spins, the inverse temperature, and the interaction strength. Our main focus is on the asymptotic behavior of the magnetization m(beta_n,K_n) for appropriate sequences (beta_n,K_n) that converge to a second-order point or to the tricritical point of the model and that lie inside various subsets of the phase-coexistence region. The main result states that as (beta_n,K_n) converges to one of these points (beta,K), m(beta_n,K_n) ~ c |beta - beta_n|^gamma --> 0. In this formula gamma is a positive constant, and c is the unique positive, global minimum point of a certain polynomial g that we call the Ginzburg-Landau polynomial. This polynomial arises as a limit of appropriately scaled free-energy functionals, the global minimum points of which define the phase-transition structure of the model. For each sequence (beta_n,K_n) under study, the structure of the global minimum points of the associated Ginzburg-Landau polynomial mirrors the structure of the global minimum points of the free-energy functional in the region through which (beta_n,K_n) passes and thus reflects the phase-transition structure of the model in that region. The properties of the Ginzburg-Landau polynomials make rigorous the predictions of the Ginzburg-Landau phenomenology of critical phenomena, and the asymptotic formula for m(beta_n,K_n) makes rigorous the heuristic scaling theory of the tricritical point.
Spin dependence of the tricritical point in the mixed-spin Blume-Capel model on three-dimensional lattices: Metropolis and Wang-Landau sampling approaches
Spin Fluctuation Theory for Quantum Tricritical Point Arising in Proximity to First-Order Phase Transitions: Applications to Heavy-Fermion Systems, YbRh2Si2, CeRu2Si2, and beta-YbAlB4
Dynamic Phase Transition in the Kinetic Spin-3/2 Blume-Capel Model: Phase Diagrams in the Temperature and Crystal-Field Interaction Plane
我们分析，在平均场的方法内，动力学自旋3/2的Blume-Capel模型的Glauber型随机动力学和随时间变化的振荡外部磁场的定态。动态相变点是通过研究动态磁化强度作为温度的函数的行为，以及计算Liapunov指数。在温度和晶场相互作用平面上构造了相图。我们发现了五种基本类型的相图的约化磁场振幅参数（h）的不同值，其中他们提出了一个无序，两个有序相和共存相区。相图还显示出0<h<1.44的动态双临界点，1.44<h 5.06的动态三临界点<5.06 and two dynamic tricritical points for h>。
A Phase Prediction Method for Pattern Formation in Time-Dependent Ginzburg-Landau Dynamics for Kinetic Ising Model without a priori Assumptions on Domain Patterns
Dynamic Phase Transition in a Time-Dependent Ginzburg-Landau Model in an Oscillating Field
从理论和数值两方面研究了在时间振荡外场中低于临界温度的Ginzburg-Landau模型。当外力的频率或振幅改变时，观察到非平衡相变。这种转变将空间均匀的恢复振动与破坏振动分开。在相变附近，发展了微扰理论，发现了在非线性破碎相的开关现象。我们的结果证实了本跃迁与振荡场中动力学Ising系统的蒙特卡罗模拟结果的等价性，证明了这两种情况下的非平衡相变都属于零场中平衡态伊辛模型的普适类.这一结论与对称性论证[G.格林施泰因角Jayaprakash和Y.他，物理牧师莱特。55，2527（1985）]和最近的数值结果[G. Korniss，C.J. White，P. A. Rikvold和M. A. Novotny，Phys. Rev. E（已提交）]。此外，基于Ornstein-Zernike近似的热噪声局域磁化结构函数的理论结果与一维数值结果吻合较好。
