Topics in Statistic Mechanics

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Adv. Sta. Phy. Homework 2 Blume-Capel Model Li,Zimeng PB06203182 stand for the interaction of the spins with the crystal field at each site i. If we set a=0, it becomes the Blume-Capel model (1.1.1) with B=0. Here . is well known for the normal Ising model, while zero occupation, or the site with spin annealed.(see Sec. 1.1) 2. Phase Transition Diagram can be viewed as Ising model with vacancies, or Considering with B=0. It is easily drawn that when , and the model returns to normal Ising model because is limited value, thus the Hamiltonian is rewritten as Ising Hamiltonian: Replace with (see Sec. 1.2), we rewrite the Hamiltonian of the simplified Blume- Capel model as: The ordered states with have energy per site , while the state with has zero energy.The ground state of an ordered state must thus guarantee the condition that , and we therefore conclude that there is a zero-temperature transition from the state with to ordered state with at . This is a first order transition, in the sense that there are discontinuities in the magnetization and the derivative of the energy with respect to .Also the correlation length is zero at this point, since there are no fluctuations.The first order transition must persist for some length since there is a convergent radius for the perturbation. Therefore, along the boundary of the ordered state and the disordered state, there is a change from the first order transition to the second order transition.And the transition point is called the tricritical point.The figure is shown below: (We can compare it to the tricritical Ising model Fig 1.1.1)
Adv. Sta. Phy. Homework 2 Blume-Capel Model Li,Zimeng PB06203182 Fig 2.1 3. Mean Field Solution of the Blume-Capel model Considering simplified Blume-Capel model Hamiltonian , where is a factor when the sum of different sites is repeated twice. 3.1 One Dimension Case The central idea of mean field theory is to approximate the interacting case by a simpler noninteracting partition function. We first rewrite where
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Topics in Statistic Mechanics

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