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Unformatted text preview: Chapter 6 Phase transition II: Ising model (A) In this chapter we discuss the phase transition of Ising model, using the meanfield approxima tion. The Ising model’s partition function can be exactly calculated in one and twodimensional lattices. It provides a platform for understanding phase transitions and critical phenomena. The Ising model can indeed be used to describe some physical systems such as alloys with order disorder transitions and uniaxial ferromagnetism. 6.1 Model The Ising model of spins in a lattice is given by the Hamiltonian ˆ H = J ∑ ⟨ ij ⟩ σ z i σ z j − 1 2 µ H ∑ i σ z j , (6.1) where ϵ is the coupling constant between nearestneighbor spins (denoted as ⟨ ij ⟩ ), µ is the magnetic moment of a spin, H is the external magnetic field, and the Pauli matrices are σ x = ( 0 1 1 0 ) , σ y = ( − i i ) , σ z = ( 1 − 1 ) . (6.2) This Hamiltonian is invariant by reversing all the spins. Such symmetry with two symmetry operations (one keeping all spins unchanged and one reserving all) is called Z 2 symmetry. In the case of ferromagnetic coupling, J < 0, the spins have lower energy when the neigh bors are polarized parallel. Otherwise in the case of antiferromagnetic coupling, J > 0, the spins prefer to being antiparallel to lower their energy. The Hamiltonian is diagonal. The eigen states are  ± 1 , ± 1 , . . . , ± 1 ⟩ (6.3) with eigen energy U = JN ++ + JN −− − JN + − − µ H 2 ( N + − N − ) , (6.4) where N ++ ( N −− ) is the number of nearestneighbor pairs with both of the two spins at the state  + ⟩ ( −⟩ ), N + , − is the number of nearestneighbor pairs with the two spins in opposite states, and 73 74 CHAPTER 6. PHASE TRANSITION II: ISING MODEL (A) N ± is the number of spins in the state ±⟩ . The evaluation of the partition function Z N = Tr e − β ˆ H ( N ) , (6.5) however, is nontrivial. The Ising model can be mapped equivalently to some other models including the order disorder transition model for alloys and the lattice gas model. In the orderdisorder transition model, two kinds of atoms are mixed in a lattice. To be specific, we consider ZnCu alloy which is a bcc lattice. Each atom could be in either a center position or a vertex position. At hightemperature, each kind of atoms have the same probability in di ff erent kinds of positions. The alloy is in a disordered phase. When the temperature is below a critical value T c , a certain kind of atoms have more probability to occupy the center positions than the vertices, the alloy enters into an ordered phase which can be experimentally verified by, e.g., Xray di ff raction. The energy of a certain configuration of di ff erent types of atoms in the lattice is approximately determined by the interaction between nearest neighbors, U = ϵ AA N AA + ϵ BB N BB + ϵ AB N AB , (6.6) where N AA and N BB are the numbers of nearestneighbor pairs of A and Batoms, respectively, and N AB are the number of heteroatom nearestneighbor pairs.are the number of heteroatom nearestneighbor pairs....
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 Spring '10
 DrKwong
 Energy, Statistical Mechanics, Critical phenomena, Phase transition, Renormalization group

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