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Ch6 Ising Model A

# Ch6 Ising Model A - Chapter 6 Phase transition II Ising...

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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 mean-field approxima- tion. The Ising model’s partition function can be exactly calculated in one- and two-dimensional 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 nearest-neighbor 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 anti-ferromagnetic coupling, J > 0, the spins prefer to being anti-parallel 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 nearest-neighbor pairs with both of the two spins at the state | + ⟩ ( |−⟩ ), N + , − is the number of nearest-neighbor 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 non-trivial. 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 order-disorder 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 high-temperature, 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., X-ray 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 nearest-neighbor pairs of A- and B-atoms, respectively, and N AB are the number of hetero-atom nearest-neighbor pairs.are the number of hetero-atom nearest-neighbor pairs....
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Ch6 Ising Model A - Chapter 6 Phase transition II Ising...

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