Ch6 Ising Model A

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

This note was uploaded on 10/14/2010 for the course PHYS 2051 taught by Professor Drkwong during the Spring '10 term at CUHK.

Page1 / 10

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online