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Unformatted text preview: Chapter 7 Phase Transition III: Landau Theory 7.1 Order parameter In Landau theory of phase transition, a phase transition is signaled by the emergence of a ther- modynamic quantity, which is zero above the transition temperature and grows (or jumps) to be non-zero below the transition temperature. The emergence of an order parameter is accom- panied by certain symmetry breaking. Di ff erent systems and di ff erent phase transitions have di ff erent kinds of order parameters, which could be a scalar, a complex number, a vector, a projected vector [which is defined equivalent to its inversion (i.e., v ⇔ − v )], a tensor, or more complex quantities. A few examples of order parameters for di ff erent phase transitions are listed below Phase transition Symmetry broken Order parameter order-disorder reflection ( Z 2 ) ρ 1 − ρ 2 uniaxial FM reflection ( Z 2 ) M liquid-gas reflection ( Z 2 ) ρ l − ρ g Heisenberg FM rotation [ S O (3)] M superfluid U 1 gauge SF wavefunction | ψ | e i ϕ superconductivity U 1 gauge gap | ∆ | e i ϕ BEC U 1 gauge BEC wavefunction | ψ | e i ϕ nematic liquid crystal S O (3) / Z 2 rotation orientation (3 nn − 1) ferroelectricity lattice symmetry polarization P 7.2 Free energy Suppose the Hamiltonian of a system in an external field h is H = H − h ˆ m , (7.1) the Gibbs free energy is G = Tr ( ρ H ) + k B T Tr ( ρ ln ρ ) . (7.2) The order parameter is m = ⟨ ˆ m ⟩ = Tr( ρ ˆ m ) , (7.3) 83 84 CHAPTER 7. PHASE TRANSITION III: LANDAU THEORY which is conjugate to the external field. The Helmholtz free energy is A = G + hm = ⟨ H ⟩ + k B T Tr ( ρ ln ρ ) , (7.4) which is the internal energy minus the entropy times the temperature. The order parameter and the external field are related to through the free energies by m = − ∂ G ∂ h , and h = + ∂ A ∂ m . (7.5) In the mean field theory, if the order parameter is known (by doing experiments and / or insights), one can replace in the Hamiltonian the microscopic operator ˆ m with its mean-field average and neglect the fluctuation. Thus the free energies can be obtained as an analytical function of the order parameter. Let us first consider the case in absence of external field, in which the two free energies are the same. Suppose the system at high temperature is symmetric at m = 0, the free energy should have a minimum at m = 0. Also, for the moment, we assume a symmetric form of free energy G ( m , T ) = G ( − m , T ). Thus G ( m , t ) = A ( m , t ) = a ( T ) + 1 2 b ( T ) m 2 + 1 4 c ( T ) m 4 + 1 6 d ( T ) m 6 + ··· . (7.6) In presence of external field, G ( m , t ) = − hm + a ( T ) + 1 2 b ( T ) m 2 + 1 4 c ( T ) m 4 + 1 6 d ( T ) m 6 + ··· . (7.7) It can be shown that the free energy of the Ising model with mean-field approximation has the above form. Such expansion is a general form of free energies in mean-field theories....
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- Spring '10
- Critical phenomena, Phase transition, order parameter, Landau theory