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Mean Field Theory of Phase Transitions 1

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Mean Field Theory of Phase Transitions 1 Contents Contents i List of Tables iii List of Figures iii 7 Mean Field Theory of Phase Transitions 1 7.1 References .............................................. 1 7.2 The van der Waals system ..................................... 2 7.2.1 Equationofstate ...................................... 2 7.2.2 Analytic form of the coexistence curve near the critical point ............ 5 7.2.3 History of the van der Waals equation ......................... 8 7.3 Fluids, Magnets, and the Ising Model .............................. 10 7.3.1 Lattice gas description of a fluid ............................. 10 7.3.2 Phase diagrams and critical exponents ......................... 12 7.3.3 Gibbs-Duhem relation for magnetic systems ...................... 13 7.3.4 Order-disorder transitions ................................ 14 7.4 MeanField Theory ......................................... 16 7.4.1 h = 0 ............................................ 17 7.4.2 Specific heat ........................................ 18 7.4.3 h = 0 ............................................ 19 6 7.4.4 Magnetization dynamics ................................. 21 i ii CONTENTS 7.4.5 Beyond nearest neighbors ................................ 24 7.4.6 Ising model with long-ranged forces .......................... 25 7.5 Variational Density Matrix Method ................................ 26 7.5.1 The variational principle ................................. 26 7.5.2 Variational density matrix for the Ising model ..................... 27 7.5.3 Mean Field Theoryof the PottsModel ......................... 30 7.5.4 MeanField Theoryofthe XY Model .......................... 32 7.5.5 XY model via neglect of fluctuations method ..................... 34 7.6 Landau Theory of Phase Transitions ............................... 35 7.6.1 Quartic free energy with Ising symmetry ........................ 35 7.6.2 Cubic terms in Landau theory : first order transitions ................ 37 7.6.3 Magnetization dynamics ................................. 39 7.6.4 Sixth order Landau theory : tricritical point ...................... 40 7.6.5 Hysteresis for the sextic potential ............................ 42 7.7 Mean Field Theory of Fluctuations ................................ 45 7.7.1 Correlation and response in mean field theory ..................... 45 7.7.2 Calculation of the response functions .......................... 46 7.7.3 Beyondthe Ising model .................................. 49 7.8 Global Symmetries ......................................... 51 7.8.1 Symmetries and symmetry groups ........................... 51 7.8.2 Lower critical dimension ................................. 53 7.8.3 Continuous symmetries ................................. 55 7.8.4 Random systems : Imry-Ma argument ......................... 56 7.9 Ginzburg-Landau Theory ..................................... 58 7.9.1 Ginzburg-Landau free energy .............................. 58 7.9.2 Domain wall profile .................................... 59 7.9.3 Derivation of Ginzburg-Landau free energy ...................... 59 7.9.4 Ginzburg criterion ..................................... 63 7.10 Appendix I : Equivalence of the Mean Field Descriptions ................... 65 7.10.1 Variational Density Matrix ................................ 66 7.10.2 Mean Field Approximation ................................ 67 7.11 Appendix II : Additional Examples ............................... 68 7.11.1 Blume-Capel model .................................... 68 7.11.2 Ising antiferromagnet in an external field ....................... 70 7.11.3 Canted quantum antiferromagnet ............................ 73 7.11.4 Coupled order parameters ................................ 75 List of Tables 7.1 van der Waals parameters for some common gases ....................... 4 7.2 Critical exponents .......................................... 21 List of Figures 7.1 Pressure versus volume for the van der Waals gas ........................ 3 7.2 Molar free energy f(T, v) of the van der Waals system ..................... 5 7.3 Maxwell construction in the (v,p) plane ............................. 7 7.4 Pressure-volume isotherms for the van der Waals system, with Maxwell construction . 8 7.5 Universality of the liquid-gas transition ............................. 9 7.6 [The lattice gas model ....................................... 11 iii iv LIST OF FIGURES 7.7 Comparison of liquid-gas and Ising ferromagnet phase diagrams .............. 14 7.8 Order-disorder transition on the square lattice ......................... 15 7.9 Ising mean field theory at h = 0 .................................. 18 7.10 Ising mean field theory at h = 0.1 ................................. 19 7.11 Dissipative magnetization dynamics m˙ = f (m) ....................... 22 − ′ 7.12 Magnetization hysteresis ...................................... 23 7.13 Variational field free energy versus magnetization ....................... 29 7.14 Phase diagram for the quartic Landau free energy ....................... 36 7.15 Behavior of the quartic free energy f(m)= 1 am2 1 ym3 + 1 bm4 .............. 38 2 − 3 4 7.16 Fixed points for ϕ(u)= 1 ru2 1 u3 + 1 u4 and flow u˙ = ϕ (u) ............... 40 2 − 3 4 − ′ 1 2 1 4 1 6 7.17 Behavior of the sextic free energy f(m)= 2 am + 4 bm + 6 cm ............... 41 7.18 Sextic free energy ϕ(u)= 1 ru2 1 u4 + 1 u6 for different values of r ............. 43 2 − 4 6 7.19 Fixed points ϕ (u ) = 0 and flow u˙ = ϕ (u) for the sextic potential ............. 44 ′ ∗ − ′ 7.20 A domain wall in a one-dimensional Ising model ........................ 53 7.21 Domain walls in the two and three dimensional Ising model ................. 54 7.22 A domain wall in an XY ferromagnet .............................. 55 7.23 Imry-Ma domains and free energy versus domain size ..................... 56 7.24 Mean field phase diagram for the Blume-Capel model ..................... 69 7.25 Mean field solution for an Ising antiferromagnet in an external field ............. 71 7.26 Mean field phase diagram for an Ising antiferromagnet in an external field ......... 72 7.27 Mean field phase diagram for the model of eqn. 7.338 ..................... 75 7.28 Phase diagram for the model of eqn. 7.350 ............................ 79 Chapter 7 Mean Field Theory of Phase Transitions 7.1 References – M. Kardar, Statistical Physics of Particles (Cambridge, 2007) A superb modern text, with many insightful presentations of key concepts. – M. Plischke and B. Bergersen, Equilibrium Statistical Physics (3rd edition, World Scientific, 2006) An excellent graduate level text. Less insightful than Kardar but still a good modern treatment of the subject. Good discussion of mean field theory. – G. Parisi, Statistical Field Theory (Addison-Wesley, 1988) An advanced text focusing on field theoretic approaches, covering mean field and Landau-Ginzburg theories before moving on to renormalization group and beyond. – J. P. Sethna, Entropy, Order Parameters, and Complexity (Oxford, 2006) An excellent introductory text with a very modern set of topics and exercises. Available online at http://www.physics.cornell.edu/sethna/StatMech 1 2 CHAPTER 7. MEAN FIELD THEORY OF PHASE TRANSITIONS 7.2 The van der Waals system 7.2.1 Equation of state Recall the van der Waals equation of state, a p + (v b)= RT , (7.1) v2 − where v = NAV/N is the molar volume. Solving for p(v, T ), we have RT a p = . (7.2) v b − v2 − Let us fix the temperature T and examine the function p(v). Clearly p(v) is a decreasing function of volume for v just above the minimum allowed value v = b, as well as for v . Butis p(v) a monotonic →∞ function for all v [b, ]? ∈ ∞ We can answer this by computing the derivative, ∂p 2a RT = . (7.3) ∂v v3 − (v b)2 T − Setting this expression to zero for finite v, we obtain the equation1 2a u3 = , (7.4) bRT (u 1)2 − where u v/b is dimensionless. It is easy to see that the function f(u) = u3/(u 1)2 has a unique ≡ 27 − minimum for u > 1. Setting f ′(u∗) = 0 yields u∗ = 3, and so fmin = f(3) = 4 . Thus, for T > Tc = 8a/27bR, the LHS of eqn. 7.4 lies below the minimum value of the RHS, and there is no solution. This means that p(v,T >Tc) is a monotonically decreasing function of v. At T = Tc there is a saddle-node bifurcation. Setting vc = bu∗ = 3b and evaluating pc = p(vc, Tc), we have that the location of the critical point for the van der Waals system is a 8a p = , v = 3b , T = . (7.5) c 27 b2 c c 27 bR For T < Tc, there are two solutions to eqn. 7.4, corresponding to a local minimum and a local maximum of the function p(v). The locus of points in the (v,p) plane for which (∂p/∂v)T = 0 is obtained by setting eqn. 7.3 to zero and solving for T , then substituting this into eqn. 7.2. The result is a 2ab p∗(v)= . (7.6) v2 − v3 Expressed in terms of dimensionless quantities p¯ = p/pc and v¯ = v/vc, this equation becomes 3 2 p¯∗(¯v)= . (7.7) v¯2 − v¯3 7.2. THE VAN DER WAALS SYSTEM 3 Figure 7.1: Pressure versus molar volume for the van der Waals gas at temperatures in equal intervals from T = 1.10 Tc (red) to T = 0.85 Tc (blue). The purple curve is p¯∗(¯v). Along the curve p = p (v), the isothermal compressibility, κ = 1 ∂v diverges, heralding a thermo- ∗ T − v ∂p T dynamic instability. To understand better, let us compute the free energy of the van der Waals system, F = E TS. Regarding the energy E, we showed back in chapter 2 that − ∂ε ∂p a = T p = , (7.8) ∂v ∂T − v2 T V which entails a ε(T, v)= 1 fRT , (7.9) 2 − v where ε = E/ν is the molar internal energy. The first term is the molar energy of an ideal gas, where f is the number of molecular freedoms, which is the appropriate low density limit. The molar specific heat ∂ε f is then cV = ∂T v = 2 R, which means that the molar entropy is T cV f s(T, v)= dT ′ = R ln(T/T )+ s (v) . (7.10) T 2 c 1 Z ′ We then write f = ε Ts, and we fix the function s (v) by demanding that p = ∂f . This yields − 1 − ∂v T s (v)= R ln(v b)+ s , where s is a constant. Thus2, 1 − 0 0 a f(T, v)= f R T 1 ln T/T RT ln(v b) Ts . (7.11) 2 − c − v − − − 0 1 There is always a solution to (∂p/∂v)T = 0 at v = ∞. 2Don’t confuse the molar free energy (f) with the number of molecular degrees of freedom (f)! 4 CHAPTER 7.
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