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PHYSICS 140B : STATISTICAL PHYSICS PRACTICE MIDTERM EXAM Consider a four-state ferromagnetic Ising model with the Hamiltonian X X X Si Sj − H Si , Si Sj − J2 Ĥ = −J1 hhijii hiji i where the first sum is over all nearest neighbor pairs and the second sum is over all next nearest neighbor pairs. The spin variables Si take values in the set − 32 , − 12 , + 12 , + 32 . (a) Making the mean field Ansatz Si = m + (Si − m), where m = hSi i is presumed independent of i, derive the mean field Hamiltonian ĤMF . You may denote z1 as the number of nearest neighbors and z2 as the number of next nearest neighbors of any site on the lattice. [15 points] (b) Find the mean field free energy F (m, T, H). [15 points] ˆ ˆ (c) Adimensionalize, writing θ = kB T /J(0) and h = H/J(0). Find the dimensionless free ˆ energy per site f = F/N J(0). [15 points] (d) What is the self-consistent mean field equation for m? [15 points] (e) Find the critical temperature θc . [15 points] (f) For θ > θc , find m(h, θ) assuming |h| 1. [15 points] (g) What is the mean field result for |Si | ? Interpret your result in the θ → ∞ and θ → 0 limits. Hint : We don’t neglect fluctuations from the same site. [10 points] 1