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PHYSICS 140B : STATISTICAL PHYSICS MIDTERM EXAM 9:00 am – 9:50 am / Friday Feb. 13 / WLH 2008 Consider a four-state ferromagnetic Ising model with the Hamiltonian X X Si Sj − H Si , Ĥ = −J i hiji where the first sum is over all links of a lattice of coordination number z. The spin variables Si take values in the set {−1 , 0 , 0 , +1}. Note that there are two distinct states, each with Si = 0, and a total of four possible states on each site. Taking the trace for a single site means we sum over the four independent states, one with S = +1, two with S = 0, and one with S = −1. (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 . [15 points] (b) Find the mean field free energy F (m, T, H). [15 points] (c) Adimensionalize, writing θ = kB T /zJ and h = H/zJ. Find the dimensionless free energy per site f = F/N zJ. [15 points] (d) What is the self-consistent mean field equation for m? [15 points] (e) Find the critical temperature θc . Show that when h = 0 the graphical solution to the mean field equation depends on whether θ < θc or θ > θc . [15 points] (f) For θ > θc , find m(h, θ) assuming |h| 1. [15 points] (g) What is the mean field result for hSi2 i? Interpret your result in the θ → ∞ and θ → 0 limits. Hint : We don’t neglect fluctuations from the same site. [10 points] 1