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MIT OpenCourseWare http://ocw.mit.edu 8.334 Statistical Mechanics II: Statistical Physics of Fields Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.� � � � � � 8.334: Statistical Mechanics II Spring 2008 Test 3 Review Problems The test is ‘closed b ook,’ but if you wish you may bring a one-sided sheet of formulas. The intent o f this sheet is as a reminder of important formulas and definitions, and not as a compact transcription of the answers provided here. The test will be composed entirely from a subset of the following problems. Thus if you are familiar and comfortable with these problems, there will be no surprises! ******** 1. Continuous spins: In the standard O(n) model, n component unit vectors are placed on the sites of a lattice. The nearest neighbor spins are then connected by a b ond J~si · ~sj. In fact, if we are only interested in universal properti es, any generalized i nteraction f(~si ·~sj) leads to the same critical behavior. By analogy with the Ising model, a suitable choice is exp [f(~si · ~sj)] = 1 + (nt)~si · ~sj, resulting in the so called loop model. (a) Construct a high temperature ex pansion of the loop model (for the partition function Z) in the parameter t, on a two-dimensional hexagonal (honeycomb) lattice. (b) Show that the limit n → 0 describes the configurations of a single self-avoiding polymer on the lattice. ******** 2. Potts mod el I: Consider Potts spins si = (1, 2, · · · , q), interacting via the Hamiltonia n −βH = K <ij> δsi,sj . (a) To treat this problem gra phically at high temperatures, the Boltzmann weight for each bond is written as exp Kδsi,sj = C(K) [1 + T (K)g(si, sj)] , with g(s, s ′) = qδs,s ′ − 1. Find C(K) and T (K) . (b) Show that q q q g(s, s ′ ) = 0 , g(s1, s)g(s, s2) = qg(s1, s2) , and g(s, s ′ )g(s ′ , s) = q 2(q − 1). s=1 s=1 s,s ′ (c) Use the above results to calculate the free energy, and the correlation function hg(sm, sn)i for a one–dimensional chain. 1� � � � � �� � � � (d) Calculate the partiti on function on the square latt ice to order o f T4 . Also calculate the first term in the low–temperat ure expansion of this problem. (e) By comparing the first terms in low- and high–temperature series, find a duality rule for Potts models. Don’t worry about higher o rder graphs, they will work out! Assuming a single transition temperature, find the value of Kc(q). (f) How do the higher order terms in the high–temperature series for the Potts model differ from t hose of the Ising model? What is the fundamental difference that sets apart the graphs for q = 2? (This is ultimately the reason why only the I sing model is solvable.) ******** 3. Potts model II: An alternative expansion is obtained by starting with exp [Kδ(si, sj)] = 1 + v(K)δ(si, sj), where v(K) = eK − 1. In this case, the sum over spins does not remove any gra phs, and all choices o f distributing bonds at random on the lattice are acceptable. (a) Including a magnetic field h i δsi,1, show that the partition function takes the form bsZ(q, K, h) = v n c × q − 1 + e hnc , all graphs clusters c in graph where nbc and nsc are the numbers of bonds and sites in cluster c. This is known as the random cluster expansion. (b) Show t hat the limit q → 1 describes a percolation problem, in which bonds are randomly distributed on the la ttice with probability p = v/(v +1). What is the percolation threshold on the square lattice? (c) Show that in the li mit q → 0, only a single connected cluster contributes to leading order. The enumeratio n of all such clusters is known as listing branched lattice animals. ******** 4. Potts duality: Consider Potts spins, si = (1, 2, · · · , q), placed on the sites o f a square lattice of N sites, interacting with their nearest-neighbors through a Hami ltonian −βH = K δsi,sj . <ij> (a) By comparing the first terms of high and low temperat ure series, o r by any other method, show that the partition function has the property � � � �2N eK − 1 Z(K) = qe 2NKΞ e −K= q −N e K + q − 1 Ξ , eK + (q − 1) 2� � � � for some function Ξ, and hence locate the critical point Kc(q). (b) Starting from the duality ex pression for Z(K), derive a similar relation for the internal energy U(K) = hβHi = −∂ ln Z/∂ ln K. Use this to calculate the exact value of U at the critical point. ******** 5. Anisotropic Random Walks: Consider the ensemble of all ra ndom walks on a square lattice starting at the origin (0,0). Each walk has a weight of txℓx × tyℓy , where ℓx and ℓy are the number of steps taken along the x and y directions respectively. (a) Calculate the total weight W (x, y), of all walks terminating at (x, y). Show that W is well defined only for t = (tx + ty)/2 < tc = 1/4. (b) What is the shape of a curve W (x, y) = constant, for large x and y, and close to the transition? (c) How does the average number of steps, hℓi = h ℓx + ℓyi, diverge as t approaches tc? ******** 6. Anisotropic Ising Mode l: Consider t he anisotropic Ising model on a square lattice with a Hamiltonian −βH = Kxσx,yσx+1,y + Kyσx,yσx,y+1 ; x,y i.e. with bo nds of different strengths along the x and y directions. (a) By following the method presented in the text, calculate the free energy for this model. You do not have to write down every step of the derivation. Just sketch the steps that need to be modified due to anisotropy; and calculate the final answer for ln Z/N. (b) Find the critical boundary in the (Kx, Ky) plane from the singularity of the free energy. Show t hat i t coincides with the condition Kx = K˜y, where K˜indicates the standard dual interaction to K. (c) Find the singular part of ln Z/N, and comment on how anisotro py affects critical behavior


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