1 Physics 127b - Final Exam Instructions: This exam must be taken in one sitting over the period of 5 hours from start to • end. Please note on the top of your solution the starting andfinishing time. The exam contains four problems. To obtain a perfect score you need to answer • only three out of the four problems. If you attempt more than 3 problems, please indicate which problems you would like to be graded. you may use any personal notes, course hand-outs, and Goldenfeld’s book, in • addition to a textbook of your choice. Indicate the additional textbook on the cover of your exam. If you need to make any assumptions due to lack of information in the text, • specify clearly your assumptions. Your solution is due in Scott’s mailbox by Thursday, March 19th, 5pm. • Do not discuss the exam with your colleagues before they (and you) handed in • the solution. 2 1. A one-dimensional magnetic spin-1 system is described by the following Hamil- tonian: ˆ= J SzSz + λ(Sz)2 h Sz (1) H − i i+1 i − i �i �i �i where the spin operatorS z can take the values 1,0, 1. i − (a) What is the transfer matrixT for this model? (8) ——- In the following consider the case ofh = 0, and a chain ofN sites, with open boundary conditions. So the transfer matrix may not be the most important tool, but you are free to use a method of your choice. (b) What is the partition function of this model? (9) (c) Find the entropy per site of the chain as a function of temperatureT . (7) (d) What is the correlation function S S ? (9) � i j� 2. Wilson-Fisherfixed point of anm 8 free-energy. Consider the following free energy for an Ising-like spin system:

d 1 2 1 2 1 8 L[m] = d x ( m) + a0(T T c)m + wm hm (2) F � �2 ∇ 2 − 8 − � wherea 0,w, are parameters, andh is the external magneticfield. When the temperatureT drops belowT c the system magnetizes through a second order phase transition. (a) (8) What is the saddle-point equation form(x)? (b) (10) Within meanfield theory, what are the critical exponentsβ,γ,δ, and ν associated with the transition? (note thath is nonzero only when one evaluatesδ). (10) (c) (8) What is the upper critical dimension for this model, i.e., the dimension above which meanfield theory applies, and the Gaussianfixed point holds? (You may either use RG scaling arguments, or the ginzburg criterion directly without deriving it) (d) (7) In class we carried out an approximate RG analysis of the Landau free energy for the Ising model which neglected the decimation step, and focused on the rescaling stage. Applying the same RG analysis to the problem at hand, show that below the upper critical dimension, thefixed point in which w is a non-zero constant is a criticalfixed point (i.e., has a critical manifold, and an unstable direction corresponding to the tuning parameter). What is the correlation-length exponentν within this approximation? 3 3. Formation of charge-density waves. Consider a square lattice in d-dimensions, where each lattice site can have either zero or one electrons. For simplicity assume that the electrons are spinless. Electrons on neighboring lattice sites have a repulsive interaction. Also, the lattice is in contact with a particle bath that has chemical potentialµ. The Hamiltonian describing this situation is:

ˆ= V ninj µni (3) H i, j − i ��� � wheren i is the occupation of sitei, and can ben i = 0, 1. (a) (8) Treat the Hamiltonian in mean-field theory neglecting charge- fluctuations from site to site. What is the effective single-site Hamiltonian in this approximation? (b) (7) Using the uniform mean-field assumption above,find a self consistent equation for the average occupation numbern= n . Show that forµ= � i� V d, the lattice is halffilled: · n=1/2 (4)

(c) (3) Show that whenµ=V d the Hamiltonian, up to an additive constant, · can be written as: 1 1 ˆ= V ni nj (5) H i, j � − 2� � − 2� ��� ——- Since the square lattice is bipartite, the electrons on the lattice may reduce the interaction energy byfilling one sublattice (say sublattice-A) more than the other (say sublattice-B), i.e.,n A > nB. This effect is called dimerization. (d) (8) Employ the mean-field approach andfind self-consistent equations for the averagefillings n , n . � A� � B� (e) (7) Using your mean-field results above, at what temperature would there be a dimerization transition? 4. Correlation length in afirst order transition. The Landau free energy of a magnetic substance is given by:

d 1 2 1 2 1 3 1 4 L[m] = d x ( m) + rm λ(T)m + um (6) F � �2 ∇ 2 − 3 4 � 4 withr 0 andu> 0 are both constants.λ(T ) is the onlyT -dependent ≥ parameter, and it is positive all through the problem.m is a real function of the space variablex. AsT is lowered,λ(T ) increases, and is a tuning parameter for afirst order transition. (a) (8) At whatm are the minima of this free energy located, as a function of λ (givenr andu)? Assume thatm is uniform. Note that there are two regimes with respect toλ. (b) (6) Out of the minima found above, which is a global minimum? Denote the magnetization at the global minimumm ;findm and [m ] as a function 0 0 F L 0 ofλ (givenr andu).

(c) (6) If your calculation so far is correct,m 0 experiences a jump at aλ PT . What isλ PT ? Draw the phase diagram for the system in ther-λ plane (in the r,λ> 0 range). 1 3 (d) (7) The 3λm term makes the system undergo afirst order phase transition, rather than a second order one. Nevertheless we can associate a correla- tion length with this system by comparing the gradient term to the second derivative of the free energy with respect tom at the global minimum. By expanding the free energy to second order around the global minimumm 0 (i.e., express the free energy in terms ofΔm=m m , and keeping only − 0 quadratic terms),find the correlation length ,ξ(λ) as a function ofλ (for a givenu andr). (e) (6) Atr=λ = 0 thefirst-order transition line terminates. Show that the correlation lengthξ(λ PT ), evaluated on the transition line, diverges as: 1 ξ(λ) . (7) ∼ (λ)χ whenλ andr both approach zero. What is the critical exponentχ?