PHY 214B: Phase Transitions & Critical Phenomena, UCI Spring Quarter 2016, HW #1 S.A. Parameswaran

Due in my FRH 4129 mailbox (listed under Ashok-Parameswaran) by 4 pm Thursday, April 14, 2016. 1. Coupled Orders and Multicritical Points. Consider a theory with two scalar order parameters φ1 , φ2 , each transforming under an independent Ising symmetry Si , so that Si φi (x) = −φi (x). (a) Based on the usual arguments of stability, symmetry and analyticity, write down the simplest individual LandauGinzburg theories for each order parameter, and explain the conditions so that it is sufficient to only keep terms up to quartic order in each. (Take care to choose independent parameters for each theory, and use the same notation and factor-of-two conventions as in class to connect to the rest of the problem.) (b) Now, argue that the simplest term that couples the order parameters given constraints of symmetry and analyticity takes the form Z βHc = dd x 2u12 φ21 φ22 (1) (c) For the case t1 = t2 and u1 = u2 = u12 , this model reduces to a well-known model; what is it? (It is often useful to identify such special points or lines that have extra symmetry.) (d) Assuming that the fields coupling to each order parameter vanish (i.e., h1,2 = 0), determine the mean-field phase t2 −t1 2 2 structure in the r − g plane, where r ≡ t1 +t 2 ,g ≡ 2 . Your answer will be different for the cases u1 u2 < u12 2 and u1 u2 > u12 ; explain why. (e) Draw the mean field phase diagram for the two cases in part (d) above, identifying the order of each transition. An n-critical point is defined as one where n second-order lines meet; identify the bi- and tetra-critical points in each phase diagram. Such multicritical points are considered ‘fine tuned’ and hence less ‘natural’ or ‘non-generic’ as they are not intersected by tuning a single parameter (we will give a more formal definition of this when we study the renormalization group.) (f) Sketch the expected behavior of the two order parameters in the two cases above, for the following lines: fixed small and large r, varying g; fixed small and large g, varying r; and lines that pass through the multicritical points. (g) Now, an experimentalist is studying a system that has two possible orders. The first is an orientational order, for instance a tetragonal distortion that breaks the fourfold rotational symmetry of the square lattice down to a twofold one (i.e., reduces the symmetry from that of a square to a rectangle). The other is a more standard uniaxial magnetic order. Using the results of this problem, what can you say about the possible critical behavior the experimentalist can observe? [This result is illustrative of a general principle: within Landau theory it is ‘unnatural’ for one order to turn on at the precise point where the other turns off.] (h) How would the answers to this problem change if the two order parameters transformed under the same symmetry, i.e. Sφi (x) = −φi (x) ? 2. Kardar, Problem 2.4. (Hint: Problem 2.1 is solved in the text and may be useful as a warm-up)