PHY 214B: Phase Transitions & Critical Phenomena, UCI Spring Quarter 2016, HW #3 S.A. Parameswaran

Due in my FRH 4129 mailbox (listed under Ashok-Parameswaran) by 4 pm Thursday, May 5, 2016. 1. Kardar, Problem 4.3. 2. Kardar, Problem 4.4. 3. The Kosterlitz Equations. The RG equations for two variables, K and y0 (as usual, b = e` ) are given by dK = −4π 3 K 2 a4 y02 + O(y04 ) d` dy0 = (2 − πK)y0 + O(y03 ). d`

(1)

Analyze these equations and sketch the RG flows in the (K −1 , y0 ) plane (you may assume both K −1 and y0 are positive and so work only in the upper right quadrant.) Show that y0 changes from being relevant to being irrelevant at a critical value of Kc−1 (determine this), leading to a line of fixed points at y0 = 0 and K −1 < Kc−1 ; along this line, y0 is irrelevant and K −1 is exactly marginal and so there is no flow. We will return to this when studying the Coulomb gas formulation of the 2D XY model (Hint: it helps to rewrite the first RG equation into an equivalent one for K −1 .)