PHY 214B: Phase Transitions & Critical Phenomena, UCI Spring Quarter 2016, HW #2 S.A. Parameswaran
Due in my FRH 4129 mailbox (listed under Ashok-Parameswaran) by 4 pm Thursday, April 21, 2016. 1. Gaussian Integrals. Parts of this problem are from Kardar (pp 43-45) but please show all your work: without a firm knowledge of Gaussian integral identities, it will be very difficult to follow the lectures on the renormalization group. q R∞ K 2 h2 2π 2K (a) The simplest Gaussian integral involves a single variable φ: I1 = −∞ dφ e− 2 φ +hφ = . Show that Ke 1 ; compute the corresponding cumulants hφic ≡ hφi, hφ2 ic ≡ hφ2 i − hφi2 . All higher hφi = h/K, hφ2 i = h2 /L2 + K cumulants vanish for the Gaussian distribution. (b) Consider a multivariable Gaussian integral " # Z ∞ Y N X X Kij IN = φi φj + hi φi . dφi exp − +i,j (1) 2 −∞ i=1 i
where Kij = Kji . By reducing this to N one dimensional integrals by diagonalizing Kij , show that r −1 X Kij (2π)N exp hi hj IN = det K 2 i,j
(2)
P (c) The joint characteristic function of the φi s is given by he−i J kj φJ i; derivatives of this (with respect to the ki s) yield moments of the distribution, and cumulants are obtained by derivative of its logarithm. Show that −1 P X X K ij −1 he−i J kj φJ i = exp −i Kij hi kj − ki kj (3) 2 i,j i,j P −1 −1 and use this to confirm that hφi ic = j Kij hj and hφi φj ic = Kij . P 1 (d) Show that if A = i ai φi , then hexp(A)i = exp hAic + 2 hAic . Use this to verify the result (computed in class) for order-parameter correlations of the phase-fluctuating superfluid. 2. Domain Wall Arguments for the Ising Model. In class, we argued that the neglect of fluctuations is particularly egregious when there are ‘soft’ transverse (Goldstone) modes, since they can destroy order in low dimensions, below the lower critical dimension (dl = 2). For the Ising model, n = 1 and so there are no Goldstone modes, so this argument fails. Indeed, the Ising model does have a finite-temperature phase transition for d = 2. However, for d = 1, fluctuations again destroy the order – this comes from the thermal population of domain walls. In this problem we will explore the role of domain walls in d = 1, 2. We will often switch back and forth between continuum field theories and lattice models; the latter are often useful for things like Pdomain wall energetics. Consider the lattice Ising model, where the energy of a configuration is given by H = −J hi,ji si sj where h. . .i denotes nearest neighbors. First, focus on the d = 1 case, and assume we have an N -site chain. (a) Determine the energy of a domain wall; does it depend on the location? Is this similar or different to the domain walls considered in the Landau-Ginzburg theory for an n = 1 magnet? (b) Now, estimate the entropy of a state with a single domain wall, and hence compute its free energy. Explain why this result suggests that in reality the system is disordered for all T > 0. (c) Repeat this discussion for an N × N d = 2 lattice; now the domain wall energy depends on its length, L, and its entropy is a bit trickier to compute. Show that the system is stable against domain formation if T < Tc ≈ 2J/(kB ln 2), surprisingly close to the exact result of Tc = 2.269815 . . . J/kB . A more sophisticated version of this argument was used by Peierls to prove the existence of a phase transition in this model. (Hint: there are N possible starting points for the wall, and you can use the fact (stated without proof here) that the entropy of a non-looping chain of length L is at least 2L . To determine how L depends on N , use the fact that at each step the wall has two choices, to move “ahead” or “deflect” (left or right of where it is heading, or ahead; assuming these are equiprobable, the average length of the domain wall is ∼ 2N . Argue that the standard deviation will be small for large N – using the similarity to a random walk – and use this to compute the entropy.) 3. Scaling Arguments in Quantum Mechanics. Show that the spectrum of a quantum-mechanical particle moving in a logarithmic central potential, V (r) = −V0 log r/a has a mass-independent excitation spectrum; all the dependence on the mass can be removed by redefining the zero of energy.
