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 Problem Set # 2 Due: 2/29/08 Fluctuations 1. Coupling to a ‘massless’ field: Consider an ~n-component vector field m(x) coupled to a scalar field A(x), through the effective Hamiltonian dd K (∇~tm 2 u(~ e m + L βH = x m)2 + ~ + m 2)2 + 2 ~2A2 (∇A)2 ,2 2 2with K, L, and u positive. (a) Show that there is a saddle point sol ution o f the form ~ = eℓ and A(x) = 0, and m(x)mˆfind m for t > 0 and t < 0. (b) Sketch the heat capacity C = ∂2 ln Z/ ∂t2, and discuss its singularity as t → 0 in the saddle point approximation. (c) Include fluctuations by setting ~ =ˆ et,m(x)m + φℓ(x) eℓ + φt(x)ˆA(x) = a(x), and expanding βH to quadratic order in φ and a. (d) Find t he correlation lengths ξℓ, and ξt, for the longitudinal and transverse components of φ, for t > 0 and t < 0. (e) Find the correlation length ξa for the fluctuations of the scalar field a, for t > 0 and t < 0. (f) Calculate the correlation function ha(x)a(0)i for t > 0. (g) Compute the correction to the saddle point free energy ln Z, from fluctuations. (You can leave the answer in the form of integrals involving ξℓ, ξt, and ξa.) (h) Find the fluctuation corrections to the heat capacity in (b), again leaving the answer in the form of integrals. (i) Discuss the behavior of the integrals appearing above schematically, and state their dependence on t he correlation length ξ, and cutoff Λ, in different dimensions. (j) What is the critical dimension for the validity of saddle point results, and how is it modified by t he coupling to the scalar field? ********� � � � � 2. Random magneti c fields: Consider the Hamiltonian βH = dd x K (∇m)2 + tm 2 + u m 4 − h(x)m(x) ,2 2 where m(x) and h(x) are scalar fields, and u > 0. The random magnetic field h(x) results from frozen (quenched) impurities that a re independently distributed in space. For simplicity h(x) is assumed to be an independent Gaussian variable at each point x, such that h(x) = 0, and h(x)h(x ′ ) = Δδd(x − x ′ ), (1) where the over-line indicates (quench) averaging over all values of the random fields. The above equation implies that the Fourier transformed random field h(q) satisfies h(q) = 0, and h(q)h(q ′ ) = Δ(2π)dδd(q + q ′ ). (2) (a) Calculate the quench averaged free energy, fsp = min{Ψ(m)}m, as suming a saddle point solution with uniform magnetization m(x) = m. (Note that with this a ssumption, the random field disappears as a result of averaging and has no effect at this stage.) (b) Include fluctuations by setting m(x) = m + φ(x), and expanding βH to second order in φ. (c) Express the energy cost of the above fluctuations in terms of the Fourier modes φ(q). (d) Calculate the mean hφ(q)i, and the variance |φ(q)|2 c, where h· · ·i denotes the usual thermal expectation value for a fixed h(q). (e) Use the above results, in conjunction with Eq.(2), to calculate the quench averaged scattering line shape S(q) = h|φ(q)|2i. (f) Perform the Gaussian integrals over φ(q) to calculate the fluctuation corrections, δf [h(q)], to the free energy. � � � � � �� ∞ Reminder : dφdφ∗ exp − K |φ|2 + h∗φ + hφ∗ =2π exp |h|2 2 K 2K−∞ (g) Use Eq.(2) to calculate the corrections due to the fluctuations in the previous part to the quench averaged free energy f. (Leave t he corrections in the form of two integrals.)� (h) Estimate the singular t dependence of the integrals obtained in the fluctuation correc-tions to the free energy. (i) Fi nd the upper critical dimension, du, for the validity of saddle point critical b ehavior. ******** 3. Long–range interactions: Consider a continuous spin field ~s(x), subject to a long–range ferromagnetic interactio n dd xdd y ~s(x) · ~s(y) ,|x − y|d+σ as well as short-range interactions. (a) How is the quadratic term in the Landau-Ginzburg expansion modified by the pres-ence of this long-range interaction? For what values of σ is the long-range interaction dominant? (b) By estimating the mag nitude of thermally excited Goldstone modes (or otherwise), obtain the lower critical dimension dℓ below which there is no long–range order. (c) Find the upper critical dimension du, above which saddle point results provide a correct description of the phase transition.