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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.� � � � � � � � � � � � � IV.E Perturbative RG (First Order) The last section demonstrated how various expectation values associated with the Landau–Ginzburg Hamiltonian can be calculated perturbatively in powers of u. However, the perturbative series is inherently divergent close to the critical point and cannot be used to characterize critical behav ior in dimensions d ≤ 4. Wilson showed that it is possible to combine perturbative and renormalization group approaches into a systemati c method for calculating critical exponents. Accordingly, we shall extend the RG calculation of Gaussian model in sec.III.G to the Landau–Ginzburg Hamiltonian, by treating U = u � ddxm4 as a perturbation. 1. Coarse Grain: This is the most difficult step of the RG procedure. As before, subdivide the fluctuations into two components as,   ~˜m(q) for 0 < q < Λ/b ~ = (IV.28) m(q) .  ~σ(q) for Λ/b < q < Λ In the partition function, � � Λ Z = ~ σ(q) exp − ddq t + Kq2 � | ˜ + |σ(q)|2� ~ σ(q)] Dm˜(q)D~m(q)|2 − U[m˜(q), ~ ,(2π)d 20 (IV.29) the two sets of modes are mixed by the operator U. Formally, the result of i ntegrating out {~σ(q)} can be written as � � Λ/b Z = ~ − ddq t + Kq2 m(q)|2Dm˜(q) exp | ˜ × (2π)d 20 �� (IV.30) exp − nV � Λ ddq ln � t + Kq2�� e −U[m,~~˜σ] � ≡ � ~−βH˜[~˜.Dm˜(q)e m]2 Λ/b (2π)dσ Here we have defined the partial averages �� Λ � � hOiσ ≡D~Zσ(σ q)O exp − Λ/b (2dπdq )d t +2 Kq2 |σ(q)|2 , (IV.31) with Zσ = D~σ(q) exp{−βH0[~σ]}, being the Gaussian partitio n function associated with the short wavelength fluctuations. From eq.(IV.30), we obtain β˜H[ ~˜= V δfb 0 + � 0Λ/b (2dπdq )d t +2 Kq2 | ˜� e −U[~˜σ] � σ . (IV.32) m]m(q)|2 − ln m,~59� � � � � � � � � � � � � � � � The final expression can be calculat ed perturbatively as, ln � e −U � = − hUiσ+1 � U2− hUi2 � +· · · +(−1)ℓ ×ℓth cumulant of U +· · · . (IV.33) σ 2 σσ ℓ! The cumulants can be computed using the rules set in the previous sections. For example, at the first order we need to compute � � �� ddq1ddq2ddq3ddq4 U m, ~˜σ (2π)dδd(q1 + q2 + q3 + q4) ~ = u σ (2π)4d . (IV.34) �� � � � � � � �� ~˜σ(q1) · m˜(q2) + ~ m˜(q3) + ~ · ~ σ(q4)m(q1) + ~~ σ(q2) ~ σ(q3) m˜(q4) + ~σ The following types of terms result from expanding t he product: [1] 1 m˜(q1) · ~˜~˜· m˜(q4)~ m(q2) m(q3) ~σ [2] 4 σ(q1) · ~˜~˜· m˜(q4)~ m(q2) m(q3) ~σ [3] 2 σ(q1) · ~ ~˜· m˜(q4)~ σ(q2) m(q3) ~σ . (IV.35) [4] 4 σ(q1) m˜(q2) ~ · ~˜~ · ~ σ(q3) m(q4) σ [5] 4 σ(q1) ~ σ(q3) ~˜~ · σ(q2) ~ · m(q4) σ [6] 1 h~σ(q1) · ~σ(q2) ~σ(q3) · ~σ(q4)iσ The second element in each line is the number of terms with the a given ‘symmetry’. The total of these coefficients is 24 = 16. Since the averages hOiσ, involve only the short wavelength fluctuations, only cont ractions w ith ~σ appear. The resulting int ernal momenta are integrated from Λ/b to Λ. Term [1] has no ~σ factors and evaluates to U[m˜]. ~ The second and fifth terms i nvolve an odd number of ~σs and their average is zero. Term [3] has one contraction and evaluates to − u × 2 ddq1 · · · ddq4(2π)dδd(q1 + · · · δjj(2π)dδd(q1 + q2) m˜(q3) · m˜(q4)(2π)4d + q4) t + Kq12 ~ ~ = − 2nu Λ/b ddq |m˜ (q)|2Λ ddk 1 . 0 (2π)dΛ/b (2π)d t + Kk2(IV.36) Term [4] also has one contraction but there is no closed loop (the factor δjj) and hence no factor of n. The various contractions of 4 ~σ in term [6] lead to a number of terms with 60� � � ~� � � � � � m. Summing up all no dependence on ~˜We shall denote the sum of these terms by uV δfb 1 . terms, the coarse g rained Hamiltonian at order o f u is given by β˜H[ ~˜=V � b + uδfb 1�� Λ/b (2dπdq )d t˜+2 Kq2 | ˜m] δf0 + m(q)|2 0 , (IV.37) + u Λ/b ddq1(2ddπq)32d ddq3 m~˜(q1) · m~˜(q2)m~˜(q3) · m˜(−q1 − q2 − q3) 0 where � Λ ddk 1 t˜= t + 4u(n + 2) . (IV.38) Λ/b (2π)d t + Kk2The coarse grained Hamiltonian is thus described by the same 3 parameters t, K, and u. The other two parameters in the coarse g rained Hamiltonian are unchanged, i.e. K˜= K, and u˜ = u. (IV.39) 2. Rescale by setting q = b−1q ′ , and 3. Renormalize, m˜= z ~′ , to get ~ m (βH) ′ [m ′ ] =V � δfb 0 + uδfb 1� + � Λ (2ddπq )′ db−d z 2 t˜+ Kb2 −2q ′2 |m ′ (q ′ )|2 0 . Λ ddq1′ ddq2′ ddq3 ′ 4b−3d m ′ m ′ m+ uz ~′ (q · ~′ (q ′ m ′ (q · ~′ (−q ′ ′ ′ (2π)3d 1)2) ~3)1 − q2 − q3) 0 (IV.40) The renormalized Hamiltonian is characterized by the tri plet of interactions (t ′, K′ , u ′ ), such that t ′ = b−d z 2t, ˜K ′ = b−d−2 z 2K, u ′ = b−3d z 4 u. (IV.41) As in the Gaussian model there is a fixed point at t ∗ = u ∗ = 0, provided that we set b1+d ′ z = 2 , such that K = K. The recursion relations for t and u in the vicinity of this point are given by   Λ ddk 1  tb ′ = b2 t + 4u(n + 2) Λ/b (2π)d t + Kk2 . (IV.42)   ′ ub = b4−d u While the recursion relation for u at this order is identical to that obtained by dimensional analysis; the one for t is different. It is common t o convert the discrete recursion relations to continuous differential equations by setting b = eℓ, such that for an infinitesimal δℓ, t ′ b ≡ t(b) = t(1 + δℓ ) = t + δℓ dt + O(δℓ2) , u ′ b ≡ u(b) = u + δℓ du + O (δℓ2). dℓ dℓ 61� � � � � � Expanding eqs.( IV.42) to order of δℓ, gives � �    t + δℓ dt = (1 + 2δℓ) t + 4u(n + 2 ) Sd 1Λdδℓ dℓ (2π)d t + KΛ2. (IV.43)  du


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