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4.6. Consider the linearized RG transformation (14.3.12), with (a11 a12) a21 a22 (3) such that (a11922-a12a21) #0. We now introduce the "generalized coordinates" u₁ and u₂ through equations (14.3.13); clearly, u₁ and u₂ are certain linear combinations of the system parameters ki and k2. (a) Show that the slopes of the lines u₁ = 0 and u₂ = 0, in the (k₁, k₂)-plane, are a21 21-a11 a21 λ2 - a22 m1 = λ₂-a11 a12 and m2 λ1-922 a12 respectively; here, ₁ and 2 are the eigenvalues of the matrix A. Verify that the product 1m2 21/12 and hence the two lines are mutually perpendicular if and only if a12 = 21. (b) Check that, in the special case when a12 = 0 but a21 #0, the above slopes assume the simple form m₁ = and m₂ = a21/(a11-a22) whereas, in the special case when a21 = 0 but a12 #0, they become m1 = (a22-a11)/a12 and m2 = 0; note that Figure 14.7 pertains to the latter case. (c) Examine as well the cases for which either an or a22 is zero; Figure 14.6 pertains to the latter of these cases. 4.7. Check that the critical exponents (14.4.38) through (14.4.40), in the limit n→∞, agree with the corresponding exponents for the spherical model of Section 13.5 with d < 4.