10. Matric models: In some situations, the order parameter is a matrix rather than a vector. For example, in triangular (Heisenberg) antiferromagnets each triplet of spins aligns at 120, locally defining a plane. The variations of this plane across the system are described by a 3 x 3 rotation matrix. We can construct a nonlinear o model to describe a generalization of this problem as follows. Consider the Hamiltonian K BH = / a'x te dx tr [VM(x). VM"(x)] where M is a real, N * N orthogonal matrix, and 'tr' denotes the trace operation. The condition of orthogonality is that MMT = M M = I, where I is the N ~ N identity matrix, and MT is the transposed matrix, M = Mji. The partition function is obtained by summing over all matrix functionals, as Z= ? = DM(x)3 (M (x)m (x) 1) e-BH1()] (a) Rewrite the Hamiltonian and the orthogonality constraint in terms of the matrix ele- ments Mij (i, j = 1,...,N). Describe the ground state of the system. (b) Define the symmetric and anti-symmetric matrices = { (M+M") = 01 } (M MT) = -2 Express BH and the orthogonality constraint in terms of the matrices o and T. (c) Consider small fluctuations about the ordered state M(x) = I. Show that o can be expanded in powers of T as o= +.... Use the orthogonality constraint to integrate out o, and obtain an expression for BH to fourth order in 1. Note that there are two distinct types of fourth order terms. Do not include terms generated by the argument of the delta function. As shown for the nonlinear o model in the text, these terms do not effect the results at lowest order. (d) For an N-vector order parameter there are N - 1 Goldstone modes. Show that an orthogonal N * N order parameter leads to N(N 1)/2 such modes. (e) Consider the quadratic piece of BH. Show that the two point correlation function in Fourier space is (27)d84(q + q') (Tij(q) #ki(q')) = Kg [8ik Ojl - Diljk) We shall now construct a renormalization group by removing Fourier modes M(9), with q in the shell A/b