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Unformatted text preview: Physics 127B – Homework Set 7: (due March 3) Problem 1: “b=3” decimation RG for 1d Ising antiferromagnet Consider the one-dimensional Ising spin Hamiltonian ¯ H =- H/ ( k B T ) given by ¯ H = K N summationdisplay i =1 s i s i +1 + h N summationdisplay i =1 s i + g N summationdisplay i =1 (- 1) i s i + NC . (1) We now want to consider the antiferromagnetic case K < 0, as well as the ferromagnetic case K > 0. For the antiferromagnet we introduce the “staggered field” g which couples to the antifer- romagnetic order parameter - the staggered magnetization M stagg = ∑ i (- 1) i s i . To implement the renormalization for the antiferromagnet we need to consider the “ b = 3” spin decimation where at each step the number of spins is reduced by a factor of 3 by tracing over adjacent pairs of spins, i.e. retain spins at i = 3 n , trace over spins at 3 n + 1 and 3 n + 2. (If you want, you can try the b = 2 scheme to see what goes wrong.) As in class we can use a slightly modified partition function ¯ Z N = 1 2 N summationdisplay states e ¯ H where the sum over states is given by summing over all s i = ± 1, and the extra factor of 1 / 2 N just means that- k B T ln ¯ Z N = F + Nk B T ln 2, i.e....
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