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# hmwk8q2 - 2. Derive a general expression for S-&amp;amp;lt;...

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Problem set 8 1. The simplest many-body model of magnetism is the Ising model described by the Hamiltonian H = J X <ij> σ i σ j + h X i σ i (1) where σ i can take on values of ± 1, and the symbol < ij > denotes that spin i and spin j are nearest neighbors. It is a classical model of easy-axis magnetism. In one and two-dimensions it can be solved exactly. Read the paper by K. Wilson [ ScientiFc American, 241, 158 (Aug, 1979)], and then use with the xising and xrenor codes which are in the xtoys and RG subdirectories, respectively, to study the system near the transition. Use xrenor to study the RG ±ow of the system for a few temperatures just above and just below T c . Make sure that the system has achieved equilibrium before you renormalize. Repeat this several times for each temperature. Plot the RG ±ow of the average magnetization. Is the estimate of T c from Orlando’s code too high or too small? Why (hint, consider Fnite size e²ects).
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Unformatted text preview: 2. Derive a general expression for S-&lt; S z i &gt; for a spin-S quantum antiferromagnet, in the quadratic spin-wave approximation. Examine this equation in one, two, and three dimensions for simple cubic lattices and small wavevectors k . If S-&lt; S z i &gt; diverges, then the spin-wave theory fails. or what dimensions and temperatures is this the case (i.e. consider the form of your equation for T = 0 and T 6 = 0 for one, two and three dimensions. Hint, consider the sum over k for small k )? Why does the theory fail in these cases (the reason has to do with the validity of the starting point, reconsider the results of problem one)? or the three-dimensional case, you may want to look at T. Oguchi, Phys. Rev. 117 117, (1960); however, note that this author goes into signiFcantly more detail than is required here. 1...
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