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Unformatted text preview: Physics 212: Statistical mechanics II, Fall 2006 Lecture XV Let us get some practice drawing phase diagrams of different systems in order to understand the renormalization-group ideas more generally. As an extension of the Ising model, consider a version with annealed/thermal vacancies. There are now three possible states for each site: spin up, spin down, or vacant, which we denote as σ = 0. Incidentally, if the vacancies are not thermal variables but rather static or “quenched”, then the physics is different (and much more difficult). We keep the spin interaction Hamiltonian but add an energy Δ to favor or penalize vacancies: H =- J X h ij i σ i σ j + Δ X i σ 2 i- H X i σ i . (1) This is known as the Blume-Capel model. Here the sign of Δ is chosen so that positive Δ makes vacancies favorable. Assume H = 0 and consider the phase diagram. At Δ =-∞ , vacancies are suppressed and we wind up with the original Ising model. At Δ = + ∞ , all we have are vacancies and the ground state is just vacant. Another simple limit is zero temperature, when we need only to find the ground state of the Hamiltonian. This will contain all spins aligned, and the only question is whether the state of all spins up or down has lower energy than the state of all vacancies. With the Hamiltonian as written above, this is a competition between energy per site Δ- Jz/ 2 (all spins up or down) and 0. At zero temperature, this is a first-order transition as a function of Δ /J . We know that the transition is second-order at Δ =-∞ , and occurs at the nonzero transition temperature T c of the Ising model. This means that at some Δ there is a change between first-order transitions and second-order transitions: the point that separates the transition line is known as a tricritical point. Note that reaching a tricritical point requires “tuning” of one more experimental parameter or “knob” than an ordinary critical point. Now we return to the Blume-Capel model and try to understand the renormalization-group flows. We said before that this model has a line of first-order transitions, a tricritical point T , and a line of second-order transitions. Where do the RG flows go in the phase diagram? At every one of the second-order transitions, the correlation length is infinite, so we cannot guess the direction of the RG flows just from the requirement that the correlation length decrease in the rescaled problem. If we are willing to assume that the ordinary 2D Ising model transition at Δ =-∞ has only the relevant directions that we already know (temperature and magnetic field), then the...
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