phys212ln21

# phys212ln21 - Physics 212 Statistical mechanics II Fall...

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Unformatted text preview: Physics 212: Statistical mechanics II, Fall 2006 Lecture XXI In the previous lecture, we discussed how models with continuous order parameters, like the XY model, show different physics than models with discrete order parameters, like the Ising model. We can think of these physically important models as points in the ( d,n ) plane: then n = 1 corresponds to the Ising model, n = 2 the XY model, n = 3 the Heisenberg model. The Kosterlitz- Thouless-Berezinskii case, which has a line of critical points, is at d = 2 ,n = 2. For continuous order parameters n > 1, there is no finite-temperature transition below 2 dimensions, while for the Ising model ( n = 1) there is no finite-temperature transition in one dimension. Continuous spins also allow a number of important phenomena related to “topological defects”: the simplest example of this idea, which we discuss today, is a vortex in an XY model, which is a good description of real vortices in superfluid. We will focus on the 2D system, where the vortex is easily visualized and also controls the phase diagram. In general a topological defect refers to a configuration of a continuous model that cannot smoothly relax to a uniform configuration because of a topological invariant (usually a winding number or some generalization thereof). (There are also topological configurations that do not involve defects; an example is “Shankar’s monopole” in the A phase of superfluid helium-3.) Vortices are the simplest example. The local spin variable in the 2D Ising model is a unit vector on the circle. Suppose that we are at low temperature so that the spin moves only slightly from one site to the next. Then, in going around a large circle, we can ask how many times the spin winds around the unit circle, and define this as the “winding number” n ∈ Z . Note that if the winding number is nonzero, then the continuum limit must break down at some point within the circle, as otherwise we would have the same angular rotation 2 πn around circles of smaller and smaller radius, implying larger and larger gradients and hence infinite energy (we will calculate the energy of a vortex below). This will also let us see from a fairly simple calculation how there can be continuously varying exponents in the power-law correlations of the 2D XY model at low temperature. The assumption we’ll need to make is that “vortices” are unimportant at sufficiently low temperature, so that the 2...
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phys212ln21 - Physics 212 Statistical mechanics II Fall...

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