phys212ln16 - Physics 212 Statistical mechanics II Fall...

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Unformatted text preview: Physics 212: Statistical mechanics II, Fall 2006 Lecture XVI The Ising model that has been considered in most of the lectures so far has a discrete degree of freedom on each site (the spin is either up or down), or in other words, a discrete order param- eter. Many physical systems are modeled by continuous degrees of freedom, and problems with a continuous order parameter have several very different properties, even though their mean-field theory (and hence their behavior above the upper critical dimension) is the same. We now start to discuss systems where the local spin-like degree of freedom may be continuous, and where there may be no lattice (the problem is also spatially continuous). This lecture shows that there are new The term “order parameter” refers to the variable that orders spontaneously below a second- order transition, e.g., the magnetization in the Ising model. One famous example of a transition modeled by a continuous order parameter is the Heisenberg magnet (where the spin is modeled by a unit vector in 3D): E =- J X h ij i s i · s j- X i H · s i . (1) Note that even though the spin per site lives in 3D, the dimensionality can be one, two, three, etc.: it is important to distinguish between the order parameter dimensionality and the spatial dimensionality . The Heisenberg model can be thought of as a special case of the general model with three different couplings J x ,J y ,J z . Here the anisotropy is not on the bond variables but on the couplings: on each bond the energy is- ( J x s ix s jx + J y s iy s jy + J z s iz s jz ) . (2) This general anisotropic version is sometimes known as the XYZ model, and the Heisenberg model is sometimes referred to as the XXX model since J y = J z = J x . Many magnets fall into the XXZ category: J x = J y 6 = J z . When J x > J z , the magnet is called easy-plane , since the spins like to lie in a plane. When J z > J x , the magnet is known as easy-axis . Now that we have the RG language, we can make some simple statements that can be justified using the techniques mentioned before. Suppose J z > J x , the easy-axis case. It turns out that the system then flows to the pure Ising fixed point with J x = 0. Similarly, if J x > J z , the system flows to the pure XY fixed point with J z = 0. The Heisenberg and XY fixed points are quite different from the Ising fixed point, especially in low spatial dimension. The easy-plane case is especially important because it describes superconductors and superflu- ids. In fact, the most precisely known scaling behavior in nature is observed for this universality class in three dimensions. On Earth, the superfluid transition in He 4 has been observed over four decades, and experiments in zero gravity have added another 1.5 decades or so. The reason why these transitions are described by the XY universality class is that the superfluid or superconducting order parameter has a quantum phase. For the superconducting case, we write this asorder parameter has a quantum phase....
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This note was uploaded on 08/01/2008 for the course PHYSICS 212 taught by Professor Moore during the Fall '06 term at Berkeley.

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phys212ln16 - Physics 212 Statistical mechanics II Fall...

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