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Phases and the Renormalization Group

It's usually a good idea when introducing a subject to start with the basics. In statistical mechanics, probably the most basic model we can study is the Ising model. So I'm going to start there with a little review. The Ising model has the simple Hamiltonian

equation50

where the angular brackets indicate a sum over nearest neighbors, J is the exchange constant, and h is a uniform applied field. Let's concentrate on the ferromagnetic case, where J>0. To study the equilibrium properties we need to calculate averages using the statistical weight,

equation53

where T is the temperature (I've taken tex2html_wrap_inline2672 ), and the partition function

equation57

The Ising ferromagnet has two equilibrium phases. Above some critical temperature tex2html_wrap_inline2674 , the system is in a paramagnetic or disordered phase. There tex2html_wrap_inline2676 , the susceptibility tex2html_wrap_inline2678 is finite. At low temperatures, the magnet orders, and there is a broken symmetry. The system spontaneously chooses some particular orientation of the magnetization tex2html_wrap_inline2680 , and tex2html_wrap_inline2682 .

Physically, we can think of the system as being composed of fluctuating domains of spins. For tex2html_wrap_inline2684 , there are, roughly speaking, finite domains of parallel spins, the size of which constitutes a correlation length tex2html_wrap_inline2686 . At the temperature is lowered towards tex2html_wrap_inline2674 , the size of these domains grows and finally diverges when tex2html_wrap_inline2690 . In the low temperature phase, the entire system is essentially in a single macroscopic domain. Within this domain, there may be some small (finite) clusters of flipped spins, which dynamically reverse themselves occasionally. These local reversals of spins simply lower the magnitude of the equilibrium magnetization, but below tex2html_wrap_inline2674 do not renormalize it to zero. As we cool further, for tex2html_wrap_inline2694 , even these small spin-flips are suppressed, and the zero temperature magnetization is perfect.

It will be useful to look at this from the point of view of the renormalization group (RG) (a nice and detailed introduction to the RG is contained in Ref. [1]). We'll come back to this again and again throughout the course. Let me remind you of the idea behind the RG. In physics we study some very complicated models, with many parameters and detailed interactions. For instance, we could wonder about our Ising model in the presence of 2nd and 3rd nearest neighbor interactions, etc. Often time, however, we are really interested in answers to some relatively "simple" questions about those models. By "simple", I really mean having to do with the long-distance and low-energy properties of the system. For the Ising system, these long-wavelength properties include the equilibrium magnetization, the large-distance behavior of the spin-spin correlations, and the susceptibility.

All these quantities are definable in terms of low momentum modes or spatially averaged collections of the spin fields. To study just these quantities, we can imagine doing a sort of coarse-graining. That is, we can systematically eliminate the short-distance degrees of freedom from our theory, for instance by summing over them in the partition function Z. This may be difficult to actually do in practice, but we can, at least in principle, obtain an effective theory that just described the remaining average quantities. We can then repeat the procedure, getting each time a theory that works on longer and longer length scales. As we do this, the Hamiltonian itself changes, of course, and we have a sort of "flow" in the space of models.

We can think of this flow as an evolution like any other dynamical system. In particular, there will be certain fixed points (FPs) in the model space, which represent special theories which are scale invariant. Some of these are "sinks", i.e. fixed points with no unstable directions. If our original theory is in some sense near to such an attractive fixed point, the RG flow will take its effective long-wavelength theory closer and closer to the FP. The asymptotic behavior of all the correlation functions in the original model is thus determined by the FP properties. This kind of absolutely stable FP therefore describes a phase. There are other FPs, which have a single unstable direction in model space. These therefore separate distinct regions of model space, and in fact describe transitions between different phases.

Let's imagine doing this coarse-graining for the Ising model. We'll just divide up the lattice into blocks, say of size tex2html_wrap_inline2698 , and define the average spin in each block as a new spin. In the paramagnetic phase, recall that we have finite clusters of oriented spins. As we define new average spins, these get smaller and smaller, until they reach the size of a single block. Then the average spin in each block rapidly iterates to zero. At that point, we can think of the (average) spin in each block flipping back an forth dynamically, and independently of every other spin. Essentially, at this point the equilibrium magnetization appears to be zero. This is the picture of the disordered fixed point. It is essentially a tex2html_wrap_inline2700 fixed point, in the sense that the average spins are completely disordered. If we do the same thing in the ordered phase, we eventually coarse-grain beyond the size at which the local spin-flips occur, and at that point the system appears perfectly ordered. This is by analogy a T=0 fixed point.

   figure66
Figure 1: Schematic RG flow diagram for the Ising model.

So we can draw a schematic RG flow diagram (see Fig. 1). There are two stable fixed points, one at T=0 and another at tex2html_wrap_inline2700 . Somewhere in between is a critical fixed point, describing the ferromagnetic Ising phase transition. What do we learn from this? Well, we learn that many of the long-distance properties of the ferromagnet are universal, because they are described by a particular attractive fixed point. What's more, because it is a zero temperature FP, most of the long-distance properties of the ordered phase are adequately described (at least qualitatively) by ignoring (or handling perturbatively) thermal fluctuations. This is in fact a generic feature of the RG: almost all ordered phases are described by T=0 fixed points. We'll talk quite a bit more about the RG and look at some much more concrete examples later. For much of this course, though, the RG will primarily serve as a general framework, and the concepts of FPs, etc., will be probably more useful than any explicit calculations.


next up previous contents
Next: Coarsening in the pure Up: Prologue: Ising Model Previous: Prologue: Ising Model

Leon Balents
Thu May 30 08:21:44 PDT 1996