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Coarsening in the pure model

Now let's discuss coarsening. Imagine taking the system at a temperature tex2html_wrap_inline2710 and rapidly (let us say instantaneously for our purposes) quenching it at t=0 to a temperature tex2html_wrap_inline2714 . The system begins in the paramagnetic state, with equal amounts of up and down spins, randomly arranged. After the quench, the new equilibrium state is ferromagnetic. It must, however, find a way to evolve dynamically from the low temperature disordered state into the fully polarized state. From the RG perspective, since the ferromagnetic state is described by a T=0 fixed point, it's actually natural to try and understand this coarsening process by completely ignoring thermal fluctuations, i.e. working as if tex2html_wrap_inline2718 .

At t=0, there are then equal amounts of up and down spins in a random pattern with a scale set by the correlation length at tex2html_wrap_inline2722 . The system evolves from this point deterministically. If there is a small bubble of down spins in a largely up spin domain, it will flip to lower the local energy. Likewise, the large domain itself may grow or shrink depending upon its environment. For simplicity, let's think about a spherical droplet of down spins, of radius r, in the midst of a much larger up spin domain. The energy of the droplet, relative to a uniform domain, is


where the surface tension tex2html_wrap_inline2726 . From this, we can determine the total force on the bubble:


This is negative because the bubble tends to collapse. It represents the total force, so that the force per unit area


The local force is thus proportional to the local curvature of the surface. If we take a local equation of motion for the bubble wall,


we easily find that


The bubble thus fully collapses in a time tex2html_wrap_inline2728 .

In reality, the system has a much more complicated set of domains and spatial structure, but roughly speaking, there is a characteristic length scale R(t), below which the system appears ordered (i.e. all domains with sizes below R(t) have collapsed). Since the local forces are determined by the curvature tex2html_wrap_inline2734 , we still expect that this characteristic scale tex2html_wrap_inline2736 . This is a fairly rapid (power-law) coarsening, and we consequently expect, among other things, a scaling for the spin-spin correlations after the quench:


where F is a ``universal'' scaling function.

There are many more interesting questions to be asked about coarsening, such as the nature of this function, and anyone who is interested should certainly look at the review article by Allan Bray[2].

next up previous contents
Next: Random bond Ising model Up: Prologue: Ising Model Previous: Phases and the Renormalization

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