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Next: Elastic manifolds Up: Prologue: Ising Model Previous: Coarsening in the pure

Random bond Ising model

How does this simple picture change in a disordered system? For concreteness, let's consider a simple ``random bond'' Ising model,


where the exchange constants satisfy


with J>0 and tex2html_wrap_inline2742 random. I'll discuss short-range correlated disorder, where the tex2html_wrap_inline2742 are independent random variables of different links, taken, without loss of generality to have zero mean. Let's also assume that these random variables are narrowly (e.g. exponentially or bounded) distributed, with a variance


The last requirement guarantees the stability of the ordered phase, since flipping any spin still costs a positive energy because it breaks a number of bonds, essentially all of which are ferromagnetic. In the case of an unbounded but narrow distribution, there will be exponentially rare regions in which spins are flipped in the ground state, reducing the average magnetization and likewise reducing tex2html_wrap_inline2674 , but not destabilizing the ordered phase. The critical behavior at the transition can, however, be modified.

To understand coarsening, we need to consider the behavior of a domain wall in such a random magnet. We will see that it has a much more dramatic effect than on the true equilibrium properties. The basic physical reason for this is clear. In the random system, there are preferred positions for the domain wall, in which the bonds that are broken by the wall are chosen to lie on ``weak'' links.

To study this, consider the following toy model. Take a large cube of size tex2html_wrap_inline2748 , forcing the spins to point up at x=0 and down at x=L. At T=0, the system must put in a domain wall to accomodate the anti-periodic boundary conditions. This is the simplest example of a topological defect, which is defined as a configuration of an order parameter in which the states at infinity are non-trivially ``twisted''. In fact, such topological defects often lead to elastic models of the kind we will discuss here. Coming back to the interface, there are an enormous choice of possible conformations for the wall to take. The system must therefore ``solve'' an extremely non-trivial optimization problem to decide where it should go. I would like to describe this as a balance between the elastic energy, which tries to minimize its area (i.e. keep the total number of broken bonds fixed), and the disorder, which tries to distort the wall to take advantage of spatially separate weak links.

It is easy to see that the dynamics of such a domain wall will also be dramatically modified. Because an optimized wall has found at least a local energy minima, an applied force from, for instance, curvature forces or an applied field, will have to overcome the restoring forces near this minima in order for the wall to move. There are therefore substantial barriers to the wall's movement, and we expect the resulting motion to be reduced. In fact, we will see that this slowing down of the coarsening process is incredibly drastic!

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