From: israel@math.ubc.ca (Robert Israel) Subject: Re: Ising model, was: probability question, take 2 Date: 16 Jun 1999 00:52:09 GMT Newsgroups: sci.math To: Randy Poe Keywords: Statistical mechanics In article <3765D047.2BF1B358@dgsys.com>, Randy Poe writes: > Been a long time since I saw statistical mechanics, but > I believe an Ising model is a spin model. Everything > has one of two states, up or down. The energy of this > thing depends on the spin states. Any given pair of > spins has an energy which is higher when the spins are > aligned than when they are opposite (this models the > magnetic interactions of actual electron spins). No, it's _not_ the magnetic force that causes this. If you had a lattice of magnetic dipoles free to rotate, they might tend to a configuration something like ... -> -> -> -> ... ... <- <- <- <- ... ... -> -> -> -> ... Since opposite magnetic poles attract, neighbouring magnets like to be in opposite directions, not the same direction, if they are oriented perpendicular to the vector from one to the other. Instead, the interaction is due to complicated quantum-mechanical effects. There's a fairly simple calculation to show that (in a model with one electron per site) the neighbouring spins should prefer to be opposite, due to "exchange interactions" relating to the Pauli exclusion principle - basically, their wave functions can overlap better if the spins are opposite. These effects are much stronger than the magnetic interaction. Why in a few cases (ferromagnets) the neighbouring spins prefer to be aligned is a very complicated question - but in any case, the magnetic interaction is not the answer. A more realistic model for ferromagnets is the Heisenberg model, where two neighbouring spins S1 and S2 have an interaction energy -J S1 . S2 (J a constant, "." the dot product). The Ising model was proposed as a simplified version of this. It is more realistic as a model for a binary alloy (atoms of two types A and B corresponding to "+" and "-" spins). > Now you can arrange these things in a regular lattice, > and do things like apply an "external magnetic field", > modeled as an external spin which interacts with all > the elements in the same way. You can have models > where only nearest neighbors interact, or in which > interactions extend out to 2nd neighbors, etc. > Very common animals in statistical mechanics, I've > seen these in math only in combinatorics, because > the problem of searching for the minimum energy > state is one of combinatorial optimization. It > seems to me a paper showed the existence of > magnetic domains (large regions of aligned > spins), which is interesting because real materials > do this. A very important aspect that you're leaving out is that we're not just looking for minimum-energy configurations, which would be trivial for the Ising model. Instead we're usually looking for the statistical properties of the equilibrium states at positive temperature (in the limit of infinite volume). In a system with a finite number of configurations, each configuration should have probability proportional to exp(-E/(kT)) where E is the energy of the configuration, k Boltzmann's constant and T the temperature. Various methods can be used to extend this to a notion of an equilibrium state in an infinite system. One important result is that if the temperature is low enough, the Ising model (on a lattice of two or more dimensions) exhibits spontaneous magnetization: although the model itself is symmetric under interchange of "+" and "-", there are equilibrium states in which this symmetry is broken and the spins tend to be of one type over the whole lattice. This is believed to be true also for the Heisenberg model in three or more dimensions, but AFAIK there is still no rigorous proof of that (for the quantum-mechanical Heisenberg model). BTW, I can't resist a plug: I was the first to prove the existence of long-range summable interactions in one dimension for which Heisenberg models have spontaneous magnetization: see R. Israel, "Existence of Phase Transitions for Long-Range Interactions", Commun. math. Phys. 43 (1975) 59-68, or my book "Convexity in the Theory of Lattice Gases", Princeton U. Press 1979. Real magnetic domains arise from the long-range character of the magnetic field: on the microscopic level it's not important, but on a larger scale it is. Because of this long-range magnetic force, in a large block of material it becomes energetically unfavourable to have spins aligned over the whole block; instead, the block becomes divided into separate domains, in each of which the spins are aligned in a certain direction, with different directions in different domains. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2