Phases in the Ising Model

At low temperatures, the magnetization should align well with the external field. Does positive external field favor red, or white? If you start in a red state (say, by pushing M=1), you'll find that it is metastable at low temperatures until quite a large negative external field. This is analogous to the supercooling of water (which is metastable at temperatures where ice is the lower free-energy state too). See the preset on nucleation!

At high temperatures, the magnetization at zero field should be near zero. At low temperatures, the magnetization will, even at zero field, be near plus or minus one, at least in equilibrium. (It can sometimes take a while to get there: see the preset on Coarsening. A small "symmetry-breaking" external field will speed up the equilibration: turn it off after it settles down.) Can you roughly locate the phase transition, say within +-0.2? You'll find that the location is hard to pin down, because the "clumps" of red and white become rather large even in the paramagnetic, unmagnetized state: when the clumps get to the size of the system (see the discussion of diverging correlation lengths in the preset on the Critical Point) it's impossible to say whether it's ordered or disordered! You can make the system bigger using the "configure..." button: only in the "thermodynamic limit" when the height and width go to infinity does the critical point become precisely defined.

Draw a rough phase diagram in the H, T plane (external field and temperature), showing (i) the red phase, (ii) the white phase, (iii) the disordered, paramagnetic phase, (iv) the phase boundary between red and white, and (v) the critical point. In iron, the critical temperature where the magnetization disappears is at 1043 degrees Kelvin (and the critical external field is zero). In water, the triple point (where the boundary between water and water vapor ends) is at high temperatures and pressures (pressure here corresponds to the external field). Above this pressure, H20 liquid and gas smoothly transform into one another as temperature is varied.

Other Ising Model Presets

Description of the Model.
Phase Diagram
Magnetization M(T)
Domain Coarsening

Last modified: June 1, 1997

James P. Sethna,,

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).