Thursday, 6 June 2013

The Potts Model

jQuery UI Slider - Multiple sliders

The last post was about the Ising model and how to simulate it. The Ising model is the simplest type of spin model and has several generalizations. You can make different shaped lattices; triangular, honeycomb, union-jack. You can make the couplings in the horizontal and vertical directions different, or you can make every coupling on every link different; this type of model is called a spin glass. You can increase the number of dimensions; one and two are exactly solved, three and four are interesting and unsolved analytically; five and higher are somewhat less interesting. This is because of the large number of neighbours each spin has in $N$ dimensions - $2N$. The more neighbours a spin has the more accurate it is to describe the spin as interacting with a fixed background and mean field theory becomes more and more accurate. You could even put several Ising models on the same lattice and make them interact - this is the Ashkin-Teller model. There are many more.

Here I will consider a simple generalization - more than two states. A spin can point in $N_c$ directions. If two neighbouring spins point in the same direction this decreases the energy, if they point in opposite directions the energy increases, the Ising model has $N_c = 2$.

Let's look at the possible states two neighbours can be in. For the Ising model we have \[ \left( \uparrow \uparrow \, , \, \downarrow \downarrow \right) \] decreasing the energy and \[ \left( \uparrow \downarrow \, , \, \downarrow \uparrow \right) \] increasing it, ie. two increasing and two decreasing. For $N_c = 4$ it goes like, \[ \left( \uparrow \uparrow \, , \, \downarrow \downarrow \, , \, \leftarrow \leftarrow \, , \, \rightarrow \rightarrow \right) \] and \[ \left( \begin{array}{ccc} \uparrow \downarrow, & \uparrow \leftarrow, & \uparrow \rightarrow, \\ \downarrow \uparrow, & \downarrow \leftarrow, & \downarrow \rightarrow, \\ \leftarrow \downarrow, & \leftarrow \uparrow, & \leftarrow \rightarrow, \\ \rightarrow \downarrow, & \rightarrow \uparrow, & \rightarrow \leftarrow \end{array} \right) \] four decreasing the energy and twelve increasing it. This means if you pick a random configuration it is three times as likely that a given pair will increase the energy than decrease it. In the energy-entropy battle the Potts model has more ways to mix up the configurations and so it is easier for entropy to dominate. This means, among other things, that the critical temperature is lower. If $N_c$ is increased enough ($\geq 4$) even the nature of the transition changes, from an Ising-like second order to a water-steam type first order.

Below you can tune both the temperature, $\beta$, and the number of spin orientations allowed, $N_c$, symbolised by the colour of a pixel.

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