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I am currently working on an open source project called Spinify. The Spinify

project aims at producing an efficient simulation algorithm for the Ising

model on surfaces with non trivial metric tensor. The project is still in its

infancy, but I firmly believe that some very interesting results can come out

of it.

The project home page is at http://bitbucket.org/loicseguin/spinify/wiki/Home" [Broken].

Volunteer developers and testers are needed in order

to accelerate the development of the project. The software is written in C++

and released under a BSD license. If you are interested in joining the project,

send an email to the spinify-discuss@googlegroups.com mailing

list and I will communicate with you.

The current roadmap is a little vague, but among other things, lots of testing

and code review is needed. Many more measures could be implemented and it

would be interesting to study the feasibility of including some routines to try

and calculate critical exponents for the model around the critical temperature.

The only curved surface on which simulations can be run is the sphere.

The next objective is to allow simulations on the torus and other surfaces of

greater genus. I also work on a GUI. Up to now, I am writing the

GUI in Objective-C and Cocoa, but I think it might be better to use some

cross-platform toolkit such as Qt, Tk or wxWidget.

Now, lets explain in a little bit more detail what Spinify is about. First of all,

the Ising model is a very simple model of how magnetic

materials become magnetized. A simple block of matter can be modelled as

an arrangement of atoms, each of which has a spin. Spin is a quantum

property of matter and, in the case of the Ising model, atoms can have

a spin value of either +1 or -1. The reason for this is outside the scope

of this wiki, but anyone interested in that can find more about it in any

good introductory quantum mechanics book.

If a material has magnetic properties (such as iron), the potential

energy of two neighbouring atoms is lower if their spin is the same. Thus,

since potential energy tends to be minimized, atoms try to get their spins

all in the same direction. And since spin is associated with the magnetic

field generated by the atom, all the magnetic fields add up together and

form a macroscopic magnetic field.

The Ising model makes many simplifying assumptions, the most important of

which is that atoms only interact with their closest neighbours. This is

reasonable since we know that the strength of a magnetic field decreases

with the square of the distance. The Ising model has been studied in one

and two dimensions extensively (the one dimensional Ising model is often

an exercise in introductory statistical physics books; the two dimensional

model on a square lattice has been solved by Onsager).

The Spinify project aims at simulating the model on surfaces such as the

sphere, the torus and the klein bottle. These surfaces have a non trivial

metric tensor which is just a complicated way of saying that they are

curved. This makes it hard, or sometimes even impossible, to embed a regular

lattice on the surface. To circumvent this problem, we suggest to generate

random lattices on the surface and then to run the simulation on that

lattice.

The project is motivated by conformal field theories which make

interesting predictions on the behaviour of the Ising model on

different surfaces in the limit where the number of atoms on the

lattice tends to infinity. However, there are no proofs that the

discrete model tends to the continuous model considered by CFT.

The project started as an undergraduate research project at Université de

Montréal in Canada a couple of years ago. But it has evolved a lot in the

meantime.

Thanks for you interest and have a great day,

Loïc Séguin-C.

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# Ising model simulator on curved surfaces

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