# Ising model simulator on curved surfaces

1. Jun 12, 2010

### loicseguin

Hi,

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.

Volunteer developers and testers are needed in order
to accelerate the development of the project. The software is written in C++
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.

Last edited by a moderator: May 4, 2017