Sounds like a simple standard thing, I checked a couple of books and papers but just couldn't find much:
Where can I find a plot of magnetization vs temperature (or normalized interaction strength) for finite constant external field for the 2D or 3D Ising model? I heard there is a theory for infinitesimal external field strength but has there been some progress on larger fields?
I'm not sure off hand where to go for a specific plot, but I would recommend searching Reviews of Modern Physics for older reviews. The Ising model has been exhaustively studied for a very long time. To my knowledge there are very few analytical results (e.g. Onsager's solution), and most investigations use computation to investigate the model and its phase transitions. Generalizations have included adding more degrees of freedom (e.g. Potts models) or varying the topology of the underlying lattice (e.g., Cayley trees or Bethe lattice).
I suggest reading chap. 14 and on of the following course notes (and references therein) to get an overview of the work done on this problem (Mean-field theory, Landan theory, Ornstein-Zernicke theory, RG).
http://www.physics.mcgill.ca/~grant/559/notes0.91.pdf
Look at http://works.bepress.com/miron_kaufman/3/ and references therein.
For Francisco Lima:
(1) Take a square lattice:
(2) Add bonds to neighbors in direction (+1,+1) to achieve the connectivity of the triangular lattice.
(3) Then delete one fourth of the sites according, for instance, to the rule: Delete (x,y) if (and only if) x and y are both even numbers: you have built up the Kagome lattice. (Please check the recipe!)
Within this picture, it is straighforward to apply periodic boundary conditions as in square lattices.
(Of course the lattices must be of size 2m * 2n (with m, and n being positive integer numbers).
Does this answer your question?
For Francisco:
I see the problem. I have not used kagomé lattices so far, so I have never written code for it. My suggestion is to look at the system as composed by primitive cells of size 2*2,
in each of this "primitive" cells only three sites are considered. Each site has one or two
nearest neighbors (NN) in the same primitive cell, and three or two in NN primitive cells. Therefore, depending on the site, say (2i,2j) (2i,2j+1), (2i+1,2j) [Position (2i+1,2j+1)has been removed in primitive cell [i,j] ] one has to look at the sites belonging to primitive cells which are NN of [i,j] to decide which pairs of sites are NN.
Good luck
I do not kow if I have understood your explanation, ... , but it seems to me that for reaching the honeycomb lattice from the triangular lattice built as I explained above (i.e. by adding bonds in the direction 11 to the square lattice) you need to consider a square unit cell of size 3x3 (i.e. 9 initial sites), then by removing one third of the initial sites you get 6 sites per unit cell that by translation define the topology of the honeycomb lattice.
Hi Thomas! Coming back to your original question: I suggest that you write a (simple) Ising MC program and generate the curves numerically. It's very quick to do at reasonable temperatures (J/kT not too large). The curves are smooth so they are also easy to sketch. Actually, they are boring and that's why you don't see them published anywhere :).
Hi Tapio. I recently also want to write a MCMC program to simulate Ising model. I found that lots of introductory documents focus on updating rule. But how about initial distribution? Can I set it randomly?
Hi Shu-Chang! You should not start from a random initial condition if your final state has a lot of (local) order (not to mention below T_c for H=0). The closer you start from the final state, the faster the MC algorithm will converge. As a general rule, it's a good idea to generate the initial state from a Boltzmann distribution at the same T where you're running your simulation. If this is unpractical then just make an educated guess on what your configurations should look like :).
Hi Shu-Chang!
I think that you should start from different initial conditions (random, fully ordered) and
check if your program reaches the same (or equivalent) results in the different cases (it should!),
Dear All, one should never start an MC simulation below T_c from the fully disordered state. The reason is that due to ergodicity breaking one of the degenerate ground states (spins up or down) is randomly selected, if there's no external field. The approach to an ordered equilibrium state is algebraically slow (this is called "domain growth") and in practice you'll never the reach equilibrium state because eventually the domain walls get pinned!
Dear All, I partially agree with previous comment from Tapio. In fact I was implicitely considering Ising model at zero field. On the other hand, for the particular case of Ising and Potts models, there are very powerful cluster algorithms (Swendsen-Wang, Wolff) to carry out Monte Carlo simulation that overcome the problems pointed out by Tapio.
You're absolutely right Noe. Equilibration problems may occur when simple spin flip or exchange dynamics is used but can be overcome with special update schemes.
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