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What progress has been made towards sampling from the 2D lattice Ising model with the following Hamiltonian:

$H=-J\sum_{\langle i,j \rangle}S_iS_j - \sum_i b_iS_i$

Where the first sum runs over all nearest-neighbour positions on a lattice. Especially, my emphasis is on very large lattice sizes where Markov-chain Monte Carlo methods become unwieldy.

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  • $\begingroup$ The title of the question is confusing. Is your question about sampling algorithms or about an Onsager-like exact evaluation of the partition function? There's some detail about the latter on the Wikipedia page en.wikipedia.org/wiki/Ising_model#Onsager.27s_exact_solution $\endgroup$
    – Igor Khavkine
    Commented Apr 29, 2013 at 13:03
  • $\begingroup$ Onsager's solution is for when the external field is zero and thus does not apply in this case. My emphasis is on sampling algorithms. An exact partition function calculation is a very difficult problem for this case. $\endgroup$
    – Alin
    Commented Apr 29, 2013 at 19:18
  • $\begingroup$ How about coupling from the past? It should work pretty well ... $\endgroup$
    – Vincent Beffara
    Commented Apr 29, 2013 at 21:22

1 Answer 1

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The entropic sampling technique can efficiently sample quite large lattices, several thousands spins. It has been applied to your Hamiltonian, in a uniform magnetic field ($b_i$ independent of $i$), in several works:

Systematic enumeration of configuration classes for entropic sampling of Ising models, Bruno Jeferson Lourenço, Ronald Dickman

Complete high-precision entropic sampling, Ronald Dickman, A. G. Cunha-Netto

Entropic sampling dynamics of the globally-coupled kinetic Ising model, Beom Jun Kim, M.Y. Choi

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  • $\begingroup$ Thanks, I appreciate the reply. Has there been any progress to solve the Hamiltonian I gave? $\endgroup$
    – Alin
    Commented May 7, 2013 at 2:52

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