While working on a completely unrelated quantum computing problem, I ran into a quantity that can be mapped to a partition function of spins on a triangular lattice. It is not quite an Ising model, though, since interaction happens between triplets of spins, some configurations are not allowed, and I don't explicitly have the Hamiltonian but only the weight of every configuration (I guess one could just pick a random value for the temperature and define a Hamiltonian, but it would probably just be a more complicated mess).

I have been trying to understand if I can use Montecarlo methods to get the partition function, but I am getting lost in the literature. From what I understood, my best shot would be to use something like the Wang-Landau algorithm to estimate the density of states and use that to get the partition function, but I couldn't find any paper detailing such a procedure so if someone could give me some good literature to look at (or any suggestions) that would be very helpful.


P.S. As far as I understand, Wang-Landau gives the density of states up to a normalization constant. However, in my specific case I have some conditions on this fictitious "partition function" that should allow me to infer that constant.

  • $\begingroup$ Hi, you might want to look at my answer here to at least get started: physics.stackexchange.com/questions/550504/… $\endgroup$
    – Godzilla
    Jul 23 '20 at 16:38
  • $\begingroup$ What information do you have that lets you infer the normalization constant for the density of states, but doesn't let you infer the partition "function" (really just a single number, it sounds like) directly? $\endgroup$
    – Daniel
    Jul 29 '20 at 21:38
  • $\begingroup$ @Daniel I have dependence on a continuous parameter $\epsilon \in [0,1]$ and I know that the "partition function" is 1 in the limit of $\epsilon \rightarrow 0$. Since the density of states is an integer in this case, then this should allow me to get the normalization constant. $\endgroup$ Jul 31 '20 at 18:18
  • $\begingroup$ Ok! For $\epsilon$ near 0, you can use Monte Carlo to sample from the $\epsilon = 0$ distribution, and these samples will allow you to estimate the small-$\epsilon$ partition function pretty efficiently. Is an answer at small $\epsilon$ good enough for you? $\endgroup$
    – Daniel
    Jul 31 '20 at 19:00

The Ising model is fairly well studied, a good start might be this paper. Many numerical methods exist as well, which may be used, starting with metropolis-hastings and cluster algorithms like the Wolff/Swendsen-Wang algorithm. Frustrated systems and transverse field systems can be efficiently sampled using QMC algorithms like Determinant quantum MC (DQMC) and Stochastic Series Expansion (SSE).

Also, as far as I can remember the 2D triangular lattice is exactly solvable assuming there is no transverse field. I believe in that case you are looking for a solution using transfer matrix method.

  • $\begingroup$ I will look into these in the next days and let you know, thank you! $\endgroup$ Jul 31 '20 at 18:20

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