Monte-Carlo and $O(n)$ models for non-integer n

$O(n)$ lattice statistical models can be generalized to non integer values of n, starting from their (expanded and resumed in graphs) partition function: $$Z = \sum_{\text{loop configurations}} n^{\# \text{loops}} x^{\text{total length}}$$

The sum is over all possible loops or graphs configuration, $\#\text{loops}$ is the number of loops present in a specific configuration, and $\text{total length}$ the sum of their individual length.

Though, as far as I know, only the n integer cases admit a local Hamiltonian description. I would like to do some Monte-Carlo simulations in the range $1<n<2$, but I don't see how to implement the Metropolis algorithm for instance without knowledge of an explicit Hamiltonian to calculate the $\text{min}(1,\text{e}^{-\beta\Delta H})$ acceptance ratio. Is there a known to circumvent that? Has it already been done? Using other algorithms (I would like to know about the use of Wolff algorithm for example)?

• I know very little about loop models, and nothing about the specific Monte Carlo methods that one might use to simulate them. They sound cool enough, though. Is journals.aps.org/pre/abstract/10.1103/PhysRevE.88.021301 or arxiv.org/abs/1011.1980 of any use? – alarge Jun 6 '14 at 9:24
• @amlrg: Hi, thank you for the references, I'm still reading them (and the references within) but I can already tell you that it helped me making a few steps forward. I'd give you the bounty but your answers needs to be in a proper answer post for that and not an underlining comment. Please do so, so I can reward you ;) – Learning is a mess Jun 13 '14 at 14:19

Your idea is good but it cannot be the solution as the O(n) model correspond as the invariance group of a sphere in n dimensions (a remark on the notation, n=1 is the Ising model and n=2 is the O(2) model). In particular, for values between 1 and 2 you have fractional dimensions (dimension = n(n-1)/2), the configuration space of the model should be a fractal.

From this reasoning, the problem can be recasted in finding a representation for the sphere (and the scalar product associated) for fractional dimensions.

It is evident this is very different from actual O(n) symmetry, but one can hope that this model still belong to the O(n) universality class.

The crossover between n=1 and n=2 is particularly interesting because in two dimensions you have a phase transition at n=1 that disappears at n=2.

I've kept looking for an answer to my question and though I'm convinced in it's not the final: could the $Z_n$ models be of any help to me?

They are located between the Ising model ($n=2$) and the $O(2)$ model ($n=\infty$) so I'm quite hoping there could be a way to connect them to $O(n)$ models with $1<n<2$. Is it only wishful thinking?

• You should add this as an addendum to your original question. – JamalS Jun 6 '14 at 9:37