# What is the minimum non-integer dimension for which the XY model shows a phase transition? (if well-defined)

I know that XY statistical model for $d=2$ doesn't show a regular phase transition , while the $3d$ has, I was wondering what is the behaviour for $2< d < 3$.

If it is simpler one could consider another model in its class of universality, like it's done for the $2+\epsilon$ expansion.

The simplest hypothesis is that exist a $d_{min}$ between 2 and 3 such that for values of dimension greater than $d_{min}$ there is phase transition.

Assuming this number has a meaning for every statistical model, I would like to ask if it is universal with respect to the simmetries of the order parameter.

Going further, what is the behaviour of the critical temperature around $d_{min}$, is that universal?

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Depending how you set up your model in fractional dimensions, there will be long-range order as soon as $d>2$. The existence of long-range order is closely related to the transience of the random walk on the underlying graph. See this paper for one direction and this one for the other direction. Note that it is not clear how to make the latter paper rigorous, but it should be convincing for a theoretical physicist. – Yvan Velenik May 14 '13 at 15:35
I've quickly looked at the interesting articles you linked, that could be considered the answer to my question, the value $d_{min}=2$ is "universal" with respect to O(n) hamiltonians (so automatically XY model), but it appears that the situation is still not clear for $n=1$, it would be also interesting to know the behaviour of the critical temperature, but I think it's a difficult problem. – Ikiperu May 14 '13 at 22:18
Of course, this universal behavior does not extend to $O(1)$: this is just the Ising model, which has a phase transition in dimension $2$ (and should have one for any dimension $d>1$, if you set up the model correctly). The difference is due to the fact that the symmetry of the latter is discrete, while it is continuous for $O(N)$ with $N\geq 2$. – Yvan Velenik May 15 '13 at 6:38
Ok, in fact I was about to ask for the Ising model but I felt that this was the result because one can say that there is phase transition in 1d at zero temperature, so I expect that kind of result, for the XY model I thought the situation was messier. – Ikiperu May 15 '13 at 7:27