conformal invariance at higher order critical points

Question

This is a pretty naive question. I know of infrared conformal invariance in various second order phase transitions. I also know about the conformal U(1) boson at the XY critical point, an infinite order phase transition. I was always confused about the relationship between the order of the phase transition and the existence of a conformal fixed point. Do we expect CFT behavior at all critical points with order 2 and higher?

A model I'm particularly interested in right now is the Ising model on an infinite tree with, say, constant vertex degree. This model has a finite temperature, infinite order phase transition. I'd really like to know if this theory has a conformal fixed point. I'm not even sure what dimension it should live in. Probably somewhere between 1 and 2 like the tree itself...

Comments

I don't know what a conformal field theory in nonintegral dimension should mean. -

Can you give a reference for the conformal U(1) boson at the XY critical point?

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I think the theory is just free. I can't find a good reference.

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any reference would be ok. I want to see how it arises. Perhaps one can generalize the argument.

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Here are some nice notes on various formulations of the XY model http://www.cmp.caltech.edu/~motrunch/Teaching/Phy127c_Spring10/Representations_of_XYmodel.pdf

An analysis of the RG for the KT transition is here http://www.cmp.caltech.edu/~motrunch/Teaching/Phy127c_Spring10/Kosterlitz_Thouless_RG.pdf

From the first, one can see how the continuum field theory is $(\grad \phi)^2 + u cos(\phi)$ and the transition happens as u goes to zero.

My issue is that I don't have a reasonable continuum model I can map the ising model onto.

Here's the field theory version http://www.cmp.caltech.edu/~motrunch/Teaching/Phy127c_Spring10/Sine_Gordon.pdf

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It turns out, however, that the model is dual to a certain 1d spin model with an exponential interaction. I wrote about it here https://math.berkeley.edu/wp/tqft/low-t-high-t-ising-duality-on-infinite-tree-a-toy-for-adscft/ . Perhaps one can concoct a field theory for this guy...

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I was always confused about the relationship between the order of the phase transition and the existence of a conformal fixed point. Do we expect CFT behavior at all critical points with order 2 and higher?

Actually I already have this confusion for any fixed point (conformal or not): On the one hand we have phase transition of second or higher order, on the other hand we have a point in parameter space where correlation length is infinite, but beyond Landau mean field theory I don't know any better way of justifying these two being the same thing.

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I would guess that the order of the singularity of F, and hence the number of singularity-resolving perturbations of F, and hence the number of relevant operators at the fixed point, are all related in some simple way.