# conformal invariance at higher order critical points

+ 4 like - 0 dislike
947 views

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...

asked Jan 31, 2016
retagged Jan 31, 2016

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?

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

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

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

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...

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.

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.

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\varnothing$ysicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.