# SU(2) critical point and volume dependence

I am doing multi-dimensional plots of $\beta_j$ for SU(2) for infinite volume to understand the flow behavior and I was wondering, before I go too much further, if anyone knew off the top of their head whether the value of SU(2)'s critical point on the phase diagram has any volume dependence? I have tried to find out in the literature but so far I only found two papers (http://arxiv.org/abs/hep-lat/0404015, http://arxiv.org/abs/hep-lat/9509091) on SU(2) critical phenomena and I haven't been able to sort out that fact yet. Any help would be appreciated.

EDIT:

sorry, I mean in the $\beta_{\frac{1}{2}}$ and $\beta_{1}$ ("beta fundamental and beta adjoint") plane, with a Wilson action $\beta_{1/2}(\chi_{1/2}-2)+\ldots$ keeping only the first three terms in the expansion (up to 3/2). $j$ is the representation. I am doing Migdal-Kadanoff recursion and plotting the different $\beta$s against each other (this is the flow of the $\beta$s throughout the recursion). There is a first order phase transition line in the $\beta_{1/2}$vs $\beta_1$ plane that ends at a critical point, I would like to know if that line (and the critical point) moves around depending on the volume.

• This is pretty cryptic. You mean SU(2) pure gauge theory on the lattice (Wilson action?) with $\beta=1/g^2$ (or $2N_c/g^2$?). What is the index j? What do you mean by flow? (as a fct of $N_\tau$?) – Thomas Oct 17 '12 at 15:27
• I tried to make it better. Sorry 'bout that. – kηives Oct 17 '12 at 21:31
• This might not be referring to gauge theory--- it might be a Wess-Zumino model. What dimension are you in, and what's the action? – Ron Maimon Oct 18 '12 at 4:33
• @RonMaimon it's pure gauge, and in 4D. The initial action before recursion is $S_{plaq}=-[\beta_{1/2}(\chi_{1/2}-2)+\beta_{1}(\chi_{1}-3)+\beta_{3/2}(\chi_{3/2}-4)]$ – kηives Oct 18 '12 at 13:52

## 1 Answer

Well, in Monte-Carlo, yes, cf Patel & Gupta 1985 and references thereto. Your estimate of the infinite-size critical index will depend on your finite-sized approximant. Also see Meurice et al 2014, Sampling versus blocking, or Engels & Mitrjushkin 1992.

But, if you are just interested in Migdal-Kadanoff, that is just a recursion, and so the accuracy would depend on numerical implementations, not size... the size would be notionally infinite, cf Bitar et al, 1982, online.

In fact, the shape of the MK trajectories is dictated by a universal limit, the heat kernel action echoing the central limit theorem of statistics, cf Horn & Zachos 1984.

I excerpt the fundamental adjoint projection you are interested in: All (inverse) couplings find their way to the "central stream" (below the fundamental component axis, spin 1/2), and flow to the origin (confinement) along it, upon "block-spinning" devolution to long distances.

You might imagine there is a phase boundary from the upper left region to the origin, but there are no real phase boundaries. The trajectory you see close to the adjoint axis (spin 1) is actually the projection on the left wall in the following 3D figure of the dashed line: there are always windows, as in the solid phase diagram of the first three components (spins 1/2,1/3/2) you mention, .