As one might know that topology plays an important role in the classical O(2) model in d=2 (i.e. x-y model in two dimension) becuase one can have non-trivial excitations, namely, vortices. These excitations, while having a huge energy, can contribute to the partition function at high temperatures becuase the entropy associated with them is also proportionately large. One can easily verify that perturbative epsilon-expansion (i.e. an expansion in no. of dimensions) fails here. The reason which is often cited is that such an expansion destroys all information related to topology since atleast till now there is no homotopy theory for fractional dimensions.

Considering an even simpler case, the ising transition in d=2 could be looked upon as condensation of domain walls which are again topological objects. My question is: why does an epsilon-expansion in this case works out quite nicely? (note: here one does an expansion near the lower critical dimension which is one). Is it that notion of domain walls can be extended to even fractional dimension in some way?

Btw, one might like to visit the website http://www.math.princeton.edu/~aizenman/OpenProblems.iamp/index.html which contains a collection of open problems in mathematical physics. Coming to a problem related to this post (though somewhat remotely), one might be surprised to find through this link that the perturbative result of O(N > 2) model in d=2 being always in a single disordered phase at all temperatures has actually not been rigourosly proved. Infact, it is even disputed by some numerical results (though I don’t claim any understanding of these). See for example, http://prola.aps.org/abstract/PRB/v54/i10/p7177_1.

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The first difference one notices between these two cases is the fact that Ising symmetry is a discrete symmetry, whereas x-y model is not.

Of course, this doesn’t answer your question, but, I would like to know whether this distinction is crucial. For example, Potts Model interpolates between Ising and XY.Is there a way in which the two cases you have mentioned can actually be related using the above fact ?

Just to elaborate on that further, there are goldstone modes in XY model which try to destroy any long-range-order. Despite them, there is a topological transition(called the Kosterlitz-Thouless transition) in 2-d XY model which comes about because the fundamental group or the first Homotopy group of a circle is the set of integers. Ising model, of course, doesn’t have such troubles. Can this fact be important in understanding why epsilon expansion works in one case and not in another ?

Hmm, Isn’t this the same thing as saying that Mermin-Wagner is not rigorously proved beyond perturbation theory ?

P.S.: Happy New Year. 🙂 Senthil had come here few days ago and he was talking about how gauge theories arise in condensed matter systems. So, what are you upto these days ?

The distinction due to discrete/continuos symmetry might be crucial.

Ofcourse the first and higher homotopy group for ising symmetry are zero but the zeroth homotopy isn’t and that is what is responsible for domain walls (technically, pi_0(Z_2) = Z_2). I somehow have a feeling that zeroth homotopy makes sense always, irrespective of dimension being fractional, but I am not totally sure.

Regd. Mermin-Wagner, I think that Mermin-Wagner is true beyond perturbation theory and it just states that the average of correlation function S(r)S(r’) vanishes as r-r’ tends to infinity in all dimensions less than and equal to 2. So even though there is a transtion (BKT) for O(2) in d=2, both phases satisfy the condition put by M-W as is easily verified. It would be great if some more stringent theorem could be formulated which deals with the non/existence of power-law phases for O(N) model at lower critical dim say by putting conditions on how fast the decay of correlation could be for a given N.

True, indeed. And this fact is important in 1-d(near which the epsilon expansion is done).But, in 2-d, to check the possibility of topological order, we usually look at the first homotopy group(e.g. KT transition). (And in 3-d,for example, when we look for ‘t Hooft Polyakov Monopoles in Georgi-Glashow Model, we look at the second homotopy group.)

So, in that sense, there is indeed a difference between the two cases. In case of Ising, the topology enters through the homotopy related to 1-d, where the epsilon-expansion starts and we want to understand the long range order in 2-d. In case of BKT, the relevant homotopy is that of 2-d itself which leads to the quasi-long-range-order with vortices etc.

May be. I’m not sure how to go about defining homotopy groups for fractional dimensions either.May be first we have to dig out/formulate some model defined over fractrional dimensions showing something like a topological order….

That indeed might be the way to go, if one is seeking the answer to the question, why ε expansion works for Ising ?

Let me pose a more ambitious question – let us say, I am seeking some kind of an epsilon expansion for KT – of course, the usual expansion doesn’t work – so some other expansion. Again, I will repeat the fact that Potts Model interpolates between Ising and XY – now, epsilon expansion works at the Ising end, but doesnot capture the story at the KT end – so, what happens between ? Do you know of any work that addresses this question ?