42. Recalling a couple of facts about 2D and 3D Ising models
DmitryIn reply to my comment about string theory applicability for the solution of 3D Ising model, Thomas Larsson writes on the Peter Woit’s blog:
I am well aware that the 3D Ising model is Kramers-Wannier dual to the 3D Ising gauge model and admits a random surface representation. Even better, one can write down a funny
or
lattice gauge model on a 3D brick lattice, with a log action, that can be exactly mapped onto a model of self-avoiding random surfaces; the weight of each graph equals
, where
is the Euler characteristic,
is related to the coupling constant, and
the surface area.
Does this construction solve
A method has only really contributed to our understanding of a model if it helps to extract some kind of quantitative information about it, not necessarily critical exponents and not necessarily exactly. The methods that Peter O. mentioned do that, as do high- and low-temperature expansions, real-space RG and computer simulations. However, AFAIK no quantitative results about 3D Ising have come out of string theory.
So, I decided to refresh my memory concerning the subject and the problem of self-avoiding. Sorry if the post will be too lengthy, but I write it mostly for myself…
(By the way, Lubos, 3D Ising model is “mine” in the same sense as it is “yours” :-), I am cosmologist, not string theorist or condensed matter expert, I just happen to vaguely remember this staff.)
Let me first recall what happens in 2D case (exactly solved by Onsager), start with descrete case
,
.
We are generally interested in behavior of correlators of the spin variable 
but one can also consider correlation functions of disorder variables 
. Disorder variables are constructed in the following way:
1. We consider the point of the dual lattice 
, i.e., the center of one of the cells of initial lattice.
2. We put a string (or, better say, path or dislocation line) 
going from to infinity. This path can be arbitrary.
3. We change the sign of 
for all connections on the lattice our dislocation line intersects. As a result, partition function is changed (perturbed partition function depends on the dislocation line and initial point ).
Disorder variable is defined through its correlation functions:
etc.
Equations for corr. functions of are as bad as eqs. for corr. functions of . But let us construct the following object:
,
where vectors 
, 
connect the point 
and the centers of four adjacent plackets 
.
Equation for the corr. functions of 
can be found from the equality
.(1)
Also, satisfies the following condition: if we rotate by angle 
and simultaneously change the index 
, will remain invariant:
,
so we can say that is actually spin variable. In the continuous limit Eq. (1) acquires the form
where 
- nothing else but the Dirac equation (the spinor is actually four-component, not 2-component, but I dropped other 2 comp.)
This is well known free fermion representation for 2D Ising model. Where in this discussion is the problem of self-avoiding Thomas mentioned? The answer is very simple: fermion does not want to self-intersect its path due to the Pauli principle. Does it preclude me from considering continuous limit of the 2D Ising model? No way. I will easily write down amplitude of the propagation for the free fermion - self-intersecting paths will give strongly oscillating contribution into the overall function integral defining the amplitude. In the end it will give me
for the random Brownian motion of the fermionic particle instead of linear diffusion for bosonic particle
but one has no doubts that the model of free fermions is tractable and describes 2D Ising near the critical point.
No, let us proceed to 3D Ising. Dislocation lines 
now become dislocation surfaces 
and instead of free fermion 
describing critical behavior of the 2D Ising we have to deal with the construct
.
Equation for this construct is remarkably similar to (1) but very long - I will not write it here ;-)
Although it is unclear how to write its continuous limit (and this is the main difficulty today in this prolem as I understand), from its descrete form one can see that it describes free fermionic (NSR?) string (with free fermions - elementary segments).
I do not think that self-avoidance problem will important role here as well, since Pauli principle for the string segments will preclude it from self crossing (at the level of functional integral again one will get suppression of self-crossing paths as it was for the case of free fermion in 2D).
Thomas, due to the lack of my education, I do not quite understand why you mention U(N)/O(N) sigma models in connection with 3D Ising; the latter has just discrete 
symmetry. If I am wrong and there is relation, what is 
corresponding to Ising model, are you talking about large expansion or something else?
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Comments
Dmitry (Check me out!) Hi Thomas
Thanks a lot for the explanation and references. Because of the comments like this one blogging is a thing worth doing.
I will read the papers.
Cheers
Dmitry.
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2.5 |
Hi Dimitry,
I was not talking about sigma models, but about lattice gauge theory. Recall that the 3D Ising model is dual to Ising gauge theory, which is a lattice gauge theory with gauge group G = Z_2. The lattice action reads
S = J sum_P tr UUUU
where the sum runs over all plaquettes P. But this action works for any G, also O(N) and U(N) (you must add a hermitian conjugate for U(N)).
The mapping of lattice gauge theory into a gas of self-avoiding random surfaces is analogous to the first step in B Nienhuis’ celebrated solution of the O(N) model. The usual action for an O(N) spin model reads
S = J sum_ij s_i.s_j
Nienhuis replaced this by the log action
S = sum_ij log(1 + J s_i.s_j)
on a hexagonal lattice. The point is that the partition function is simple:
Z = Tr prod_ij (1 + J s_i.s_j)
One can show that the partition function can be rewritten as a gas of random, self-avoiding loops:
Z = sum_G N^c u^L,
where L is the total length, u is related to J, and c is the number of connected components. To show this it is important that every vertex in the hex lattice is connected to only three links.
My idea was to generalize Nienhuis construction to gauge theories in 3D. To this end, one needs a lattice where each link is on the border of only three plaquettes, e.g. the 3D brick lattice. Further, one replaces the action by
S = sum_P log(1 + J UUUUUU (+h.c.))
where each plaquette comes with a product over surrounding links. This is not the same model as the original one above, except for G = Z_2, but one expects that it is in the right universality class. Now one can show that the partition function equals
Z = sum_G N^chi u^A
where the sum runs over self-avoiding random surfaces with area A and Euler characteristic chi. In particular, the Ising gauge theory (and dually the 3D Ising model) has N=1.
Let us take this formula for Z seriously and consider the continuum limit. A and chi have nice, well-known expressions in terms of integrals over local fields. However, the measure is very complicated, because the sum only runs over self-avoiding surfaces (oriented and unoriented for U(N) and O(N)), and not over all surfaces. One could try to implement self-avoidance by adding a short-range repulsive potential, but such a potential will be very nonlocal in worldsheet coordinates although it is short-range in ambient space. Hence I believe that the problem with this approach is that self-avoidance makes the measure very complicated.
References:
TAL, J Phys A 20 (1987) L535
TAL, J Phys A 23 (1990) 1881
and independently by
F David, Europhys Lett 2 (1986) 577