Continuum theory from lattice theory

Question

I am looking for references on how to obtain continuum theories from lattice theories. There are basically a few questions that I am interested in, but any references are welcome. For example, you can obtain the Ising chiral CFT from a lattice theory. How does this work exactly? Intuitively is is clear that one should do something like taking the lattice spacing to zero. Is this worked out somewhere in detail for this example?

One can also image, say, quantum spin models with sites on the edges of some graph, such that the interactions do not depend on the distance between the sites. One can imagine subdividing this graph further, to obtain an inclusion of the associated algebras of observables. This leads to an increasing sequence of algebras, and one can take the direct limit of this. Can one in this way obtain a continuum theory? I suppose that one might have to impose some conditions on the dynamics of the system at each step. Is something like this done in the literature?

I'm mainly interested in a mathematical treatment of these topics.

Answer 1

To take a meaningful continuum limit, essentially, you need to be in regime where your field is smooth enough that a gradient expansion is possible. This is usually acheived by associating a very high energy cost to field configurations that take different values on nearest neigbours in the lattice.

The continuum limit of $O(n)$ models is worked out in Fradkin's book, Field Theories of Condensed Matter Systems. For the Ising model a direct continuum limit is problematic because the discrete values of the spin make it impossible to directly elevate the Ising spin to a continuum field. Usually, any continuum limits have to defined by some sort of coarse graining and working with the resulting mean magnetization. For the Ising model, this is worked out by Milchev, A., Heermann, D.W. & Binder, K. in J. Stat. Phys.44, 749 (1986)

Hope that helps.

Answer 2

The most relevant tool: the Renormalization Group. You see the lattice model at larger and larger scales, and find out which terms get more relevant, and which get more irrelevant, as you zoom out. Once you reach a fixed point, the surviving terms make up your continuous system.

Answer 3

If you are looking for a mathematical treatment for your question you need to look at the book Fernandez-Frohlich and Sokal "Random walks, critical phenomena, and triviality in quantum field theory" Springer-Verlag, 1992. It might be out of print so if you can't get it you can also try these freely accessible articles:

1. A. Sokal "An alternate constructive approach to the $\phi_3^4$ quantum field theory, and a possible destructive approach to $\phi_4^4$". Annales de l'institut Henri Poincaré (A) Physique théorique 37 no. 4 (1982), pp. 317-398. Available online here.

2. J. Frohlich and T. Spencer, "Some recent rigorous results in the theory of phase transitions and critical phenomena". Séminaire Bourbaki 24 (1981-1982), Exposé No. 586, pp. 159-200. Available online here.