# Lagrangian formulation of the Ising model as a conformal field theory

An important example of conformal field theory is the 2d Ising model, more precisely its scaling limit when the size of the lattice goes to zero. I am not an expert in the field, but this is the only description of this specific field theory I have seen in the literature.

Question. Can the above conformal field theory be described by a conformally invariant Lagrangian?

Any feedback, ideally a reference, will be helpful.

• here is one reference -- I guess there are many others. – Carlo Beenakker Dec 23 '16 at 9:17
• @CarloBeenakker: Thank you. It seems that they just claim that this is the theory of free massless Majorana fermions $(\psi,\bar \psi)$. If my understanding is correct then this answers my question. That simple... – makt Dec 23 '16 at 9:22
• Another Lagrangian description is the bosonic $\phi^4$ model. The keywords for web search on this are "Ginzburg-Landau" formulation of CFTs. – Abdelmalek Abdesselam Dec 23 '16 at 15:30
• ...The original reference is the article "Conformal symmetry and multicritical points in two-dimensional quantum field theory" by Zamolodchikov. You can find it in books.google.com/books?id=xHHFCwAAQBAJ. – Abdelmalek Abdesselam Dec 23 '16 at 15:38

In critical point 2d Ising model described by 2d Ising Conformal Field Theory.

We know quantum hamiltonian of the model and then we can study its most important properties, such as the duality transformation. This symmetry involves the order and disorder operators and we can clarify their physical interpretation. The operator mapping between the order/disorder operators and the fermionic fields is realized by the so-called Wigner–Jordan transformation: this brings the original hamiltonian to a quadratic form in the creation and annihilation operators of the fermions. So we able to diagonalize the quantum hamiltonian by means of particular fermionic fields.

In the limit in which the lattice spacing goes to zero, the Ising model becomes a theory of free Majorana fermions. They satisfy a relativistic dispersion relation and their mass is a direct measurement of the displacement of the temperature from the critical value $$T_c$$.

On the other hands, we know famous minimal model. Besides them, exist series of unitary minimal models $$\mathcal{M}_{p,p+1}$$ with central charges ($$p\geq 2$$): $$c= 1- \frac{6}{p(p+1)}$$ Conformal minimal models describe the scaling limit of an infinite number of statistical models with a discrete symmetry, among which we find the Ising model ($$\mathcal{M}_{3,4}$$) , the tricritical Ising model ($$\mathcal{M}_{4,5}$$), the Potts model ($$\mathcal{M}_{5, 6}$$), the Yang–Lee edge singularity ($$\mathcal{M}_{2,5}$$), etc. In addition, the unitary minimal models can be put in correspondence with the critical Landau–Ginzburg theories with power interaction $$\phi^{2(p−1)}$$ $$(p \geq 3)$$: as a matter of fact, they provide the exact solution of these theories at their multicritical point. Critical Landau-Ginsburg theory for Ising model is $$\phi^4$$ theory.

To conclude, exist 3 equivalent formulation of Ising model in critical point:

1) theory of free Majorana fermions

2) unitary minimal model $$\mathcal{M}_{3,4}$$

3) $$\phi^4$$ bosonic theory

Main reference Giuseppe Mussardo, Statistical Field Theory. About Ising model see chapter 7, 9, 14.