# Minimal models and Lattice models

How does one see that the minimal model M(4,3) is the Ising model ? And how can I argue out that the fields contained in M(6,5) but with the non-diagonal modular invariant partition function describes the 3 state Potts model ?

• It may be more accurate to state that the minimal model M(4,3) describes certain observables of the critical Ising model, in particular spin correlation functions. Other observables such as cluster connectivities are not described by the minimal model. – Sylvain Ribault Dec 3 '19 at 19:34
• @SylvainRibault Thank you for sharing the insight. Are these ideas described somewhere ? – symanzik138 Dec 3 '19 at 23:14
• For the relation between spin correlation functions and cluster connectivities you may consult Delfino-Viti arxiv.org/abs/1104.4323 . But I do not have a good reference for the relation with the minimal model CFT. – Sylvain Ribault Dec 4 '19 at 8:18
• @symanzik138, You can consult about scaling limit of Ising model and free fermions in chapter 2 of esc.fnwi.uva.nl/thesis/centraal/files/f1541951402.pdf – Nikita Jan 1 at 10:22

For the Ising model, we have the power of exact solutions. In particular, the one-dimensional transverse-field quantum Ising model, defined by the Hamiltonian $$H = - J \sum_{i} \sigma^z_i \sigma^z_{i+1} - h \sum_i \sigma^x_i,$$ turns out to be exactly solvable. One can use the Jordan-Wigner transformation to map it to free fermions, after which you can take the continuum limit and get a massless free fermion QFT at the critical point. Then you could calculate the central charge by, e.g., identifying the stress-energy tensor for the theory and computing its two-point function. Once you know that $$c = 1/2$$, then the "usual" methods of CFT tell you that this is the M(4,3) minimal model (and in this case there is only one modular-invariant choice of field content).
Of course, we believe that if we add more generic short-range interaction terms to the above model which preserve its symmetries, then there should still be a critical point, and this should still flow to the same CFT. One could argue that these related models would be given by adding extra field content to the Ising CFT, in which case the only possible perturbations are either irrelevant or they cause the system to flow to a gapped phase (this follows from the $$c$$-theorem for example).
I believe that for all possible Virasoro minimal models and modular-invariant field content, there exists some integrable lattice model which can be argued to represent it (the restricted solid-on-solid models). I don't think that the three-state Potts model can be analyzed in as much detail as these, but already in the first paper stressing the importance of modular invariance, Cardy noted that the assignment followed from some exact results. There is also a $$\mathbb{Z}_3$$ symmetry if one chooses the non-diagonal field content as opposed to the diagonal one, which obviously also lends evidence to the Potts assignment. But I am not enough of an expert on integrability to know if stronger results exist.