Uma abordagem tensorial para o estudo de dualidades entre modelos de spin na rede

Abstract

In this work, we study the dualities between spin models in two-dimensional lattices from a tensorial approach. In this approach, we associate tensor components to the vertices and links so that the partition function Z is constructed by a contraction of the indices of the tensor components thereby making Z a scalar under change of basis of the group algebra C [G] used to de ne the tensors. Having obtained this, and noting that the values of the components x the studied model, we obtain a di erent model for each basis transformation proposed. These di erent models, however, have the same partition function since Z is invariant under these transformations. In fact we can obtain several models all dual to each other in this manner. We then focus on Zn spin models, which include the Ising model, the Potts model and AshkinTeller-Potts model. Exploring a speci c basis transformation, we are able to rederive Kramers and Wanniers' duality for the Ising model. With analogous arguments, we also show that Potts models with n = 3 and n = 4 are self-dual whereas this property is lost for n ≥ 5. The Ashkin-Teller-Potts model is shown to be self-dual for all n ∈ N.