The properties of the pure-site clusters of spin models, i.e., the clusters which are obtained by joining nearest-neighbor spins of the same sign, are here investigated. In the Ising model in two dimensions it is known that such clusters undergo a percolation transition exactly at the critical point. We show that this result is valid for a wide class of bidimensional systems undergoing a continuous magnetization transition. We provide numerical evidence for discrete as well as for continuous spin models, including
SU(N) lattice gauge theories. The critical percolation exponents do not coincide with the ones of the thermal transition, but they are the same for models belonging to the same universality class.
The Ashkin-Teller model is a two-layer lattice model where spins in each layer interact ferromagnetically with strength
J, and the spin-dipoles (product of spins) interact with neighbors with strength $\ensuremath{\lambda}.$ The model exhibits simultaneous magnetic and electric transitions along a self-dual line on the $\ensuremath{\lambda}\text{\ensuremath{-}}J$ plane with continuously varying critical exponents. In this article, we investigate the percolation of geometric clusters of spins and spin-dipoles denoted, respectively, as magnetic and electric clusters. We find that the largest cluster in both cases becomes macroscopic in size and spans the lattice when interaction exceeds a critical threshold given by the same self-dual line where magnetic and electric transitions occur. The fractal dimension of the critical spanning clusters is related to order parameter exponent ${\ensuremath{\beta}}_{m,e}$ as ${D}_{m,e}=d\ensuremath{-}\frac{5}{12}\frac{{\ensuremath{\beta}}_{m,e}}{\ensuremath{\nu}},$ where
d=2 is the spatial dimension and $\ensuremath{\nu}$ is the correlation length exponent. This relation determines all other percolation exponents and their variation with respect to $\ensuremath{\lambda}.$ We show that for magnetic percolation, the binder cumulant, as a function of ${\ensuremath{\xi}}_{2}/L$ with ${\ensuremath{\xi}}_{2}$ being the second-moment correlation length, remains invariant all along the critical line and matches with that of the spin-percolation in the usual Ising model. The function also remains invariant for the electric percolation, forming a new superuniversality class of percolation transition.