Critical properties of two tensor models with application to the percolation problem

Two models having $p$-dimensional Cartesian tensor order parameters are introduced. In the first the tensor is constrained to be symmetric and traceless, and in the second it is constrained to be diagonal and traceless. The three-dimensional form of the first can be used to describe the isotropic to nematic phase transition in liquid crystals. The second model is a continuum generalization of the Ashkin-Teller-Potts model, which describes the percolation problem when $p=1\mathrm{}$. Both models have cubic invariants which according to Landau mean-field theory, give rise to first-order phase transitions. These models are studied near four dimensions when the cubic invariant is small using the $\epsilon$ expansion. A new fixed point, stable in $6-{\epsilon }^{\prime }$ dimensions, is located and its properties studied. The percolation exponents to second order in ${\epsilon }^{\prime }=6-d$ follow from perturbations about this fixed point for the $p=1\mathrm{}$ Ashkin-Teller-Potts model and are $\eta =-\left(\frac{1}{21}\right)\epsilon -\left(\frac{206}{3^3{7}^3}\right){\epsilon }^2$ and $\frac{1}{\nu }=2-\left(\frac{5}{21}\right)-\left(\frac{23}{7^3{3}^3{2}^2}\right){\epsilon }^2$.