Theory of the Landau critical point.I. Mean-field theory, scaling theory, and critical exponents to orderε2

The Landau critical point is investigated for a system with an order parameter ${\psi }_{\mathrm{ij}}$, a symmetric traceless tensor of rank 2 and order 3. The nematic-isotropic transition of a liquid crystal is described by such an order parameter. Mean-field theory predicts that the ordered phase can be of two types: uniaxial and biaxial. The appearance of these phases is determined by the competition between the terms $\mathrm{Tr}{\psi }^3$ and ${\left(\mathrm{Tr}{\psi }^3\right)}^2$ in the thermodynamic potential. The extended scaling hypothesis is formulated for the shape of the phase diagram. The critical exponent ${\lambda }_{m,n}$ for the field conjugate to ${\left(\mathrm{Tr}{\psi }^2\right)}^m{\left(\mathrm{Tr}{\psi }^3\right)}^n$ is computed to order $\epsilon =4-d$. The cases $2m+3n=5 \mathrm{and} 6$ are computed to order ${\epsilon }^2$. The results are: ${\lambda }_{1,1}=-1-\left(\frac{23}{26}\right)\epsilon +\left(\frac{10441}{8788}\right){\epsilon }^2+O\left({\epsilon }^3\right)$, ${\lambda }_{0,2}=-2-\left(\frac{4}{13}\right)\epsilon +\left(\frac{5037}{4394}\right){\epsilon }^2+O\left({\epsilon }^3\right)$, and ${\lambda }_{3,0}=-2-\left(\frac{31}{13}\right)\epsilon +\left(\frac{15783}{4394}\right){\epsilon }^2+O\left({\epsilon }^3\right)$.