Lifshitz point ($K(p)\sim p_{\perp }^2+p_{\parallel }^4$)

A multicritical point where disordered, spatially uniform ordered and spatially modulated phases meet. The quadratic stiffness in one or more directions vanishes, so a positive higher-derivative term stabilises the Landau-Ginzburg theory. For a uniaxial case the kernel at the point has $p_{\perp }^2+p_{\parallel }^4$, with the coefficient of $p_{\parallel }^2$ tuned to zero.

For kernel ${\kappa }^{-1}{p}_{\perp }^2+{\mu }^{-1}{p}_{\parallel }^4+m^2$ with positive stiffnesses, preserve both derivative coefficients by $q_{\perp }=b{p}_{\perp }$ and $q_{\parallel }=b^{1/2}{p}_{\parallel }$. The Fourier field multiplier is $\tilde{Z}=b^{(2D+3)/4}$, the mass becomes $bm$, and a uniform source becomes $b^{(2D+3)/4}h$. The anisotropic effective dimension is $D-1/2$ and the Gaussian exponents are $\alpha =9/4-D/2$, $\Delta =(2D+3)/8$.