Low-temperature renormalization group for the Lifshitz point

Abstract

The momentum-shell technique is employed to derive the recursion relations for the Lifshitz point above the lower critical dimensionality. For the uniaxial point ($m=1\mathrm{}$) the expansion is made in $\epsilon =d-2.5\mathrm{}$ and the critical exponents are calculated. The results are ${\lambda }_{\mathrm{t}}={\nu }_4^{-1\mathrm{}}=2\mathrm{}\epsilon$, ${\eta }_2=\frac{\epsilon }{\mathrm{}(n-2\mathrm{})\mathrm{}}$, ${\eta }_4=\frac{2\epsilon }{\mathrm{}(n-2\mathrm{})\mathrm{}}$, ${\beta }_k=\frac{1}{2}+\frac{3\epsilon }{4(n-2\mathrm{})\mathrm{}}$, where $n$ is the number of components of the spin. The critical behavior in the presence of long-range power-law interactions is also studied.