Surface Critical Exponents in Terms of Bulk Exponents

Abstract

The surface exponents associated with critical phenomena in semi-infinite systems are derived exactly in terms of bulk exponents. Results are ${\gamma }_{1,1}=\nu -1\mathrm{}$, ${\gamma }_1=\nu +\frac{(\gamma -1)}{2}$, ${\beta }_1=\frac{(3-\alpha )}{2}-\nu$, ${\Delta }_1=\frac{(1-\alpha )}{2}$, ${\eta }_{\parallel }=\frac{1}{\nu }$, and ${\eta }_{\perp }=1-\frac{(\gamma -1)}{2\nu }$, where $\nu$, $\gamma$, $\alpha$, etc., are the standard bulk exponents. The bulk-surface crossover exponent is ${\varphi }_s=1-\nu$. For $\nu >1\mathrm{}$ no surface phase can exist.