Critical Behavior of a Semi-infinite System:n-Vector Model in the Large-nLimit

Abstract

The $n$-component Ginzburg-Landau-Wilson model for a semi-infinite system is solved exactly at $T=T_c$ in the limit $n\to \infty$. In the scaling regime the spin-spin correlation function is $G(\overrightarrow{\rho }, z, z^{\prime }, T_c)=\mathrm{const}{\{{[{\rho }^2+{(z-z^{\prime })}^2]}^{-1\mathrm{}}-{[{\rho }^2+{(z+z^{\prime })}^2]}^{-1\mathrm{}}\}}^{\frac{\mathrm{}(d-2\mathrm{})\mathrm{}}{2}}$, for dimensionalities $d$ in the range $2<d<\mathrm{}4\mathrm{}$, where $z$, $z^{\prime }$ are the distances of the two spins from the surface and $\overrightarrow{\mathrm{p}}$ is their separation parallel to the surface. The critical exponents ${\eta }_{\perp }$ and ${\eta }_{\parallel }$ are $\frac{\mathrm{}(d-2\mathrm{})\mathrm{}}{2}$ and ($d-2\mathrm{}$), respectively.