Nonuniversal and Anomalous Decay of Boundary Spin Correlations in Inhomogeneous Ising Systems

A triangular, half-infinite, nearest-neighbor Ising system is studied whose coupling strengths close to the boundary are weaker than those in the bulk. The deviations decay as $-\frac{A}{m^y}$ with the distance $m$ from the boundary. By means of a repeated star-triangle transformation the boundary spin-spin correlation $g\left(\mathcal{r}\right)$ is studied for critical bulk coupling. For $y=1\mathrm{}$ the behavior is $g\left(\mathcal{r}\right)\sim \frac{1}{{\mathcal{r}}^{\eta }}$ as $r\to \infty$, with a nonuniversal exponent $\eta$ depending on $A$. For $0<y<\mathrm{}1\mathrm{}$ anomalous decay $g\left(\mathcal{r}\right)\sim \exp [-{\left(\frac{r}{\xi }\right)}^{1-y}]$ is found. Expressions for $\eta$ and $\xi$ in terms of $A$ and $y$ are given.