Interplay of order and disorder in two-dimensional critical systems with mixed boundary conditions

Abstract

In spin systems such as the Ising model, the local order and disorder can be characterized by the order-parameter and energy density profiles $⟨\sigma ({\mathbf{r}}_1)⟩$ and $⟨\epsilon ({\mathbf{r}}_2)⟩$, respectively. Does increasing the order at ${\mathbf{r}}_1$ always decrease the disorder at ${\mathbf{r}}_2$? Does increasing the disorder at ${\mathbf{r}}_2$ always decrease the order at ${\mathbf{r}}_1$? The answer to these questions is contained in the cumulant response function $⟨\sigma ({\mathbf{r}}_1)\epsilon ({\mathbf{r}}_2){⟩}^{(\mathrm{cum})}$. This correlation function vanishes in the unbounded bulk but not in systems with fixed-spin boundary conditions. Using the universal operator-product expansion of $\sigma ({\mathbf{r}}_1)\epsilon ({\mathbf{r}}_2)$ and exact results for the Ising model, we analyze $⟨\sigma ({\mathbf{r}}_1)\epsilon ({\mathbf{r}}_2){⟩}^{(\mathrm{cum})}$ in two-dimensional critical systems defined on the $x-y$ plane with mixed $+$ and $-$ boundary conditions. Particularly interesting behavior is found when either of the operators $\sigma$ or $\epsilon$ is located on a ``zero line" in the $x-y$ plane, along which $⟨\sigma (\mathbf{r})⟩$ vanishes. Results for half-plane, triangular, and rectangular geometries are presented.

arXiv comment37 pages, 2 figures