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

In spin systems such as the Ising model, the local order and disorder can be characterized by the order-parameter and energy density profiles 〈σ(r_{1})〉 and 〈ε(r_{2})〉, respectively. Does increasing the order at r_{1} always decrease the disorder at r_{2}? Does increasing the disorder at r_{2} always decrease the order at r_{1}? The answer to these questions is contained in the cumulant response function 〈σ(r_{1})ε(r_{2})〉^{(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 σ(r_{1})ε(r_{2}) and exact results for the Ising model, we analyze 〈σ(r_{1})ε(r_{2})〉^{(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 σ or ε is located on a "zero line" in the x-y plane, along which 〈σ(r)〉 vanishes. Results for half-plane, triangular, and rectangular geometries are presented.