Effect of midgap states on the magnetic exchange interaction mediated by a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math> -wave superconductor

We theoretically study the indirect interaction between two ferromagnetic contacts located on surface of a $d$-wave superconductor. When magnets are connected to $\{010\}$ edge superconductor we find an oscillating RKKY that varies in sign as distance magnetic is varied. However, when coupling $\{110\}$ superconductor, presence midgap states qualitatively changes results. The ground state system then found always favor alignment this configuration most strongly suppresses states, leading larger condensation energy which dominates over intrinsic interaction.