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Properties of the $π$ state induced by impurities in a d-wave superconductor

We study the properties of a quantum impurity embedded in a superconducting host of a d-wave symmetry. The superconductor is described by the extended negative $U$ Hubbard model while the impurity introduces a repulsive interaction to the system around the central site. We discuss the influence of this repulsion on the local properties (density of states, electron pairing and spontaneous current around the impurity) of the superconductor. We show the condition of $π$ - like behaviour, defined as two subsystems having a phase difference of $π$, in the system by using a proper combination of attractive pairing bond interaction and repulsive one between electrons located at the impurity and nearest neighbour sites.