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Superconducting $π$-ring metamaterials

We develop the concept of fractal metamaterials which consist of arrays of nano and micron sized rings containing Josephson junctions which play the role of "atoms" in such artificial materials. We show that if some of the junctions have $π$-shifts in the Josephson phases that the "atoms" become magnetic and their arrays can have tuned positive or negative permeabilty. Each individual "$π$-ring" - the Josephson ring with one $π$-junction - can be in one of two energetically degenerate magnetic states in which the supercurrent flows in the clockwise or counter-clockwise direction. This results in magnetic moments that point downwards or upwards, respectively. The value of the total magnetization of such a metamaterial may display fractal features. We describe the magnetic properties of such superconducting metamaterials, including the magnetic field distribution in them (i.e. in the network that is made up of these rings). We also describe the way that the magnetic flux penetrates into the Josephson network and how it is strongly dependent on the geometry of the system.