We consider the Bethe ansatz integrable Russian doll (RD) model of superconductivity with time-reversal symmetry breaking, which exhibits a cyclic renormalization group. By obtaining an exact solution for the renormalization group flows, we investigate the phase structure in the one-pair sector, which includes localized, fractal, and delocalized phases. We show that the quantum number $Q$, arising from the Bethe ansatz equations, counts the number of cycles and parametrizes the towers of states. Using the action of the renormalization group on the eigenstates, we demonstrate that $Q$ serves as an order parameter, providing a mechanism for the formation of the fractal phase in the deterministic systems and an example of the interplay between fractality and cyclic RG.

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• Order parameters
• Quantum phase transitions
• Superconductivity
• Bethe ansatz
• Integrable systems
• Renormalization group