A magnetic impurity in a BCS superconductor induces the formation of a Shiba state and drives a local quantum phase transition. We generalize this concept to a one-dimensional superconductor with fractionalized excitations, where the dominant instability is superconducting. In this framework, conduction electrons fractionalize into gapless charge and gapped spin excitations. We show that magnetic impurity interacts exclusively with the spin degrees of freedom and induces a quantum phase transition. Furthermore, charge excitations influence dynamical observables, giving rise to the phenomenon we term the “fractionalized Shiba state.” At zero temperature, the tunneling spectrum exhibits universal power-law scaling with an exponent of $-1/2$ at half filling, stemming from the gapless charge modes that form a standard Luttinger liquid. Extending this analysis to finite temperatures reveals that the spectral features retain universal behavior at the critical point.

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• Kondo effect
• Superconductors
• Luttinger liquid model
• Single impurity model