We propose carbon nanotubes (CNTs) with magnetic impurities as a versatile platform to achieve unconventional Kondo physics, where the CNT bath is gapped by the spin-orbit interaction and surface curvature. While the strong-coupling phase is inaccessible for the special case of half-filled impurities in neutral armchair CNTs, the system in general can undergo quantum phase transitions to the Kondo ground state. The resultant position-specific phase diagrams are investigated upon variation of the CNT radius, chirality, and carrier doping, revealing several striking features, e.g., the existence of a maximal radius for nonarmchair CNTs to realize phase transitions, and an interference-induced suppression of the Kondo screening. We show that by tuning the Fermi energy via electrostatic gating, the quantum critical region can be experimentally accessed.
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