Multi-scale competition in the Majorana-Kondo system

A side-coupled Majorana zero mode in Kondo systems realizes a simple yet nontrivial hybridization that profoundly alters the low-energy physics, making such setups promising candidates for detecting Majorana zero modes. Recently, we demonstrated that the low-energy behavior of this system can be captured by a spin-charge-entangled screening process with an $A\otimes N$ boundary condition. Here, we investigate the evolution of both the screening cloud and the boundary condition in the presence of competing terms that could break either the spin-charge-entangled ${\rm SU}_{\bf L}(2)$ rotation symmetry or the topological degeneracy. We introduce a temperature-dependent spatial integral of the screening cloud, which can be obtained from the numerical renormalization group. This quantity serves as a proper observable that unambiguously captures the properties of the screening process across temperatures. A clear crossover from conventional Kondo spin screening to spin-charge-entangled screening is observed. Taking into account the overlap between Majorana zero modes, the $A\otimes N$ boundary condition reduces to a normal one, yet the spin-charge-entangled screening is protected by the ${\rm SU}_{\bf L}(2)$ symmetry. On the other hand, perturbation that breaks the ${\rm SU}_{\bf L}(2)$ symmetry can destroy the screening singlet, while leaving the low-temperature $A\otimes N$ boundary condition intact.