Real Space Renormalization of Majorana Fermions in Quantum Nano-Wire Superconductors

We develop the real space quantum renormalization group (QRG) approach for majorana fermions. As an example we focus on the Kitaev chain to investigate the topological quantum phase transition (TQPT) in the one-dimensional spinless p-wave superconductor. Studying the behaviour of local compressibility and ground-state fidelity, show that the TQPT is signalled by the maximum of local compressibility at the quantum critical point tuned by the chemical potential. Moreover, a sudden drop of the ground-state fidelity and the divergence of fidelity susceptibility at the topological quantum critical point are used as proper indicators for the TQPT, which signals the appearance of Majorana fermions. Finally, we present the scaling analysis of ground-state fidelity near the critical point that manifests the universal information about the TQPT, which reveals two different scaling behaviors as we approach the critical point and thermodynamic limit.