Braiding induced by the finite-size effect in one-dimensional topological superconductors

We investigate the transport properties of Majorana zero modes (MZMs) and Majorana Kramers pairs (MKPs) in one-dimensional topological superconductors, respectively. An effective model is established for the braiding of MZMs and MKPs. We employ ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave topological superconductors to embody the effective model for the braiding of MKPs by utilizing the finite-size effect and minimal locally tunable coupling parameters. We show how to construct the state initialization and readout via gate control. We also use this method for the braiding of MZMs in $s$-wave topological superconductors. Our proposal presents a promising avenue for experimentally verifying the non-Abelian statistical properties of MZMs and MKPs, with implications for topological quantum computing.