We construct two-band effective models describing Majorana edge states in a short Kitaev chain. We derive an analytical formula for the effective Hamiltonian as a function of model parameters. Then, we discuss the robustness of Majorana edge states as a function of model parameters. We have found an even-odd effect on the robustness of Majorana states, which is characteristic of short Kitaev chains. It is experimentally observable as a differential conductance in quantum dot systems. We also study effects of coupling to an environment based on non-Hermitian Hamiltonians derived from the Lindblad equation. It is found that the Majorana zero-energy edge states acquire nonzero energy such as $E\propto \pm {\left(i\mathrm{\Gamma }\right)}^L$ for the local dissipation, where $\mathrm{\Gamma }$ is the magnitude of the dissipation and $L$ is the length of the chain. The even-odd effect is manifest for small $L$ in this formula. On the other hand, the Majorana zero-energy edge states acquire nonzero energy such as $E\propto \pm i\mathrm{\Gamma }$ for small $\mathrm{\Gamma }$ irrespective of the length $L$ for the global dissipation.

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• Majorana bound states
• Majorana fermions
• Symmetry protected topological states
• Topological phases of matter
• Topological quantum computing
• Topological superconductors
• Non-Hermitian systems