Probability Distribution of Majorana End-State Energies in Disordered Wires

One-dimensional topological superconductors harbor Majorana bound states at their ends. For superconducting wires of finite length L, these Majorana states combine into fermionic excitations with an energy epsilon(0) that is exponentially small in L. Weak disorder leaves the energy splitting exponentially small, but affects its typical value and causes large sample-to-sample fluctuations. We show that the probability distribution of epsilon(0) is log normal in the limit of large L, whereas the distribution of the lowest-lying bulk energy level epsilon(1) has an algebraic tail at small epsilon(1). Our findings have implications for the speed at which a topological quantum computer can be operated.