A one-dimensional spin-orbit coupled nanowire with proximity-induced pairing from a nearby s-wave superconductor may be in a topological nontrivial state, in which it has a zero-energy Majorana bound state at each end. We find that the topological trivial phase may have fermionic end states with an exponentially small energy, if the confinement potential at the wire's ends is smooth. The possible existence of such near-zero-energy levels implies that the mere observation of a zero-bias peak in the tunneling conductance is not an exclusive signature of a topological superconducting phase, even in the ideal clean single channel limit.

A recent experiment [Mourik et al., Science 336, 1003] on InSb quantum wires provides possible evidence for the realization of a topological superconducting phase and the formation of Majorana bound states. Motivated by this experiment, we consider the signature of Majorana bound states in the differential tunneling conductance of multi-subband wires. We show that the weight of the Majorana-induced zero-bias peak is strongly enhanced by mixing of subbands, when disorder is added to the end of the quantum wire. We also consider how the topological phase transition is reflected in the gap structure of the current-voltage characteristic. [5,6], superconducting order is induced in an InSb quantum wire by proximity to a Nb lead attached alongside the wire. At the other end, the quantum wire is contacted to a normal lead via a gate-induced tunnel junction. Evidence for the formation of Majorana bound states is found through measurements of the differential conductance, which exhibits a zero-bias peak when a magnetic field is applied in certain directions. Similar results were also obtained for normal-metal-superconductor structures based on InAs quantum wires [7].

We present a solution of Kitaev's spin model on the honeycomb lattice and of related topologically ordered spin models. We employ a Jordan-Wigner-type fermionization and find that the Hamiltonian takes a BCS-type form, allowing the system to be solved by Bogoliubov transformation. Our fermionization does not employ nonphysical auxiliary degrees of freedom and the eigenstates we obtain are completely explicit in terms of the spin variables. The ground state is obtained as a BCS condensate of fermion pairs over a vacuum state which corresponds to the toric-code state with the same vorticity. We show in detail how to calculate all eigenstates and eigenvalues of the model on the torus. In particular, we find that the topological degeneracy on the torus descends directly from that of the toric-code, which now supplies four vacua for the fermions, one for each choice of periodic vs antiperiodic boundary conditions. The reduction of the degeneracy in the non-Abelian phase of the model is seen to be due to the vanishing of one of the corresponding candidate BCS ground states in that phase. This occurs in particular in the fully periodic vortex-free sector. The true ground state in this sector is exhibited and shown to be gapped away from the three partially antiperiodic ground states whenever the non-Abelian phase is gapped.

The spectral properties of Kitaev's honeycomb lattice model are investigated both analytically and numerically with the focus on the non-abelian phase of the model. After summarizing the fermionization technique which maps spins into free Majorana fermions, we evaluate the spectrum of sparse vortex configurations and derive the interaction between two vortices as a function of their separation. We consider the effect vortices can have on the fermionic spectrum as well as on the phase transition between the abelian and non-abelian phases. We explicitly demonstrate the 2 n -fold ground state degeneracy in the presence of 2n well separated vortices and the lifting of the degeneracy due to their short-range interactions. The calculations are performed on an infinite lattice. In addition to the analytic treatment, a numerical study of finite size systems is performed which is in exact agreement with the theoretical considerations. The general spectral properties of the non-abelian phase are considered for various finite toroidal systems.

One-dimensional p-wave superconductors are known to harbor Majorana bound states at their ends. Superconducting wires with a finite width W may have fermionic subgap states in addition to possible Majorana end states. While they do not necessarily inhibit the use of Majorana end states for topological computation, these subgap states can obscure the identification of a topological phase through a density-of-states measurement. We present two simple models to describe low-energy fermionic subgap states. If the wire's width W is much smaller than the superconductor coherence length ξ, the relevant subgap states are localized near the ends of the wire and cluster near zero energy, whereas the lowest-energy subgap states are delocalized if W ξ. Notably, the energy of the lowest-lying fermionic subgap state (if present at all) has a maximum for W ∼ ξ. The search for Majorana fermions has attracted a great deal of interest in the last few years [1]. Notably their nonlocal properties and non-abelian braiding statistics make Majorana fermion systems attractive candidates for fault tolerant quantum computation [2][3][4]. The present wave of interest is driven by a number of proposals that suggest ways of realizing and manipulating Majorana states in solid state systems, most prominently interfaces of s-wave superconductors and topological insulators [5,6], half-metallic ferromagnets [7-9], or semiconductor films or wires [10][11][12], where the latter stand out because Majorana manipulation require a mere series of gate operations [13]. In all these proposals, the proximity coupling to the s-wave superconductor effectively turns the normal metal into a p-wave superconductor, which is well known to harbor Majorana fermions at its ends or edges [14][15][16][17].Majorana bound states at ends of what is effectively a pwave superconducting wire can be analyzed most straightforwardly if these wires are strictly one dimensional, with only a single propagating mode at the Fermi level in the absence of superconductivity [10,11]. Nevertheless, Majorana end states can also exist in a quasi one-dimensional geometry. The effect of multiple transverse channels, present in most realistic realizations, has been addressed in Refs. [12,[18][19][20][21]. Specifically one sees that a complex p-wave superconductor in a strip geometry undergoes a series of oscillatory quantum phase transitions between topologically trivial and topologically nontrivial phases (without and with Majorana end states, respectively) as the strip width W or chemical potential µ are varied. Both with and without Majorana end states, a range of subgap states is found [19], analogous to the sub-gap states in vortex cores of bulk superconductors [22]. Although the mere presence of sub-gap states does not prohibit the use of Majorana end states for topological quantum computation [23], the presence of low-lying sub-gap states clearly obstructs an unambiguous experimental verification of the Majorana states.The purpose of this paper is to systematically analyse the energi...