We study nanowires with proximity-induced s-wave superconducting pairing in an external magnetic field that rotates along the wire. Such a system is equivalent to nanowires with Rashba-type spin-orbit coupling, with strength proportional to the derivative of the field angle. For realistic parameters, we demonstrate that a set of permanent magnets can bring a nearby nanowire into the topologically nontrivial phase with localized Majorana modes at its ends. This occurs even for a magnetic field configuration with nodes along the wire and alternating sign of the effective Rashba coupling.

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We study one-dimensional topological superconductivity in the presence of time-reversal symmetry. This phase is characterized by having a bulk gap, while supporting a Kramers' pair of zero-energy Majorana bound states at each of its ends. We present a general simple model which is driven into this topological phase in the presence of repulsive electron-electron interactions. We further propose two experimental setups and show that they realize this model at low energies. The first setup is a narrow two-dimensional topological insulator partially covered by a conventional s-wave superconductor, and the second is a semiconductor wire in proximity to an s-wave superconductor. These systems can therefore be used to realize and probe the time-reversal invariant topological superconducting phase. The effect of interactions is studied using both a mean-field approach and a renormalization group analysis.

We consider Kramers pairs of Majorana bound states under adiabatic time evolution. This is important for the prospects of using such bound states as parity qubits. We show that local adiabatic perturbations can cause a rotation in the space spanned by the Kramers pair. Hence the quantum information is unprotected against local perturbations, in contrast to the case of single localized Majorana bound states in systems with broken time reversal symmetry. We give an analytical and a numerical example for such a rotation, and specify sufficient conditions under which a rotation is avoided. We give a general scheme for determining when these conditions are satisfied, and exemplify it with a general model of a quasi 1D time reversal symmetric topological superconductor.Majorana bound states (or Majorana Fermions) in condensed matter systems have been the subject of a large research effort in the last few years. Among other reasons, this effort has been motivated by a number of recent proposals for feasible experimental systems hosting Majorana bound states (MBS) [1][2][3], and by their relevance to topological quantum computation [4]. In superconducting systems, a MBS describes a localized zero energy solution of the Bogoliubov-deGennes (BdG) equation. Such a solution constitutes "half" a Fermion, and two such solutions span a fermionic mode, of two states. The zero energy solutions of the BdG equations signify a degeneracy of the superconducting many-body ground state, defining a degenerate subspace within which manipulations are possible through adiabatic variation of the Hamiltonian. A particularly interesting set of manipulations is braiding of the positions of the MBSs (while maintaining the degeneracy of the ground state), which constitutes a set of non-abelian operations referred to as gates. If the MBS are all spatially separated from one another, these gates are expected to be topologically protected. If the distance between the MBS, L, is much larger than their localization length ξ, the unitary transformation associated with their braiding is topologically stable, which means that corrections are exponentially small in the ratio L/ξ. As such, they are exponentially small in E g , the energy gap of the superconductor.The isolation of single localized zero energy solutions requires a system where time reversal symmetry (TRS) is broken, since under TRS the solutions of the BdG equation form degenerate Kramers pairs, and isolation of an odd number of localized solutions is impossible. Recently, there has been a large interest in topological superconductors respecting TRS, i.e. in absence of magnetic fields (or spontaneously broken TRS). A number of proposals for systems in hybrid materials and structures [5][6][7][8][9][10][11][12] have been put forward. Such systems accommodate Kramers pairs of MBSs, and therefore do not allow for braiding of single Majorana fermions. Since these systems allow for braiding of Kramers pairs of MBS, the question of the possible protection of quantum information in Kramers pairs ...