Local Adiabatic Mixing of Kramers Pairs of Majorana Bound States

Abstract

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 ...

Classification of topological insulators and superconductors out of equilibrium

“…The question of which topological modes are vulnerable to local adibatic mixing is exactly equivalent to our previous consideration of which bulk topological phases are destroyed out of equilibrium, since we both problems reduce to the question of whether time-reversal and chiral symmetry-breaking terms are enough to lift the equilibrium topological protection. We therefore expect that our classification (Table II) should generalize the results of 66 to all symmetry classes, in that entries which become trivial out of equilibrium indicate that the edge modes can adiabatically mix due to local perturbations, and therefore are inappropriate for qubit storage.…”

“…∈ Z 2 . As a toy model of such a system (analogous to the one used to demonstrate mixing in class DIII in 66 ), we use a semi-infinite extended Kitaev chain with beyond-nearest-neighbour hopping and pairing 39,67…”

“…Following Ref. 66, we calculate the degree of mixing between the two modeŝ γ A 1 ,γ A 2 using a non-Abelian Berry connection. Since the variation is slow with respect to E g , a Majorana zero mode Γ (which squares to 1) must remain an instantaneous zero-energy eigenstate after time evolution, which means we must have Γ(t) = cos ϕ(t)γ I η(t) + sin ϕ(t)γ II η(t) , where the Roman numerals distinguish the two instantaneous zero-modes.…”

Gapless symmetry-protected topological phase of fermions in one dimension

“…In the presence of time-reversal symmetry a different kind of topological superconductors can be realized [16][17][18][19][20][21][22][23][24][25][26]. If the bulk is fully gapped, these phases host a Kramers pair of Majorana bound states [24,[27][28][29].…”

Proximity induced time-reversal topological superconductivity in Bi2Se3 films without phase tuning

“…The zero energy excitations in this class of TSs are Kramers pairs of Majorana modes. While their braiding properties appear to be path dependent [11,12], they exhibit other exotic transport [13,14] and spin [4,[16][17][18] properties which render them objects of fundamental interest.…”