Transition from Fractional to Majorana Fermions in Rashba Nanowires

Abstract

We study hybrid superconducting-semiconducting nanowires in the presence of Rashba spin-orbit interaction (SOI) as well as helical magnetic fields. We show that the interplay between them leads to a competition of phases with two topological gaps closing and reopening, resulting in unexpected reentrance behavior. Besides the topological phase with localized Majorana fermions (MFs) we find new phases characterized by fractionally charged fermion (FF) bound states of Jackiw-Rebbi type. The system can be fully gapped by the magnetic fields alone, giving rise to FFs that transmute into MFs upon turning on superconductivity. We find explicit analytical solutions for MF and FF bound states and determine the phase diagram numerically by determining the corresponding Wronskian null space. We show by renormalization group arguments that electron-electron interactions enhance the Zeeman gaps opened by the fields.PACS numbers: 73.63. Nm; 74.45.+c Introduction. Majorana fermions 1 (MF) in condensed matter systems 2 , interesting from a fundamental point of view as well as for potential applications in topological quantum computing, have attracted wide interest, both in theory [3][4][5][6][7][8][9][10][11][12][13][14][15][16] and experiment [17][18][19] . One of the most promising candidate systems for MFs are semiconducting nanowires with Rashba spin-orbit interaction (SOI) brought into proximity with a superconductor 7-9 . In such hybrid systems a topological phase with a MF at each end of the nanowire is predicted to emerge once an applied uniform magnetic field exceeds a critical value 6-9 . As pointed out recently 20 , the Rashba SOI in such wires is equivalent to a helical Zeeman term, and thus the same topological phase with MFs is predicted to occur in hybrid systems in the presence of a helical field but without SOI 21,22 .Here, we go a decisive step further and address the question, what happens when both fields are present, an internal Rashba SOI field as well as a helical-or more generally-a spatially varying magnetic field. Quite remarkably, we discover that due to the interference between the two mechanisms the phase diagram becomes surprisingly rich, with reentrance behavior of MFs and new phases characterized by fractionally charged fermions (FF), analogously to Jackiw-Rebbi fermion bound states 23 . Since the system is fully gapped by the magnetic fields at certain Rashba SOI strengths (in the absence of superconductivity), these FFs act as precursors of MFs into which they transmute by turning on superconductivity.The main part of this work aims at characterizing the mentioned phase diagram. For this we find explicit solutions for the various bound states, which allows us to derive analytical conditions for the boundaries of the topological phases. We also perform an independent numerical search of the phases and present numerical results illustrating them. We show that the phases can be controlled with experimentally accessible parameters, such as the uniform field or the chemical potential. We formulat...

Topological phases of inhomogeneous superconductivity

“…A few comments are in order regarding the topological characterization of our Hamiltonian and end states. One may show that the total Hamiltonian, H 0 + H I , is invariant under the the (pseudo-)time-reversal operation T and particle-hole symmetry C defined by U † This suggests a strong analogy with other models such as the Shiba impurity regime in the presence of spin-orbit interaction [54] or semiconducting-superconducting Rashba nanowires in in the presence of magnetic fields [55][56][57], which fall into the same topological symmetry class and support the same topological numbers as our model.…”

Spin and charge signatures of topological superconductivity in Rashba nanowires

“…One of the hallmarks of such phases, in particular of topological superconductivity, are zero-energy modes such as Majorana fermions (MF) that emerge at the edges of the system. Various candidate materials can host such topological states [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] but one of the most promising platforms are semiconducting nanowires of InAs or InSb material, with strong Rashba spin orbit interaction (SOI), subjected to an external magnetic field and in proximity to an s-wave superconductor [22,23]. Experimental evidence has been reported for topological phases in such wires [24][25][26][27][28][29][30][31] as well as in magnetic atomic chains on superconducting substrates [32][33][34].…”

NMR response of nuclear-spin helix in quantum wires with hyperfine and spin-orbit interaction

“…12,13 Apart from material considerations, designs based on tunable one-dimensional conductors are suitable for realization of topologically nontrivial excitations such as Majorana fermions, [14][15][16][17][18][19][20] non-Abelian fractionally charged fermions, 21 and parafermions.…”