Helical Majorana modes in iron based Dirac superconductors Elio J. K¨ onig 1 and Piers Coleman 1, 2 1 Department of Physics and Astronomy, Center for Materials Theory, Rutgers University, Piscataway, NJ 08854 USA 2 Department of Physics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, UK (Dated: June 18, 2022) We propose that propagating one-dimensional Majorana fermions will develop in the vortex cores of certain iron-based superconductors in the flux phase, most notably Li(Fe1-xCox)As. A key ingredient of our proposal is the presence of bulk 3D Dirac semimetallic touching points, recently observed in ARPES experiments [P. Zhang et al., Nat. Phys. 15, 41 (2019)]. Using an effective k ⋅ p model which describes this class of material in the vicinity of the Γ − Z line, we solve the Bogoliubov- deGennes Hamiltonian in the presence of a vortex, demonstrating the development of gapless one- dimensional helical Majorana modes, protected by C4 symmetry. To expose the topological origin of these modes, we use semiclassical methods to evaluate a topological index for arbitrary dispersion beyond the k ⋅ p approximation. This allows us to relate the helical Majorana modes in a vortex line to the presence of monopoles in the Berry curvature of the normal state. We highlight various experimental signatures of our theory and discuss its possible relevance for quantum information applications and the solid state emulation of the early universe. Can iron-based superconductors (FeSC) sustain exotic, fractionalized excitations? Recent experimental [1–6] and theoretical [7, 8] advances suggest an affirmative an- swer. Exploiting this, we here propose the previously un- noticed emergence of dispersive, helical Majorana states in the flux phase of certain FeSCs. Twelve years ago, two major, independent discover- ies revolutionized condensed matter physics: the ob- servation of high temperature superconductivity in the iron-pnictides [9, 10] and the discovery of topological insulators (TIs) [11]. Iron based superconductivity has since attracted immense interest, offering a major chal- lenge to our understanding of strongly correlated elec- tron materials and the possibility of practical applica- tions. This led to the discovery of a broad family of su- perconducting iron-based compounds. Typically, these are layered pnictides or chalcogenides where Fe 2+ ions are enclosed in tetrahedral cages of ligand atoms. As a consequence of the associated crystal field splitting, the electronic kinetics at the Fermi energy involves the three t 2g orbitals of the d-shall propagating within the iron planes. Early magneto-oscillation and photoemission ex- periments corroborated this picture of cylindrical Fermi surfaces [12, 13]. The discovery of band topology took place in parallel with these developments [14, 15], leading to the predic- tion of edge, boundary, and surface states in TIs and fully gapped superconductors. Topological field theories and concepts from differential geometry suddently found po- tential applications to nano-electronics and certain pro- posals for quantum computation devices. These concepts have been further generalized to topologically or symme- try protected Weyl and Dirac semimetals [16]. Remark- ably, the excitations in those materials emulate certain aspects of elementary particle physics in solid state ex- periments while at the same time, the protected touching points are of potential interest for quantum sensing ap- plications over a wide range of frequencies down to the infrared. Until recently, the research efforts on FeSCs and topo- logical states of matters have been largely disconnected. Spin-orbit coupling (SOC), which is a central element of most standard TIs and semimetals, was believed to be unimportant for FeSC. The discovery of SOC in photoe- mission spectra [17] subsequently challenged this assump- tion, leading to the proposal [7, 17] that for sufficiently small interlayer separations [1] an enhanced c-axis disper- sion of the p z orbitals leads to a band-crossing, driving the corresponding FeSC into a topological metal. As we shall explain in detail below, the band crossing implies that the corresponding FeSCs are topological and can sustain Majorana zero modes at the location where a magnetic flux line intersects the system’s surface, Fig. 1 (c). In this work we demonstrate that this topological framework [1, 7, 8] also implies, under the right condi- tions, the development of dispersive, helical Majorana fermions, Fig. 1 (e) along the cores of superconducting vortices. The observation of the latter would on the one hand provide further evidence for the correctness of the proposed topological scenario and on the other hand rep- resent the first experimental observation of these exotic excitations. Helical Majorana fermions in one dimension corre- spond to a pair of gapless counter-propagating fermionic excitations which, to date, have not been experimentally observed, see Tab. I. Generically, such modes can be trapped either inside a line defect of a three dimensional material, or at the interface between two two-dimensional topological phases. The first material-specific proposal concerning the o−vortex in the B-phase of superfluid 3 He [18] was never observed, possibly because the ener- getics of 3 He-B favors less symmetric v−vortices [19, 20], which do not support helical Majorana modes. While ad- ditional candidate systems were proposed in subsequent arXiv:1901.03692v1 [cond-mat.supr-con] 11 Jan 2019