Generic Theory for Majorana Zero Modes in 2D Superconductors

Abstract

It is well known that non-Abelian Majorana zero modes (MZM) harbor at vortex cores in a px +ipy topological superconductor, which can be realized in a 2D spin-orbit coupled system with a single Fermi surface and by proximity coupling to an s-wave superconductor. Here we show that existence of non-Abelian MZMs is unrelated to the bulk topology of a 2D superconductor, and propose that such exotic modes can be resulted in much broader range of superconductors, being topological or trivial. For a generic 2D system with multiple Fermi surfaces and gapped out by superconducting pairings, we show that at least a single MZM survives if there are only odd number of Fermi surfaces of which the corresponding superconducting orders have vortices, and such MZM is protected by an emergent Chern-Simons invariant, irrespective of the bulk topology of the superconductor. This result may enrich new experimental schemes for realizing non-Aelian MZMs. In particular, we propose a minimal scheme to realize the MZMs in a 2D superconducting Dirac semimetal with trivial bulk topology, which can be well achieved based on the recent cold atom experiments.The quest for realization of non-Abelian Majorana zero modes (MZMs), driven by the pursuit of both fundamental physics and their potential application to faulttolerant topological quantum computation [1, 2], has been spearheaded by the developments in p-wave superconductors. Early studies predicted that MZMs exist in ν = 5/2 fractional quantum Hall state [3], at the vortex cores in 2D spinless p x + ip y topological superconductors (SCs) [4], and at ends of a 1D p-wave SC [5]. More recently, it has been proposed that the hybrid systems of s-wave SC and spin-orbit (SO) coupled matters with odd number of Fermi surfaces (FSs) can favor effective p-wave pairing states, bringing the realization of MZMs to realistic solid state experiments [6][7][8][9][10][11][12][13][14][15]. Motivated by these proposals, numerous experimental studies have been performed to observe Majorana induced zero bias conductance anomalies with different heterostructures formed by s-wave SCs and semiconductor nanowires [16][17][18][19], magnetic chains [20], or topological insulators [21][22][23].By far the experimental proposals for MZMs are built on the realization of topological SCs. Note that MZMs in SCs are topological defect modes [24][25][26][27], which correspond to nonlocal extrinsic deformations in the Hamiltonian of the topological system. For example, MZMs in the chiral p x + ip y SC harbor at vortices which exhibit nonlocal phase windings of the SC order (a global deformation in the original uniform Hamiltonian). This feature tells that the MZMs at vortices are not intrinsic topological excitations, but extrinsic modes of a SC. In this regard, one may conjecture that the existence of MZMs is not uniquely corresponding to the bulk topology of a SC, and there might be much broader range of experimental systems which can host such exotic modes, besides those based on topologically nontrivial SCs.In this Letter...

Two dimensional superconductors in electrides

“…Since the loosely bounded anionic electrons distributed in the free space, one may naturally ask the question: does the superconductivity exist in 2D electride? If possible, the versatility and the large number of compounds make 2D electride an ideal platform for the realization of quantum phenomena transitions, topological state [33] and a broad application prospect.…”

Spin-orbit Coupling and Topological Phases for Ultracold Atoms

“…The deviation of C 2 from an integer is due to finite size effect. It was further shown and proposed that when considering the ring configuration of vortex lines, a 3D non-Abelian loop-loop braiding statistics can be obtained [170]. These results reveal a real physical system to explore the exotic 3D non-Abelian braiding statistics, and shall attract further studies based on realistic ultracold atom platforms.…”

“…20(a), showing that it is in the zero angularmomentum channel and well-localized at vortex cores, thus being a MZM. The robustness of MZMs against impurities can be shown straightforwardly [170]. The physical origin of the exsistence of MZMs can be viewed as a direct consequence of bulk-boundary correspondence, as illustrated in Fig.…”

“…For the vortex line located at x = y = 0 along z-axis, one can choose cylindrical coordinate (ρ, φ, z), with∆(ρ → 0) = 0, ∆(ρ → ∞) = constant, and nφ the phase winding of the PDW component. For n = ±1 the Majorana in-gap modes can be obtained analytically and satisfy [170] H eff γ z (ρ, φ, k z ) = E kz γ z (ρ, φ, k z ), (124) E kz = sgn(nv x v y )v z k z , which implies that the vortex Majorana modes are chiral, with chirality χ M = sgn(nv x v y v z ) being related to n and Weyl node chirality χ. The Majorana operator readsγ z (k z ) = d 2 r γ z (ρ, φ, k z )f (ρ, φ, k z ), witĥ γ z (k z ) =γ † z (−k z ) for real Majorana states.…”

“…An example of 2D SO coupled semimetal[170]. (a) The band structure of 2D topological Dirac metal with two Dirac points located at Q±, the gray thick loops around two Dirac points represent the Fermi surfaces, and the color represents the average value of the spin component sz .…”