Research ArticleTHEORETICAL PHYSICS

Three-dimensional Majorana fermions in chiral superconductors

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Science Advances  07 Dec 2016:
Vol. 2, no. 12, e1601835
DOI: 10.1126/sciadv.1601835
  • Fig. 1 Schematic structure of Majorana point nodes of spin-orbit–coupled chiral superconductors with total angular momentum J = 1 with an n-fold (n = 2, 3, 4, 6) rotation axis along z.

    Two types of Majorana nodes are shown: on- and off-axis nodes. Whereas the former are pinned to the rotation axis (that is, ±K), the latter appear at generic Fermi surface momenta. (A) C6-symmetric case with double Majorana nodes at ±K. (B) C4-symmetric case. (C and D) C3- and C2-symmetric cases, respectively, including a view from the top (projection on the xy plane). The gap structure of the C3-symmetric superconductor has both on- and off-axis nodes, whereas that of the C2-symmetric superconductor only has off-axis nodes. Nodes with a positive (negative) monopole charge C (see Eq. 18) are indicated by solid black (white) dots, with the monopole charge (that is, C = ±1, ±2) explicitly given. In case of C4 symmetry, the sign of the Majorana node monopole charge at ±K depends on microscopic details (see the Supplementary Materials).

  • Fig. 2 Schematic representation of Majorana arc surface Andreev bound states of nodal superconductors.

    For a given surface termination, the projections of the bulk Majorana nodes onto the surface momentum space (transparent gray planes) are connected by the surface Majorana arcs (thick blue lines). The surface Majorana arcs must start and terminate at nodes with opposite monopole charge. (A) Arc structure on a side surface of the A phase of 3He. (B) Schematic arc structure of C3-symmetric J = 1 chiral superconductor for a side surface in the y direction (see also Fig. 3). The projection of bulk Majorana nodes (coming from northern Fermi surface hemisphere) on the top surface is also shown (compare Fig. 3B).

  • Fig. 3 Majorana arc surface states.

    Plots of the zero-energy (E = 0) surface Majorana arc states in surface momentum space for chiral J = 1 superconductors with C6 symmetry (A) and C3 symmetry (B to D) and gap functions 13 and 16, respectively. (A), (C), and (D) show the Majorana arc states of a surface boundary in the xz plane (that is, semi-infinite superconductor at y > 0), whereas (B) shows the Majorana arc states of surface boundary in the xy plane (superconductor z > 0). As all panels show, the surface Majorana arcs connect the projections of the bulk Majorana nodes. The dashed circle shows the radius of the Fermi surface projection. In (A), the straight dashed blue line denotes the Fermi arcs of superfluid 3He-A for comparison. The parameters used are given by λaΔ0/μ = 0.013, λbΔ0/μ = 0.01, and λcΔ0/μ = 0.004, 0.009 in (B), (C), and (D), respectively.

  • Table 1 Classification of pairing potentials.

    Classification of pairing potentials.. Summary of the classification of pairing potentials Δq≡Δ2,q of the spin ↓ states c±K+q to lowest order in (q+, q), with q± = qx ± iqy. The potentials are classified for a given combination of (n, j), where n describes an n-fold rotation axis and j is the spin angular momentum. The chiral superconductor has total angular momentum 2j, and the effective orbital angular momentum of Δq is given by l.

    CnjJ = 2jl (mod n)Pairing Δq
    n = 2j=12J = 1l = 0∝ 1
    n = 3j=12J = 1l = −1q
    n = 4j=12J = 1l = −2,2q2,q+2
    j=32J = 3l = −2,2q2,q+2
    n = 6j=12J = 1l = 2q+2
    j=32J = 3l = 0∝ 1
    j=52J = 5l = −2q2
  • Table 2 Complete set of gap functions for chiral spin-orbit–coupled superconductors.

    Complete set of gap functions for chiral spin-orbit–coupled superconductors.. List of allowed gap function components FtJ(k) of Eq. 12 for the chiral pairing channels J = 1,2,3, (pseudo)spin angular momentum j=12,32,52, and crystal rotation symmetries Cn with n = 3,4,6. For each combination (J, j), a complete set of components is given; any other allowed gap function component FtJ(k) is generated by multiplying with fully point group symmetry invariant functions (19). Because angular momenta are only defined mod n, some entries in the table are equivalent, for example, (2,12)(1,12) under C3 symmetry, where (1,12) is the time-reversed partner of (1,12). Recall that s± = sx ±isy and sx,y,z are Pauli matrices acting on the Bloch electron (pseudo)spin.

    (J, j)Trigonal (C3)Tetragonal (C4)Hexagonal (C6)
    (1,12)k+sz,kzs+,ks,kzk+2s
    kzk2sz,(k+3k3)s+
    k+sz,kzs+,kz(k+4k4)s+
    kzk+2s,kzk2s,k3sz
    k+sz,kzs+,kzk+2s
    k5sz,kzk4s,kzk6s+
    (1,32)k+sz, k+s+, k+skzk2sz,kzk2s+,kzk2sk+sz,kzs,k3sz
    kzk2s+,kzk+2s+,kzk+4s
    k+sz,kzk2s+,kzk2s
    k5sz,kzk+4s+,kzk+4s
    (1,52)k+sz, ks+, kzs
    k+3s,kzk2sz,kzk+2s+
    k+sz,k3sz,kzk+4s+
    kzs+,kzk+2s,kzk2s
    k+sz,k5sz,kzs
    kzk+2s+,kzk4s+,kzk+6s
    (2,12)(1,12)k+s+,ks,kzk+2sz
    k+3s,k3s+,kzk2sz
    k+s+,kzk+2sz,kzk4sz
    k+3s,k3s,k5s+
    (2,32)(1,32)k+s,kzk+2sz,k+3s+
    ks+,kzk2sz,k3s
    kzk+2sz,ks+,ks
    kzk4sz,k+5s+,k+5s
    (2,52)(1,52)kzk+2sz,kzk2sz,k+s+
    k3s+,ks,k+3s
    kzk+2sz,kzk4sz,k+s
    k5s,k3s+,k+3s+
    (3,12)(0,12)(1,12)k+3sz,kzk+2s+,kzk+4s
    k3sz,kzk2s,kzk4s+
    (3,32)(0,32)(1,32)kzs+,kzs,kzk+6s+
    k+3sz,k3sz,kzk6s
    (3,52)(0,52)(1,52)k+3sz,k3sz,kzk2s+
    kzk+4s+,kzk+2s,kzk4s

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/2/12/e1601835/DC1

    section S1. Details on the symmetry analysis of low-energy gap structures

    section S2. Derivation of Majorana nodes

    section S3. Two-gap feature in DOS

    section S4. NMR calculations

    section S5. Semiclassical calculation of Majorana arc surface states

  • Supplementary Materials

    This PDF file includes:

    • section S1. Details on the symmetry analysis of low-energy gap structures
    • section S2. Derivation of Majorana nodes
    • section S3. Two-gap feature in DOS
    • section S4. NMR calculations
    • section S5. Semiclassical calculation of Majorana arc surface states

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