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

### Tables

• 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.

 Cn j J = 2j l (mod n) Pairing Δq n = 2 j=12 J = 1 l = 0 ∝ 1 n = 3 j=12 J = 1 l = −1 ∝ q− n = 4 j=12 J = 1 l = −2,2 ∝q−2,q+2 j=32 J = 3 l = −2,2 ∝q−2,q+2 n = 6 j=12 J = 1 l = 2 ∝q+2 j=32 J = 3 l = 0 ∝ 1 j=52 J = 5 l = −2 ∝q−2
• 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+,k−s−,kzk+2s−kzk−2sz,(k+3−k−3)s+ k+sz,kzs+,kz(k+4−k−4)s+kzk+2s−,kzk−2s−,k−3sz k+sz,kzs+,kzk+2s−k−5sz,kzk−4s−,kzk−6s+ (1,32) k+sz, k+s+, k+s−kzk−2sz,kzk−2s+,kzk−2s− k+sz,kzs−,k−3szkzk−2s+,kzk+2s+,kzk+4s− k+sz,kzk−2s+,kzk−2s−k−5sz,kzk+4s+,kzk+4s− (1,52) k+sz, k−s+, kzs−k+3s−,kzk−2sz,kzk+2s+ k+sz,k−3sz,kzk+4s+kzs+,kzk+2s−,kzk−2s− k+sz,k−5sz,kzs−kzk+2s+,kzk−4s+,kzk+6s− (2,12) ≅(−1,12) k+s+,k−s−,kzk+2szk+3s−,k−3s+,kzk−2sz k+s+,kzk+2sz,kzk−4szk+3s−,k−3s−,k−5s+ (2,32) ≅(−1,32) k+s−,kzk+2sz,k+3s+k−s+,kzk−2sz,k−3s− kzk+2sz,k−s+,k−s−kzk−4sz,k+5s+,k+5s− (2,52) ≅(−1,52) kzk+2sz,kzk−2sz,k+s+k−3s+,k−s−,k+3s− kzk+2sz,kzk−4sz,k+s−k−5s−,k−3s+,k+3s+ (3,12) ≅(0,12) ≅(−1,12) k+3sz,kzk+2s+,kzk+4s−k−3sz,kzk−2s−,kzk−4s+ (3,32) ≅(0,32) ≅(−1,32) kzs+,kzs−,kzk+6s+k+3sz,k−3sz,kzk−6s− (3,52) ≅(0,52) ≅(−1,52) k+3sz,k−3sz,kzk−2s+kzk+4s+,kzk+2s−,kzk−4s−

### 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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