Tri-Dirac Surface Modes in Topological Superconductors

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by  Chen Fang

Tri-Dirac Surface Modes in Topological Superconductors
Chen Fang1,2,3, B. Andrei Bernevig3 and Matthew J. Gilbert2,4,5
1Department of Physics, University of Illinois, Urbana IL 61801-3080
2Micro and Nanotechnology Laboratory, University of Illinois, Urbana IL 61801
3Department of Physics, Princeton University, Princeton NJ 08544
4Department of Electrical and Computer Engineering, University of Illinois, Urbana IL 61801 and
5Department of Electronics Engineering, University of Rome ”Tor Vergata”, Rome, Italy 00133
(Dated: January 10, 2014)
4 Weproposeanewtypeoftopologicalsurfacemodeshavingcubicdispersioninthree-dimensional
1 topological superconductors. Lower order dispersions are prohibited by the threefold rotational
0 symmetry and time-reversal symmetry. Cooper pairing in the bulk changes sign under improper
2 rotations, akin to3He-B. The surface manifestations are a divergent surface density of states at
n the Fermi level and isospins that rotate three times as they circle the origin in momentum space.
a We propose that Heusler alloys with band inversion are candidate materials to harbor the novel
J topological superconductivity.

8

] Topologicalsuperconductors(TSC)areanovelclassof traces an iso-energy contour. We then determine the re-
ll superconductors where Fermi surface topology and un- quirementonthebulksuperconductivityforthesesurface
a
conventional Cooper pairing in the bulk lead to gapless modes to appear. We show that the Fermi level in the
h
surface excitations called Majorana fermions[1] - aptly bulk must cross bands formed by the j = 3/2 states
- z
s named[2] as the particle-hole symmetry (PHS) renders (~ 1 hereafter), branching from a Γ -rep±resentation
e ≡ 8
m their on-site creation and annihilation operators equal. (denoting the four p-states with total angular momen-
Thepossibilityoffindingthesenewfermions,long-sought tum j =3/2 in a spin-orbit split system) on a cubic lat-
.
t in high energy physics, in condensed matter systems has tice. ThisrequirementismetinaseriesofHeusleralloys
a
m excited a wave of interesting studies[3–16]. that are zero gap semiconductors[25–28]. Furthermore,
the Cooper pairs are required to transform nontrivially
- In three dimensions (3D), TSCs are predicted in
d underthecubicsymmetrygroup: theychangesignunder
doped semiconductors having conduction/valence bands
n improper rotations but not under proper ones. We per-
inverted by spin-orbit coupling and Cooper pairing that
o
form a concrete model study of a spin-3/2 Fermi liquid
c is odd under inversion and invariant under time-reversal
with full O(3) symmetry and TRS, showing its leading
[ (TRS)[7, 8, 17]. The surface Bogoliubov quasiparticle
excitations of these TSCs form linear, spin-split Dirac instabilitytowardsthisnontrivialsingletCooperpairing,
1
induced by screened Coulomb repulsion. The resultant
v cones (as opposed to spin degenerate cones seen in
superconductivityisacloseanalogueofthesuperfluityin
3 graphene)[3, 5, 18]. More recently, both theoretical and
2 experimental studies show that crystalline symmetries the B-phase of 3He[29, 30], but in a Fermi liquid whose
8 constituent particle is spin-3/2. Finally, we discuss ex-
(with or without TRS) protect new classes of topolog-
1 perimentalsignaturesthatcharacterizethe new topolog-
ical phases. In Ref.[19], it has been shown that mirror
.
1 reflection symmetry protects gapless excitations in sev- ical superconductor.
0 Westartbyconsideringatwo-bandk p-theoryforthe
4 eral IV-VI semiconductors and in Ref.[20–22], the same surface states of a TSC, in the Bogoli·ubov-de Gennes
symmetry is shown to protect gapless surface modes in
1 (BdG) form, around Γ¯ where a doublet of Majorana
: superconductors possessing mirror-odd Cooper pairing.
v modes are assumed to exist. The symmetries are the
It then follows that the only way to open a gap in the
Xi system is by spontaneously breaking TRS or by forming TRS, denoted by T,PHS, by P, andthe three-fold rota-
tion C . They commute with each other, as they act on
r a surface topologically ordered state[23, 24]. 3
a differentdegreesoffreedomoftime,chargeandspace,re-
In this Letter, we show that the three-fold rotation
spectively. Thespin-orbitcouplinghasbrokentheSU(2)
symmetryandTRScanprotectanexotictypeofsurface
symmetry of spin rotation so C simultaneously rotates
3
states in TSCs whose dispersion consists of two cubic-
the position and the spin of a particle. For odd-integer
dispersing bands touching each other at Γ¯ in the surface spins, C3 = 1 due to the Berry phase brought by the
Brillouin zone (BZ). Lower order dispersions, i.e., linear 3 −
spin. Theseconstraintsresultinthefollowingirreducible
andquadratic,areexcludedbythesymmetrygroupgen-
representations of the group generators (up to an arbi-
erated by C ,T , if and only if the doublet at Γ¯ of the
3 trary unitary transformation):
surface BZ{has a}ngular momenta 3/2~ along the nor-
maldirection. Thesurfacedensityo±fstates(DOS)atthe E : T = K(iσ ), P =Kσ , C =eiσzπ/3, (1)
1/2 y x 3
Fermi level diverges due to the cubic dispersion, and, by E : T = K(iσ ), P =Kσ , C = I .
3/2 y x 3 2 2
writing down a generic k p-model, we see that the spin − ×
polarization makes three ·full rotations as a wave packet In Eq.(1), C -rotation operator is exp(iˆj 2π) with ˆj =
3 z 3 z
2
dispersion (Dirac point) has vanishing density of states
E
(DOS), the DOS at a tri-Dirac point is divergent, given
by ρ(E) = 1 δ(E (k) E)dk2 E 1/3, where the
2 3/2 − ∝ −
prefactor1/2isbecauseeachexcitationclosetotheFermi
R
level is roughly an equal weight linear superposition of
electron and hole states.
One may naturally infer from the divergence of DOS
Γ¯ at the band touching that the residual interaction be-
tween Bogoliubov excitations may be relevant and open
(a) (b)
a gap in the spectrum. As the surface states are of Ma-
joranacharacter,they containtwo species of Bogoliubov
FIG. 1: (a) The energy dispersion of the Bogoliubov excita-
tions around a tri-Dirac point. (b) The iso-energy contours excitations,namely γ1,2(k), satisfying γi(k)=γi†(−k) in
and the pseudospin structure of h3/2(k) along the contour k-space, or, γi(r) = γi†(r) as a real field in real space.
taking c2 =2c3 =1. The interaction hence must contain at least two spatial
derivatives, such as gγ (r)γ (r) γ (r) γ (r). Sim-
1 2 1 2
− ∇ ·∇
pledimensioncountingshowsthatcouplingconstantg is
σ /2 (ˆj = 3σ /2) for the E (E ). The k p-
z z z 1/2 3/2 · irrelevant,meaning that it flows to zerounder renormal-
Hamiltonian, h(k), must satisfy the following symmetry
ization towards the long wavelength limit. The surface
constraints:
states of a tri-Dirac cone are hence robust against weak
Th(k)T 1 = h( k), (2) residual interactions between quasiparticles.
−
−
With the nature of the surface states understood, we
Ph(k)P 1 = h( k),
−
− − now seek the form of the bulk superconductivity that
C3h(k+,k )C3−1 = h(k+ei2π/3,k e−i2π/3), gives rise to this new type of surface states. We begin
− −
by noticing that since the tri-Dirac point derivesits pro-
where k =k ik . Using Eqs.(1,2), we obtain
x y
± ± tection from both C -symmetry and TRS, it may only
3
h (k)=Re[c k ]σ +Im[c k ]σ , (3) appear on terminations that preserve both these sym-
1/2 1 + x 1 + y
h (k)=Re[c k3 +c k3]σ +Im[c k3 +c k3]σ , metries, i.e. at the Γ¯ point, in the surface BZ. (Other
3/2 2 + 3 − x 2 + 3 − y C3-invariant points K¯ and K¯′ which are invariantunder
wherec arecomplexnumbersthatarematerialdepen- C T do not have degeneracybecause (C T)2 =+1.)
1,2 2 2
dent. Hence, we find that while the dispersion of h is Sec∗ond, we need a pair of Majorana modes∗at Γ¯, which
1/2
linear (Dirac-like) around k = 0, the dispersion of h depends onbandstructure along the line, parameterized
3/2
is cubic, E (k) = c k3 +c k3 . Linear and second by k , in the 3D BZ that projects onto Γ¯. In the weak
order terms3/i2n k do n|o2t a+ppea3r [−se|e Fig.1(a)]. This can coupzling limit, the topology of this line depends on the
be understood by examining the continuum limit with signs of pairing amplitude at the Fermi points where the
C -symmetry: a hopping from a state of j = 3/2 line crosses the Fermi surface: it is nontrivial/trivial if
z
to∞j = 3/2 changes the total angular momentum± by thereareodd/evennumberofFermipointsthathaveneg-
z
∓
3, so in order to preserve the rotation symmetry, this ative pairing signsin the regionk >0. Finally, we need
z
∓
hopping should be “balanced” by an orbital part of k3. the Majoranamodes to haveangularmomenta 3/2. In
The wavefunction of h (k) in Eq.(3) is described±by the weak coupling limit, the surface Bogoliubo±v excita-
3/2
the pseudospin structure at each k. The pseudo-spin up tions are linear combinations of particle- and hole-states
(down) states correspond to the first (second) basis vec- on the Fermi surface in the bulk, so we need the Fermi
tor in the k p-model, i.e., the two degenerate states at level to cross bulk bands that have angular momenta
·
k=0. The pseudo-spin at any k is then given by a unit 3/2. InmanyHeusleralloys,the(111)-terminationsare
±
vector in the xy-plane, as is guaranteed by PHS, whose C -symmetric;theFermilevelisattheΓ -representation,
3 8
twocomponentsare(S ,S )=(Re[c k3 +c k3]/c k3 + branching into two sets of doublet bands having angular
c k3 ,Im[c k3 +c k3]x/cyk3 +c k32).+As3th−e m|o2m+en- momenta 3/2 (the E -branch as conduction band)
tu3m−|makes2a+full3cir−cle|e2nc+losing3 t−h|e origin clockwise, and 1/2±(the E -bra3/n2ch as valence band) along ΓL
1/2
±
the pseudo-spin has made three full rotations clockwise from Γ, respectively[27]. In a thin-film sample, chemi-
(if c > c ) or counterclockwise (if c < c ) [see cal doping may be applied through liquid gating[31] to
2 3 2 3
| | | | | | | |
Fig.1(b)]. Therefore, we refer to the degeneracy point tune the chemical potential into the j = 3/2 bands
z
±
at h (k) in Eq.(3) at k = 0 the tri-Dirac point, as in this system. The remaining question is what interac-
3/2
the evolution of the wavefunction around is topologi- tioninduces a Cooper pairingthat changessignbetween
cally equivalent to that around three Dirac points, re- the two bands having j = 3/2? As the Fermi level
z
| |
sulting in total winding number of 3. The tri-Dirac is close to the Γ -representation, it is reasonable to con-
8
±
point splits if C is broken by, for instance, mechani- sider a continuum model with TRS and full O(3) sym-
3
cal strain. While a generic band crossing with linear metry group, of which the point groups of half Heusler
3
alloys (T ) and full Heusler alloys (O ) are subgroups. corresponds to the normal s-wave pairing and is invari-
d h
The surface states obtained in this symmetry-enhanced ant under all symmetry operations; (L,S,J) = (1,1,0)
modelwouldcertainlychangeasonebreaksO(3)downto pairingisanalogoustothepairingintheB-phaseof3He,
T orO ,butthetri-Diracpointonthesurfacewouldnot invariantunder allproperrotationsbutchangessignun-
d h
be broken, because only C and TRS are needed for its der improper ones (parity odd)[8, 32]. Without accu-
3
protection. The restoration of O(3) simplifies the calcu- rate information on the material dependent interaction
lation and makes underlying physics more transparent. parameters,itis hardto explorethe completephase dia-
The normal state Hamiltonian for a Fermi gas having gramwith all types of instabilities, but since allpairings
O(3) and TRS around Γ -representation is given by the with J >0 are nodal (assuming that the FS encloses Γ)
8
following four-band k p-model and that nodal gaps are in general less energetically fa-
·
voredthanthe fullgaps,werestrictthediscussiontothe
5
H (k) = (λ + λ )k2 2λ (k S)2 µ, (4) two singlet pairings:
0 1 2 2
2 − · −
w( h2e.r5e, λ31,.28)a2rmee/L~u2ttiinngSecrPptBari[a2m7]e]tearnsd[Se.g=., ((Sλx1,,Sλy2,)Sz∼) Vˆ ≈ |∆ˆV0000|2 − 2|∆ˆ3V1120|2, (6)
− −
are the spin operators of a spin-3/2 fermion, physically
in which
realized by spin-orbital splitting of the p-orbitals. The
isotropic dispersion of H (k) is given by ǫ (k) = ∆ˆ V
(λ1 ±2λ2)k2 −µ, where0both bands are do1u/b2,l3y/2degen- V0000 ≡ k (1− V20k2)h3/2,3/2;m,m′|3/2,3/2;0,0ibmm′(k),
erate. The general form of short-ranged density inter- X
action is Vˆ = qV(|q|)n(q)n(−q), where n(q) is the ∆ˆ110 k(M) 1,1;M,Sz 1,1;0,0 (7)
FVourieVrtqr2an+sfOor(Pmq4)ofilsoctahledFenousirtiyeroptrearnatsoforramndofVt(h|qe|)in=- V2 ≡k,M,XSz,m,m′ h | i×
te0ra−ctio2n, and further expansion is ignored in the long h3/2,3/2;m,m′|3/2,3/2;1,Szibmm′(k),
pwualvseiolennVgt(hr)li=miter2(ek−Fr/≪r0,1V)0. =Foer2ar02sacnredenVe2d=Ceo2url04o,mwbherree- whjh1,ejr2e;mk1(±,m1)2|j1≡,j2;∓J,pM3i/’2ska±r,e thke(0)Cleb≡sch-√G3okrzd,anacnod-
r0 is the screening length. Decomposing the interaction efficients, expanding the eigenstate of total angular mo-
intovariouschannelsofinstability,wehave(modulonon- mentum J,M intheproductbasisof j ,m j ,m .
1 1 2 2
| i | i⊗| i
superconducting channels) The relative sign between the two terms in Eq.(6) is be-
cause the attraction (repulsion) favors the parity even
V
Vˆ = V0 (1 2k2)bmm′(k)2 (5) (odd) pairswhere the wavefunctionsofthe twoelectrons
| − V |
k,m,m′ 0 have greater (smaller) overlap in real space. The inter-
X
action leads to spontaneous superconducting ordering of
− 2V2 k1·k2b†mm′(k1)bmm′(k2), ∆ˆ and∆ˆ ifV <0(e.g.,BCSattraction)andV >0
k1,kX2,m,m′ (e0.g0.0, screen1e1d0 Cou0lomb repulsion), respectively. 2
wherebmm′(k)=cm(k)cm′( k)istheelectronpairoper- A standardmean field calculation (see Supplementary
−
atorandthespin-indexmrunsin( 3/2, 1/2,1/2,3/2). Materials) shows that when V <0, T >T , where T
0 e o e/o
− −
All pairing channels decompose into squares of irre- are transition temperatures for even/odd parity super-
ducible representationsof SO(3), following a formal pro- conductivity,meaningthataslongasV isattractive,the
0
cedure detailed in the Supplementary Materials. Gener- conventionals-wavepairingisenergeticallyfavored. The
ally, a Cooper pair of zero momentum is determined by oddparitypairingisenergeticallyfavoredifV ,V >0,in
0 2
the total spin of the two constituent electrons, S, and otherwords,theinteractionisrepulsive. (Whilethe∆
000
the angular momentum describing the relative motion channelbecomesattractivefork >V /V ,thisisbecause
0 2
between them, L. In the presence of spin-orbital cou- we did not keep higher order terms in the expansion of
pling (SOC), the conserved quantities are therefore S2, V(q). Generally, repulsive screened Coulomb potential
| |
L2, J2 and J , where J is the total angular momentum disfavors pairs of zero angular momentum.)
z
of the pair, or, J = L+S. Therefore, the order param- When the (L,S,J) = (1,1,0) phase is favored by
eters are denoted by three quantum numbers (L,S,J), repulsive interaction, there are three possible scenar-
and the ground state is 2J +1-fold degenerate. In our ios (see Fig.2), in which the Fermi level crosses: (i)
system where electrons are spin-3/2, the total spin is the E -bands, (ii) the E -bands and (iii) all four
1/2 3/2
S = 0,1,2,3; the orbital angular momentum L = 0,1 Γ bands. (The Γ representation also exists in non-
8 8
because a pairing order parameter for L 2 is at least centrosymmetric lattices so bands are generally non-
≥
quadratic in k, coming from decoupling a quartic term degenerate.) In case-(i), there is a Dirac point at the
in the expansion of V(q). As a result, the possible pair- Γ¯ with linear dispersion at its vicinity; in case-(ii), the
ings are: (L,S,J) = (0,0,0), (1,1,0), (1,1,1), (0,2,2), Dirac point is replaced by a tri-Dirac point, having cu-
(1,1,2),(1,3,2),(1,3,3)and(1,3,4). (L,S,J)=(0,0,0) bic dispersion and divergentDOS; in case-(iii), there are
4
Γ L Γ L Γ L dI/dV
E3/2
E3/2
E
f
E3/2E1/2
E1/2
E1/2
−∆ 0 ∆V
(a) (b)
E
f
Γ¯ Γ¯ Γ¯
FIG. 3: (a) Schematic plot of the differential conductance
on the surface of TSCs with Dirac (blue) and tri-Dirac sur-
face states inside the bulk superconducting gap. (b) Typical
symmetry-forbiddenscatteringvectorsontheiso-energycon-
FIG. 2: Three possible scenarios of the how the Fermi level
tour of the tri-Dirac surface states. Black arrows indicate
crosses the bulk Γ8-bands and the schematic dispersions of scatteringvectorsforbidden byTRS,andredarrowsarevec-
the surface states. Solid and dotted lines mean that the cor-
tors strongly suppressed duetothe special pseudospin struc-
responding band of has positive and negative pairing ampli-
tureof tri-Dirac surface states, over-plotted on thecontour.
tude, respectively, in the nontrivial superconducting phase.
Each band here is non-degenerate as inversion symmetry is
absent in general. In the presence of inversion symmetry,
bandsbecome doublydegenerate; thedoubletFermisurfaces erties of a Heusler alloy strongly depends on its com-
haveopposite signsof pairing ifthepairing operator changes position, and include: the superconductivity in LaPtBi
sign underinversion[8, 40].
and YPtBi[25, 28], the ferromagnetism in GdPtBi[33]
and the heavy fermion behavior of YbPtBi[34]. Since
we adopt the most general form of short range electron
twodoubletsatΓ,havingangularmomentumj = 1/2 interaction in the long wavelength limit (small q), the
z
± | |
and j = 3/2, respectively. Here we note that in case- results suggest that superconducting Heusler alloys such
z
±
(iii), all four branches of dispersion are generically lin- as YPtBi may be candidate materials. In some Heusler
ear (see Supplementary Materials for details). It is in- alloys where magnetic interactions are significant, spin-
teresting to consider the Z-index of the class DIII su- spin interaction is not negligible, and the competition
perconductors for these cases and one obtains 1, 3 between magnetism and superconductivity must be con-
± ±
and 2. In the first two cases the Chern numbers of sidered.
±
the FS associated with the E1/2,3/2-bands are 1 and Experimentally, the new TSCs can be identified by
±
3. Therefore if the sign of pairing changes between its measuringitssurfaceDOSusingSTM.ThedI/dV curve
±
twopieces,fromtheQi-Hughes-Zhangformula[9]wehave diverges as dI/dV V 1/3 in the vicinity, as opposed
−
z = 1, 3, respectively. For case-(iii), there are a pair to dI/dV V in T∝SCs with simple Dirac surface states
of sig±n-c±hanging E1/2-FS and another of sign-changing [see Fig.3(∝a)]. Quasiparticle interference on the surface
E3/2-FS, so it seems we can have either 2 or 4. An can be used to verify the pseudospin structure[35, 36].
± ±
expansion of ∆ˆ in Eq.(7) shows a relative minus sign Theiso-energycontourisgenericallyahexagramevenat
110
between the pairing on E - and E -bands, leaving lowenergy,similartothe contourinthe surfacestatesof
1/2 3/2
2 the only possibility. Therefore, only in case-(ii) does Bi Te topological insulator away from the linear Dirac
2 3
±
the phase exhibit protected tri-Dirac surface states with regime. In the latter, the strongest peaks in QPI results
cubic dispersion. To realize this scenario, the conduc- from scattering vectors indicated by the red arrows in
tion and the valence bands must touch at Γ and bend Fig.3(b)[37–39]. Thesepeaksindicatestronginterference
oppositely, making the normal state a doped zero gap between states having opposite velocity and not related
semiconductor. by TRS. On a hexagram contour, the two momenta of
Heusler alloys offer a wide spectrum of zero gap semi- the interfering states make an angle of 60◦. However,
∼
conductors, as a result of the interplay between SOC for the tri-Dirac surface states, these scattering channels
and orbital hybridization[27]. When SOC is stronger are stronglysuppressed,because the pseudospins are ex-
than hybridization (which is almost always the case in actly opposite to each other between states related by
topological insulators as HgTe), at Γ we have the Γ - a 60-degree rotation, a signature of the new topological
8
representationhigherthantheΓ -representationandthe surface states.
6
Fermi level of an undoped compound is exactly at the AcknowledgementsThe authors thank E. Fradkin
branching point of E -bands (conduction) and E - for helpful discussions. CF is supported by ONR-
3/2 1/2
bands (valence). When electron-doped, these materials N0014-11-1-0728 for salary and DARPA-N66001-11-1-
in general have E crossing the E -bands. The prop- 4110 for travel. MJG acknowledges support from
f 3/2
5
the AFOSR under grant FA9550-10-1-0459, the ONR [24] L. Fidkowski, X. Chen, and A. Vishwanath,
under grant N0014-11-1-0728, a fellowship from the arXiv:1305.5851 (2013).
Center for Advanced Study (CAS) at the University [25] G. Goll, M. Marz, A. Hamann, T. Tomanic, K. Grube,
T. Yoshino, and T. Takabatake, Physica B: Condensed
of Illinois. BAB was supported by NSF CAREER
Matter 403, 1065 (2008), ISSN 0921-4526, ¡xocs:full-
DMR-095242, ONR-N00014-11-1-0635, Darpa-N66001-
name¿Proceedings of the International Conference on
11-1-4110,DavidandLucilePackardFoundation,MURI-
Strongly Correlated Electron Systems¡/xocs:full-name¿.
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[15] V. Shivamoggi and M. J. Gilbert, Phys. Rev. B 88, Rev. B 80, 245439 (2009).
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arXiv:1308.2424 (2013). state is chosen to be |ψ2(k)i ≡ Pˆ∗Tˆ|ψ(k)i, where Pˆ is
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Y.Ando,Phys. Rev.Lett. 109, 217004 (2012). |ψ1,2(−k)i≡Tˆ|ψ1,2(k)i.Thereforethegenericpairingis
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6
Decomposition of an electron pair into the basis of irreducible representations of SO(3)
A general pairing operator of zero total momentum is represented by
∆ˆmm′ = f(k)cm(k)cm′( k). (8)
−
k
X
In a system with symmetry, it is necessaryto decompose ∆ˆmm′ into irreducible representations the symmetry group.
For SO(3), the procedure goes as follows:
(i) The orbital part, f(k), can be expanded in terms of irreducible representations in the function space:
f(k)= f (k2)YM(kˆ), (9)
LM L
L,M
X
where YM(kˆ)’s are spherical harmonics and f (k2) is a complex function depending only on the radius of k.
L LM
(ii) The spin part bmm′(k)=cm(k)cm′( k) decomposes into irreducible representations in the spin space:
−
bmm′(k)= hs,s;S,Sz|s,s;m,m′iδˆsSszS(k), (10)
SX,Sz
where δˆSz is a pair operator annihilating a pair of electrons of total spin S and total spin along z-axis S , at
ssS z
k. δˆSz (k) is even/odd in k if and only if S =even/odd, due to Fermi statistics. (In our case, there is s = 3/2,
± ssS
m,m = 3/2, 1/2,1/2,3/2,S =0,1,2,3 and S = S, S+1,...,S.)
′ z
− − − −
(iii) Then we decompose the product of two irreducible representations in the orbital part and the spin part into
irreducible representations of fixed total angular momentum and total angular momentum along z-axis:
YM(kˆ)δˆSz (k)= L,S;J,J L,S;M,S δ(J M S )∆ˆ˜Jz (k2), (11)
L ssS h z| zi z − − z LSJ
Xkˆ J=|L−S|,...,LX+S,Jz=−J,...,+J
where,
∆ˆ˜Jz (k2) L,S;M ,S L,S;J,J YM′(kˆ)δˆSz′ (k) (12)
LSJ ≡ h ′ z′| zi L ssS
MX′,Sz′ Xkˆ
is the Cooper pair operator with total angular momentum J and total angular momentum along z-axis J . The
z
δ-function in Eq.(11) is written down explicitly only to show the conservation of total angular momentum - it is
already in the definition of the C-G coefficients.
Using Eq.(10,11,12), we obtain
YLM(kˆ)bmm′(k)= hL,S;J,Jz|L,S;M,Szihs,s;S,Sz|s,s;m,m′i∆ˆ˜JLzSJ(k2). (13)
Xkˆ S,JX,Jz,Sz
Since YM(k) isodd/eveninkwhenL=odd/even,andδˆSz (k) isodd/eveninkwhenS =odd/even,L andS must
L ssS
have the same parity, or ∆ˆ˜ vanishes.
LSJ
Decomposition of the interaction
Withthetechniquedevelopedintheprevioussection,wecannowdecomposethepairingchannelsintheinteraction
into pairings that are irreducible representations of SO(3). Here we only consider an isotropic interaction expanded
to the second order in the momentum space as shown in Eq.(6).
7
The first term in Eq.(5)
Following the procedure, we rewrite kˆbmm′(k) as
Y0(kˆ) 3/2,3/2;S,S P3/2,3/2;m,m δˆSz (k) (14)
0 h z| ′i 3232S
Xkˆ SX,Sz
= Y0(kˆ)[ 3/2,3/2;0,03/2,3/2;m,m δˆ0 (k)+ 3/2,3/2;2,S 3/2,3/2;m,m δˆSz (k)]
0 h | ′i 23230 h z| ′i 32322
Xkˆ Sz=−X2,...,+2
= 3/2,3/2;0,03/2,3/2;m,m ∆ˆ˜0 (k2)+ 3/2,3/2;2,S 3/2,3/2;m,m ∆ˆ˜Jz (k2),
h | ′i 000 h z| ′i 022
Jz=−X2,...,+2
where only J =even terms are included as J =odd terms vanish by Fermi statistics.
The first term in Eq.(5) can thus be put in the form
| (1− VV2k2)bmm′(k)|2 =V0 | (1− VV2k2)∆ˆ˜J0SzS(k2)|2. (15)
mX,m′ Xk 0 S=0,2;JXz=−S,...,S Xk 0
Then after defining
∆ˆ = (1 V2k2)∆ˆ˜0 (k2)/V , (16)
000 − V 000 0
0
k
X
∆ˆJz = (1 V2k2)∆ˆ˜Jz (k2)/V ,
022 − V 022 0
0
k
X
we have decomposed the first term in Eq.(5) into irreducible representations of SO(3):
| bmm′(k)|2 =(|∆ˆ000|2+ |∆ˆJ02z2|2)/V0. (17)
mXm′ Xk Jz=X−2,...,2
The second term in Eq.(5)
The second term in Eq.(5) can be rewritten as
2V
−2V2 k1·k2bmm′(k1)b†mm′(k2)=− 32 | kY1M(kˆ)bmm′(k)|2. (18)
k1,kX2,m,m′ MX,m,m′ Xk
From Eq.(13), we have
Y1M(k)bmm′(k) = h1,S;J,Jz|1,S;M,Szih3/2,3/2;S,Sz|3/2,3/2;m,m′i∆ˆ˜J1SzJ(k2). (19)
Xkˆ S,SzX,J,Jz,M
Using the normalization conditions
h3/2,3/2;S,Sz|3/2,3/2;m,m′ih3/2,3/2;m,m′|3/2,3/2;S′,Sz′i = δSS′δSzSz′, (20)
m,m′
X
h1,S;J,Jz|1,S;M,Szih1,S;M,Sz|1,S;J′,Jz′i = δJJ′δJzJz′,
MX,Sz
we obtain
| k1k2Y1M(kˆ)bmm′(k)|2 = k1k2∆ˆ˜J1SzJ(k1)∆ˆ˜J1Sz†J(k2) (21)
MX,m,m′ Xk SX,J,JzkX1,k2
= k∆ˆ˜Jz (k)2.
| 1SJ |
SX,J,Jz Xk
8
Defining
∆Jz = k∆ˆ˜Jz (k)/V , (22)
1SJ 1SJ 2
k
X
we have
2
−2V2 k1·k2bmm′(k1)b†mm′(k2)=−3V |∆J1SzJ|2. (23)
k1,kX2,m,m′ 2 S=1,3,J=|SX−1|,...,|S+1|,Jz
A four-band k·p-model with TRS and C∞
In this Appendix we discuss the surface Hamiltonian of case-(iii) discussed in the text (see the right plot of Fig.2),
where there arefour Majoranamodes at Γ¯, two havingJ = 1/2and two havingJ = 3/2. They are forbiddento
z z
hybridize and open a gap at Γ¯ due to the rotation symmetry±. We choose the basis such±that
1 0 0 0
2
0 1 0 0
jz = 0 −02 3 0 , (24)
2
0 0 0 3
 −2
T = Kσ σ .  (25)
y y
⊕
Note that this choice is valid since σ σ is antisymmetric so T2 = 1 as required for fermions. Symmetry requires
y y
⊕ −
that
eijzθHeff(k+,k )e−ijzθ = Heff(k+eiθ,k e−iθ), (26)
− −
TH (k)T 1 = H ( k).
eff − eff
−
To the first order in k, the Hamiltonian is in the following general form:
0 c k c k 0
1 2 +
−
c k 0 0 c k
Heff(k)=c∗1k+ 0 0 − 0∗2 −, (27)
∗2
−
 0 c k 0 0 
 − 2 + 
 
whose dispersion can be solved as
E = c 2+2c 2 c c 2+4c 2k/√2. (28)
1 2 1 1 2
± | | | | ±| | | | | |
q
p
All four bands are linear in k. Therefore, rotation symmetry, or any of its subgroups, does not lead to higher order
dispersion in a four-band k p theory with four-fold degenerate Majorana states at the origin in two dimensions. C
n
·
isasubgroupofthe full U(1)rotationsymmetrygroup,andthereforedoesnotprotecthigherorderbanddispersions.
Phase diagram of the superconducting phases
We use mean-field approximation to obtain the BdG Hamiltonian, and use that to calculate the free energy and
hence the phase diagram. The interaction in Eq.(6) is mean-field decoupled as
|∆ˆ000|2 2|∆ˆ110|2 = ∆ˆ†000h∆ˆ000i+h.c. 2(∆ˆ†110h∆ˆ110i+h.c.) (29)
V − 3V V − 3V
0 2 0 2
2 ∆ˆ 2 ∆ˆ 2
000 110
+ |h i| |h i|
3V − V
2 0
∆ˆ ∆ˆ 2 2∆ˆ ∆ˆ 2
000 000 110 110
+ | −h i| | −h i| .
V − 3V
0 2
9
The mean-field decoupled BdG Hamiltonian is obtained by ignoring the last term of fluctuation:
Hˆ = c (k)H (k)c(k) 2∆o∆ˆ†110 + ∆e∆ˆ†000 +h.c.
BdG † 0
− 3V V
2 0
k BZ
X∈
2∆ 2 ∆ 2
o e
+ | | + | | , (30)
3V V
2 0
−
where we have defined ∆ ∆ˆ and ∆ ∆ˆ . The dispersion of the BdG Hamiltonian
e 000 o 110
≡ h i ≡ h i
is very easy to solve by utilizing SO(3) symmetry. Due to the symmetry, the dispersion must be
isotropic, and we hence only need to solve it at k = (0,0,k). Writing Hˆ in Nambu basis ψ(k)
BdG
≡
(c3/2(k),c1/2(k),c−1/2(k),c−3/2(k),c†3/2(−k),c†1/2(−k),c†−1/2(−k),c†−3/2(−k))T, we have
ǫ (k) 0 0 0 0 0 0 ∆ok
3/2 − √5
0 ǫ (k) 0 0 0 0 ∆ok 0
 1/2 √45 
0 0 ǫ (k) 0 0 ∆ok 0 0
 1/2 √45 
 0 0 0 ǫ (k) ∆ok 0 0 0 
HBodG(k=(0,0,k)) = ψ†(k) 0 0 0 3/∆2ok −ǫ √5(k) 0 0 0 ψ(k)(,31)
 − √5 − 3/2 
 0 0 ∆ok 0 0 ǫ (k) 0 0 
 √45 − 1/2 
 0 ∆ok 0 0 0 0 ǫ (k) 0 
 √45 − 1/2 
−∆√o5k 0 0 0 0 0 0 −ǫ3/2(k)
 
He (k=(0,0,k))
BdG
ǫ (k) 0 0 0 0 0 0 ∆e(1−VV2k02)
3/2 2
 0 ǫ (k) 0 0 0 0 ∆e(1−VV20k2) 0 
 00 1/200 ǫ1/20(k) ǫ 0(k) ∆e(10−VV20k2) ∆e(1−02VV20k2) − 002 00 
= ψ†(k) 0 0 0 ∆e3(/12−VV20k2) − ǫ 2(k) 0 0 0 ψ(k).
 0 0 ∆e(1−VV20k2) − 02 − 3/02 ǫ (k) 0 0 
∆e(1−0VV20k2) −∆e(10−2VV20k2) 002 00 00 − 1/002 −ǫ1/02(k) ǫ 0 (k) 
 2 − 3/2 
 
The mean-field transition temperatures are determined by the following gap equations:
1 Λ k2dk ǫ (k) ∂2Ee(k)
= tanh| n | n , (32)
−V0 n=X21,23Z0 2π2 2Te ∂∆e2 |∆e=0
1 Λ k2dk ǫ (k) ∂2Eo(k)
= tanh| n | n ,
3V2/2 n=X12,23Z0 2π2 2To ∂∆o2 |∆o=0
where
V
Ee(k) = ǫ2(k)+∆2(1 2k2)2/4, (33)
n n e − V
r 0
∆2
Eo (k) = ǫ2 (k)+ ok2,
1/2 1/2 45
r
∆2
Eo (k) = ǫ2 (k)+ ok2
3/2 1/2 5
r
10
are the quasiparticle dispersions of Hˆ .
BdG
We now providea simple proofthataslong asV <0,the systemalwaysenergeticallyfavorsthe conventionaleven
0
parity pairing. The free energy of the two phases is given by
2T 2∆ 2
F (∆ ,T) = dk3lncosh[Eo(k)/2T]+ | o| , (34)
o o −(2π)3 n 3V
n=1X/2,3/2Z 2
2T ∆ 2
F (∆ ,T) = dk3lncosh[Ee(k)/2T]+ | e| .
e e −(2π)3 n V
n=1X/2,3/2Z − 0
Assume V <0 and define ∆ =∆ /√ V and ∆ =√2∆ /√3V to make the two terms 2∆2/(3V ) and ∆2/( V )
0 1 e − 0 2 o 2 o 2 e − 0
have the same functional dependence. Then we have
V
Ee(k,∆ ) = ǫ2(k)+∆2 V (1 2k2)/4 (35)
n 1 n 1| 0| − V
r 0
V k2 V2k4
= ǫ2(k)+∆2(V + 2 + 2 ),
s n 1 | 0| 2 4V0
| |
V ∆2
Ee (k,∆ ) = ǫ2 + 2 2k2,
1/2 2 1/2 30
r
3V ∆2
Ee (k,∆ ) = ǫ2 + 2 2k2.
3/2 2 3/2 10
r
Obviously we have
Ee(k,∆)>Eo(k,∆) (36)
n n
for n=1/2,3/2. This means that
F (∆,T)>F (∆,T) (37)
o e
for any ∆ and T. Therefore the transition corresponding to ∆ must happen at a higher temperature.
e