Effect of hybridization symmetry on topological phases of odd-parity multiband superconductors

2.2 MB·

by  T. O. Puel

Effect of hybridization symmetry on topological phases of odd-parity multiband
superconductors
T.O. Puel1,∗ P.D. Sacramento1,2, and M. A. Continentino1
1Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud,
150, Urca 22290-180, Rio de Janeiro, RJ, Brazil and
2CeFEMA, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
(Dated: August 16, 2016)
We study two-band one-dimensional superconducting chains of spinless fermions with inter and
intra-band pairing. These bands hybridize and, depending on the relative angular momentum of
their orbitals, the hybridization can be symmetric or anti-symmetric. The self-consistent competi-
tion between intra and inter-band superconductivity and how it is affected by the symmetry of the
hybridization is investigated. In the case of anti-symmetric hybridization the intra and inter-band
pairings do not coexist while in the symmetric case they do coexist and the interband pairing is
6 showntobedominant. Thetopologicalpropertiesofthemodelareobtainedthroughthetopological
1 invariant winding number and the presence of edge states. We find the existence of a topological
0 phaseduetotheinter-bandsuperconductivityandinducedbysymmetrichybridization. Inthiscase
2 we find a characteristic 4π-periodic Josephson current. In the case of anti-symmetric hybridization
we also find a 4π-periodic Josephson current in the gapless inter-band superconducting phase, re-
g
cently identified to be of Weyl-type.
u
A
4 I. INTRODUCTION magnetic field [19, 20]. On the other hand, triplet pair-
1 ing has been found to be physically realizable in some
Multiband models for the superconducting state and systems. In Ref. [6] it was shown that odd-parity super-
n] their topological properties have received increasing at- conductivity occurs in superconducting (SC) multilay-
o tention recently [1–6]. This consideration has been im- ers, wherethisstateisasymmetry-protectedtopological
c portant to explain many important effects in topological state. Inaddition,tripletpairingisfoundin3He[21]and
r- systems. Forinstance,topologicalsemimetals[7]andchi- in Sr2RuO4[22], as well as in some rare noncentrosym-
p ral superfluidity [8] have been predicted in multiorbital metric systems [23]. Triplet pairing was also studied in
u modelswhereorbitalswithdifferentsymmetriesinteract. the context of extended Hubbard chain [24].
s
Two component fermionic systems with occupied s and Motivatedbytherecentlydiscussedtopologicalcharac-
.
at p orbitalstates wereshowntohavearichphasediagram tersofmultibandmodels[4,6],andbasedonthesimplest
m in both one and two dimensions [4]. A general connec- modelthatdescribesthetopologicalpropertiesofachain
tion between multiband and multicomponent supercon- ofspinlessfermions,westudytheKitaevmodelwithtwo
-
d ductivity has also been made [9]. Topological properties orbital-bands. We include and discuss inter- and intra-
n in three-band models were also studied [10–13]. band superconducting couplings. A characteristic fea-
o
It is well known that the Kitaev model [14–16] – anti- ture of multiband systems is the hybridization between
c
symmetric pairs of spinless fermions in 1D – is the sim- the different orbitals. This arises from the superposi-
[
plestmodelthatexhibitsatopologicalphasewithMajo- tion of the wave functions of these orbitals in different

2 rana modes in the ends of a p-wave chain, depending on sites. It can have distinct symmetry properties depend-
v
thestateofthesystem. Thetopologicalnon-trivialphase ing on the orbitals involved. If this mixing involves or-
9
presents Majorana fermions at its ends. Otherwise, the bitals with angular momenta that differ by an odd num-
0
chain is in a superconducting phase with trivial topolog- ber,hybridizationturnsouttobeanti-symmetric,i.e.,in
6
7 ical properties and has no end states[16]. An extension real space we have Vij =−Vji or in momentum, k-space,
0 of this effective spinless fermions model for a multiband V(−k) = −V(k). Otherwise hybridization is symmetric
. hybridized system comprised of the Su, Schrieffer and respecting inversion symmetry in different sites [25].
1
0 Heeger (SSH) model[17] and the Kitaev model was done The bulk-edge correspondance guarantees that in the
6 in Ref. [18], where topological properties are discussed topological phases there are subgap edge states. In the
1 showing edge states that are of Majorana and fermionic
case of a topological superconductor, zero energy Ma-
: types.
v jorana modes are predicted to appear and great effort
i Triplet superconductivity is rare in nature. Thus, the has been devoted to prove their existence. Methods that
X
pursuitofalternativestocreatetripletsuperconductivity
provide signatures of their presence have been proposed
r lead to engineering a topological insulating chain (made
and experimentally tested via for instance tunneling ex-
a
with strong spin-orbit material) in proximity of a nor-
periments[26,27],interferometry[16],pointcontactsus-
mal superconductor and in the presence of an applied
ing the Andreev reflection [28] through the detection of
zero-bias peaks [29], using the quantum waveguide the-
ory[30]whichgivesthecorrectbulk-edgecorrespondence
∗ [email protected] [31] and fractional Josephson currents [14, 16, 32]. Also
2
signatures of the Majorana states may be found in bulk is
measurements such as the imaginary part of frequency
dependent Hall conductance [33] and the d.c. Hall con- H0 =(cid:88)(cid:110)(cid:0)εAk −µ(cid:1)a†kak+(cid:0)εBk −µ(cid:1)b†kbk(cid:111), (1)
ductivity itself [34]. k
The existence of topological phases is detected in this (cid:16) (cid:17)
work numerically calculating the winding number and wherea†k b†k isthecreationoperatorofspinlessfermion
by showing the existence of edge states at the ends of at A(B)-band with momentum k. Also, µ is the chemi-
the chain. In addition, we calculate the Josephson cur- cal potential and we choose εA = −εB = 2tcos(k) ≡ ε
k k k
rentaccrossthejunctionbetweentwosuperconductorsto where t is the hopping amplitude. The hybridization
identify regimes where the periodicity of the Josephson term is
current on the phase differences between the supercon- (cid:88)(cid:110) (cid:111)
ductors (original proposal by Kitaev[14]) or the equiva- Hh = V (k)a†kbk−V (−k)b−ka†−k+h.c. , (2)
lent situation of a superconducting ring threaded by a k
magnetic flux and interrupted by an insulator changes
where V (k) = 2iV sin(k) ≡ V if the hybridization
as as,k
from the usual value of 2π to a 4π value [35]. As shown
is anti-symmetric or V (k) = 2V cos(k) ≡ V if the
s s,k
before [14, 35–45] the existence of the Majoranas at the
hybridization is symmetric, and V is the hybridization
edges allows tunneling of a single fermion at zero-bias
amplitude. Finally, the mean-field superconducting con-
leadingtoa4π−periodiccurrentincontrasttotheusual
tribution to the Hamiltonian is
Cooper pair transport accross the junction which leads
(cid:88)(cid:110)
to the usual 2π−periodic current. Experimental realiza- H = ∆ a†b† +∆ b†a†
SC k k −k k k −k
tion to detect 4π-periodic Josephson junction has been
k
presented in Ref. [46] and an application to multiband (cid:111)
+∆ a†a† +∆ b†b† +h.c. , (3)
systems has recently been presented in Ref. [47]. A,k k −k B,k k −k
This paper is organized as follows. In section II we
with ∆ = i∆sin(k) where ∆ is the superconduct-
k
define the general Hamiltonian including symmetric and
ing inter-band pairing amplitude, and ∆ =
(A,B),k
anti-symmetric hybridization. Also we proceed with the
i∆ sin(k) where ∆ and ∆ are the supercon-
(A,B) A B
self-consistent calculations of the superconducting order
ducting intra-band pairing amplitudes. We could
parametersrelatedtothecompetitionbetweentheintra-
also include a superconducting term that changes
and inter-band pairings. The topological properties of
Cooper pairs between different orbitals, which in
the model are discussed in section III. We show a gen-
terms of two particles interaction may be writ-
eralcalculationofthewindingnumberwhenparticle-hole (cid:16) (cid:17)
ten as (cid:80) g (k,k(cid:48)) b†b† a a +a†a† b b ,
symmetry is present in a 4 × 4 Bogoliubov-de Gennes k,k(cid:48) J k −k −k(cid:48) k(cid:48) k −k −k(cid:48) k(cid:48)
(BdG) Hamiltonian. Also, we calculate the energy spec- where gJ is the interaction strength. Without fluctua-
trum of a finite one-dimensional chain. The differences tion, i.e., in the BCS theory, this term appears as an
betweentrivialandtopologicalphasesarediscussedfrom additive parameter to ∆A and ∆B, thus besides enhanc-
the perspective of zero-energy states. We also make the ing the intra-band superconductivity it does not change
equivalence of the topological regimes with the 4π peri- qualitatively the topological properties of the Hamilto-
odicity of the Josephson current. Finally, in section IV nian considered here.
we present the conclusions and review the main results. In the more compact BdG form, the Hamiltonian may
be written in the Nambu representation [48] as H =
(cid:80) C†H C , where C† =(cid:0)a†b†a b (cid:1) and
k k k k k k k −k −k
H =−µΓ −ε Γ +∆ Γ
II. MODEL AND SELF-CONSISTENT k z0 k zz k yx
CALCULATIONS 1 1
+∆ (Γ +Γ )+∆ (Γ −Γ )
A,k2 y0 yz B,k2 y0 yz
We consider a two-band superconductor with hy- +Vk·I, (4)
bridization and triplet pairing in 1D, i.e., a chain of sites
where Γ = τ s , ∀ i,j = 0,x,y,z; τ and s are
supporting two orbitals, let’s say orbitals A and B. The the Pauliijmatriice(cid:15)s ajcting on particle-hole and sub-band
pairing between fermions may exist on different bands
spaces, respectively, and s = τ are the 2×2 identity
(inter-band)orineachband(intra-band)andarealways 0 0
matrices. With respect to the Hamiltonian parameters:
ofp-wavetype,inthesensethatpairsofspinlessfermions
V =−V Γ if the hybridization is anti-symmetric or
arespatiallyanti-symmetric. Theproblemcanbeviewed k as,k zy
V =V Γ if the hybridization is symmetric.
as a generalization of the Kitaev model to two orbitals. k s,k zx
In this section we present self-consistent results for
Wealsohavethehybridizationtermbetweentheorbitals
the superconducting parameters ∆, ∆ and ∆ us-
A and B that may be symmetric or anti-symmetric. The A B
ing the BdG formalism. The Hamiltonian defined
simplestHamiltonianinmomentumspacethatdescribes
in Eq. (4) can be solved using BdG transforma-
those types of superconductivity and hybridization may (cid:104) (cid:16) (cid:17)∗ (cid:105)
bewrittenasH=H0+Hh+HSC wherethekineticpart tions as ak = (cid:80)n uan,kγn,k+ vna,k γn†,−k and bk =
3
Figure 1. The first row shows the self-consistent solutions of the superconducting parameters, considering anti-symmetric
hybridization. Theorderparameterscalculatedaretheinter-band(∆)andtheintra-band(∆ )ones. Forinstance,according
A,B
totheanti-symmetrichybridization,AandBcouldbetheorbitalssandp. Fortheseresultswesetg/2=g =g =1.7. Second
A B
row shows the corresponding energy spectrum gap and the phase diagram. Phase I is a gapless inter-band superconducting
phase. II is a gapped intra-band superconducting phase. III is a topological insulating phase. IVa shows a trivial gapped
inter-bandSC.IVbisatrivialinsulatingphase. Finally,Visametallicphase. Thephasediagramissymmetricaroundµ=0.
(cid:104) (cid:16) (cid:17)∗ (cid:105)
(cid:80) ub γ + vb γ† . This transformation di- Bogoliubov coefficients, we may write
n n,k n,k n,k n,−k
awengiteohrngaψylinzeei=gset(cid:0)nhuveana,lHkueasumbnai,nlktdovntnaiha,e−nkwinavvnbte,h−feukn(cid:1)foTncrt,miownhHeskprψeinnEorn=saψErenψtahrnee, ∆=gL1 (cid:88)k (cid:88)n isin(k)(cid:104)uan,k(cid:0)vnb,−k(cid:1)∗+ubn,k(cid:0)vna,−k(cid:1)∗(cid:105),
n
(8)
the eigenstates.
∆ =g 2 (cid:88)(cid:88)isin(k)ua (cid:0)va (cid:1)∗, (9)
The self-consistent solution implies that the pairings A AL n,k n,−k
can be obtained using k n
∆ =g 2 (cid:88)(cid:88)isin(k)ub (cid:0)vb (cid:1)∗. (10)
B BL n,k n,−k
k n
1 (cid:88)
∆=g isin(k)((cid:104)a b (cid:105)+(cid:104)b a (cid:105)), (5)
L k −k k −k
A. Anti-symmetric hybridization
k
2 (cid:88)
∆ =g isin(k)(cid:104)a a (cid:105), (6)
A AL k −k Wefirstconsiderthecaseofanti-symmetrichybridiza-
k tion(V )thatoccurswhentheorbitalsangularmomenta
2 (cid:88) as
∆ =g isin(k)(cid:104)b b (cid:105), (7) have different parities, like orbitals s and p. In Fig 1 we
B BL k −k
showtheresultsforthethreeorderparameterscalculated
k
self-consistently, when g/2 = g = g = 1.7. A similar
A B
model was considered before [49] with only inter-band
pairing. The strength of the coupling g only changes the
where g, g and g are the strength of the interactions superconducting amplitude of the SC phases (inter- or
A B
between fermions in different orbitals, in orbitals A and intra-band ones), thus its choice does not change quali-
in orbitals B, respectively. At zero temperature, using tatively the results presented. It is interesting to point
the representation of fermionic operators in terms of the out that the self-consistent results for the superconduct-
4
Figure2. Firstrowshowstheself-consistentsolutionsofthesuperconductingparameters,consideringsymmetrichybridization.
The order parameters calculated are the inter-band (∆) and the intra-band (∆ ). For instance, according to the symmetric
A,B
hybridization, A and B could be the orbitals s and d. For these results we set g/2 = g = g = 1.7. Second row shows
A B
the corresponding spectral gap and the phase diagram. Phase I carries both types of pairings and has non-trivial topological
properties. Phase IIa is a gapped superconducting phase also with both inter- and intra-band pairings, but trivial topological
properties. Phase IIb is a normal insulator. The phase diagram is symmetric around µ=0.
ing order parameters may converge to different results a gap closing, while the dashed lines represent a phase
depending on the initial guesses. This is a consequence separation without closing the gap. Phase I in this fig-
ofthefirstordernatureofthequantumphasetransitions ure is a gapless superconducting phase, driven by the
between the different ground states. Therefore it is nec- inter-bandcoupling,anditwasshown[49]tobehavelike
essary to calculate the energy of the different states to Weyl superconductor. The phase II is a two-band super-
obtain the true ground state for a given set of parame- conductor with only intra-band couplings. Phase III is
ters. a topological insulator which was shown to have local-
ized states at the edges [49] of a finite chain. The phase
We note first that inter and intra-band superconduc-
IVa shows gapped superconductivity and represents the
tivity do not coexist as equilibrium states. Their coex-
strong inter-band coupling superconducting phase. The
istence implies that one of them is metastable. Second,
phase IVb is a trivial insulator and there is no SC re-
we note that the intra-band SC does not distinguish be-
maining. Finally, phase V is a normal metallic phase.
tween different bands, in the sense that the results are
All those phases are symmetric around µ=0. Since the
equal for both pairings. We note that considering any
intra- and inter-band pairings do not coexist, the phases
fixed value of the chemical potential in the region where
with no intra-band pairing are similar to the results pre-
thereisSC,whentheanti-symmetrichybridizationisin-
viously obtained [49]. The main difference results from
creased it eventually destroys the inter-band SC that is
theappearanceoftheintra-bandpairinginsomeregions
present. Ontheotherhand,ifwekeepincreasingthehy-
of the phase diagram.
bridization,itraisestheintra-bandSCuptoamaximum
value until it suppresses the SC definitely.
In Fig. 1d we show the spectral gap for the self-
consistent results. Also we show the phase diagram in B. Symmetric hybridization
the right plot of the same figure. As we can see, the
consideration of inter-band, intra-band superconductiv- Analogouslytothepreviouscase,wealsocalculatethe
ity and anti-symmetric hybridization results in a rich orderparametersself-consistentlyconsideringsymmetric
phase diagram. In this figure, the solid lines represent hybridization(V ). Thisisthecasewhentheorbitalsan-
s
5
gular momenta have equal parities, like orbitals s and d. (Z) number [50].
In Fig. 2 we show the results for the same set of values In the Z class of topological systems, the topological
g,g andg astheanti-symmetriccase. First,wenotice phases in odd-dimensional systems (or, in other words,
A B
that the intra-band SC distinguishes between different those with chiral symmetry) are characterized by the
bands, sincethereisachangeofsignbetweenthem. Un- topological invariant called winding number [50, 51].
like the anti-symmetric case, here there is a coexistence This invariant counts the number of the zero-energy
of inter- and intra-band SC. Remarkably, the inter-band statesprotectedbythetopologicalpropertyoftheHamil-
has the larger order parameter for all region of parame- tonian, and may be calculated in the usual way [51, 52].
ters. Ingeneral,thisindicatesthattheinter-bandSChas One needs to look for an hermitian matrix which anti-
highercriticaltemperature,whichturnsouttoberespon- commutes with the Hamiltonian (H), i.e., find Γ such
sibleforthesuperconductivityappearinginthematerial. that {H,Γ}=0. Considering spinless time-reversal sym-
Note that symmetric hybridization is responsible for the metryandparticle-holesymmetry(PHS)thentheChiral
emergence of intra-band SC. Very strong symmetric hy- operator that carries both symmetries is Γ . It implies
x0
bridization eventually destroys superconductivity. that the Hamiltonian anti-commutes with that operator,
In Fig. 2 we also show the spectral gap for the self- which can be used to bring the Hamiltonian to an off-
consistent results. We also show the phase diagram in diagonal form. Using the basis that diagonalizes Γ ,
x0
the right plot of Fig. 2, as in Fig. 1. As before, the solid i.e., R−1 Γ R = D, with R = Γ − Γ and D a
x0 xx zx
lines represent a gap closing, while the dashed lines rep- diagonal matrix, implies that
resentaphaseseparationwithoutclosingthegap. Phase
(cid:18) (cid:19)
I and IIa are gapped superconducting phases, with the 0 q(k)
R−1 H R= . (11)
coexistence of inter- and intra-band couplings, but dom- k q†(k) 0
inated by the inter-band one. Phase IIb is an insulating
phase and there is no SC. All those phases are symmet- Writing a generic Hamiltonian in the form
ric around µ = 0. The more interesting phase is phase
(cid:88)
I, which allows both types of couplings and shows non- Hk = hijΓij, i,j =0,x,y,z, (12)
trivialtopologicalproperties. Thisphaseischaracterized i,j
bylocalizededgestatesandfinitewindingnumber,aswill
whosecoefficientsh maybeextractedfromanygeneric
be shown in the next section. ij
Hamiltonian H through h = 1Tr(Γ H), if we apply
The robustness of the inter-band superconductivity ij 4 ij
the PHS to Eq. (12) as H =−Γ HT Γ and proceed
can be tested varying the relative amplitudes of the g, k x0 −k x0
with the block off-diagonal calculations described above
g and g parameters. Considering, for instance, the
A B we find that
case g = g ≡ g and selecting the point µ = 0 and
A B 0
V = 1, the appearance of the inter-band SC is not con- (cid:88)
s q(k)= c (h +ih )σ , j =0,x,y,z, (13)
j zj yj j
tinuous with increasing g, but goes through a first order
j
transition at some point g >g near to g =g to a value
0 0
that always has a larger amplitude than the intra-band where c = c = +1 and c = c = −1, σ are the
0 x y z x,y,z
ones. While the results of Figs. 2 consider a large g Pauli matrices and σ is the 2×2 identity matrix.
0
value, the results are qualitatively the same, as long as The winding number, W, is defined as the number of
the inter-band SC is present. revolutions of det[q(k)] = m (k)+im (k) around the
1 2
origin in the complex plane when k changes from −π to
π,
III. TOPOLOGICAL PROPERTIES ˆ
1 π ∂θ(k)
W = dk, (14)
2π ∂k
A. Winding number in the BDI class −π
with
The symmetry-protected topological systems are clas-
sified accordingly to their symmetries [50]. The Hamil- m (k)
θ(k)=argdet[q(k)]=tan−1 2 . (15)
tonian of equation (4) has particle-hole symmetry once m (k)
1
it obeys the relation H = −OH O−1 [50], where
k k
the operator written in the Nambu representation [48] For the generic case considered above we have that
isO =Γ K, inwhichO2 =+1andK appliesthecom-
plex conjxu0gate and inverts the momentum. In addition, m1(k) = (cid:88)dj(cid:0)h2zj −h2yj(cid:1)
the Hamiltonian has simplified time reversal symmetry j
for spinless fermions, H = H∗ . In the presence of and
k −k
both symmetries, the Hamiltonian belongs to the BDI (cid:88)
m (k) = d (2h h ), (16)
2 j zj yj
class of topological systems, and the one-dimensionality
j
guaranteesthatthespaceofthequantumgroundstateis
partitioned into topological sectors labeled by an integer where d =+1 and d =−1.
0 x,y,z
6
B. Edge states in a finite chain
Inordertofindtheenergyspectrumofafinitechainof
fermions through the BdG transformation we write the
Hamiltonian, Eq. (4) transformed to real space, in the
form
H=C†HC, (17)
where
C =(cid:0)a b a† b† ··· a b a† b† (cid:1)T (18)
1 1 1 1 N N N N
andtheoperatorsa†(a )andb†(b )create(annihilate)a
i i i i
fermion in the orbital A and B, respectively, at position
i in the chain. The matrix H is defined as
 
H ··· H
11 1N
H = ... ... ... , (19) Figure 3. Schematic figure illustrating the 1D superconduct-
H ··· H ing ring with a Josephson junction.
N1 NN
andiscomprisedbythefollowing(4×4)interactionma-
following boundary conditions
trices
 −e−iφ/2t(cid:48) 0 0 0 
H =−µΓ ,
Hrr,,rr+1 =−tΓzzz0−i∆2Γyx−i∆20Γy0+V (r+1), HN,1 =H1∗,N = 00 e−iφ0/2t(cid:48) eiφ0/2t(cid:48) 00 ,
HHr,r−1 ==−0tΓzz∀+r(cid:48)i∆(cid:54)=2Γry,xr++i1∆20oΓryr0−+1V, (r−1), 0 0 0 −eiφ/2t(cid:48)
r,r(cid:48) (22)
(20) where the superconducting phase difference φ across the
where V (r+1) = −V (r−1) = −iV2Γzy for anti- junction is related to the magnetic flux through the ring
symmetric hybridization, and V (r+1) = V (r−1) = by φ = 2πΦ/Φ , and Φ = h/2e is the superconduct-
0 0
V2Γzx for symmetric one. ing flux quantum. We have that t(cid:48) is the tunneling, or
IfweconsidertheBdGtransformationasthefollowing inversely proportional to a barrier amplitude, across the
junction. As mentioned above this is equivalent to the
a =(cid:80) (cid:2)u (r)γ +v∗ (r)γ†(cid:3), original proposal of the Josephson junction between two
br=(cid:80)n(cid:2)us,n(r)γn+v∗s,n(r)γn†(cid:3), (21) different superconductors with different pairing phases
r n p,n n p,n n also separated by some tunneling amplitude accross an
insulator (or metal).
it diagonalizes the Hamiltonian, H = E0+(cid:80)nEnγn†γn, We may now analyze the junction effect on a current
such that U†HU = E, where U is formed by all the flowingintheringaswechangethemagneticfluxbydis-
BdG coefficients us, vs, up and vp, and has the property creteamountsoffluxquantum,bychangingthejunction
to be unitary U†U = I. The matrix E is diagonal and phase φ by multiples of 2π. In a normal superconductor
contains the energy spectrum (En) of the system. each additional flux quantum (∆φ=2π, usually called a
pump)shouldleadthesystemtoitsinitialstate[54]. On
the other hand, the topological superconductor (TSC)
C. 4π Josephson effect changesitsparityateverypump[55],leadingthesystem
to a different final state after pumping. The reason is
In the previous section we have considered a 1D open that the TSC is allowed to have zero energy crossings in
chain, i.e., there is no connection between sites 1 and N. itsspectrumofexcitationduringthepumpandtherefore
In terms of eq. (20) we have H =H =0. Now we only returns to its initial state after a further change of
N,1 1,N
maythinkofachainasaringwithaJosephsonjunction the phase by 2π.
coupling the ends, see Fig. 3. An extra hopping term
t(cid:48) couples the end point of the ring to the first point via
someinsulatingjunction. Ifauniformmagneticfield(Φ) D. Symmetric hybridization
flows through this ring, its effect may be captured by a
Peierls substitution in the extra hopping term, t(cid:48) [53]. a. Winding number: we begin our analysis of the
Thus, the Josephson junction may be represented by the topological phases of the proposed model with the wind-
7
potential. To be sure that the phase is topological we
must calculate the winding number itself, or see if the
parametric plot of m¯ (k) and m¯ (k) contains the origin
1 2
when k ∈ [−π,π]. The results for the winding number
and the parametric plot are shown in Fig. 4 for the pa-
rameters V = 1.2, µ = −1.04. This figure shows that
s
theparametricplotwrapstheorigintwice;itmeansthat
the winding number in this case is two, W =2. The re-
sultsforthewindingnumberclearlyshowthetopological
phase, induced by symmetric hybridization, and dom-
inated by inter-band superconductivity for small values
ofthechemicalpotentialthatgrowsasthehybridization,
V , grows.
s
b. Edge states – Since we have defined the topolog-
ical region of the parameters, we may analyse the zero-
energy modes explicitly through the energy spectrum of
a finite chain. We have calculated the energy spectrum
Figure 4. In the left panel we show the winding number
calculated from the self-consistent results for symmetric hy- forachainofL=100sites,therefore,weget4Lenergies
bridization, over the phase space of parameters. In the right for the spectrum. We have checked that this size is large
panels we show the normalized parametric plot of real and enough to prevent finite size effects. We analyze the en-
imaginarypartsofdet[q(k)]. Thenumberoftimesdet[q(k)] ergy spectrum for two fixed values of chemical potential,
wraps the origin is the winding number and is illustrated in µ = 0 and µ = −1.4, and increasing the hybridization
the right side. according to the self-consistent solution of Fig. 2. The
results are shown in Fig. 5. What we immediately see
is that the zero-energy states are robust, i.e., even when
µ is non-zero they are present, which characterizes the
zero-energy modes in the superconducting phase. We
notice that those states are four-fold degenerated. We
have checked that they have wavefunctions that are lo-
calized exponentially close to the edges if the system is
large enough.
c. 4π Josephson effect: we may also analyse the
topological properties of the system via Josephson junc-
tion scheme, see Fig. 3. First, we look to the excitation
spectrum (bogoliubons) during two pumps for each su-
perconducting phase in the phase diagram. The results
are shown in the first row of Fig. 6, where 6a is for the
trivial phase IIa, whereas 6b and 6c are for the topolog-
ical phase I for two values of the chemical potential. We
may see that there are level crossings when the SC is in
itstopologicalphaseandthereisnocrossinginthetrivial
one.
To explicitly see the periodicity of the Josephson cur-
rent during the pump, we need to analyse the ground
state energy (E ) of the superconductor preserving its
0
Figure 5. Here we show the energy spectrum of the self- parity, i.e., the ground state is composed by the solid
consistent results, for two fixed values of the chemical po-
(red) lines of the excitation spectrum. Dashed (blue)
tential and increasing symmetric hybridization (µ=0 on (a)
lines carry the opposite parity. Thus, the sum over the
and µ=−1.4 on (b)).
"negative" excitation to compute E needs to follow the
0
excitation when it crosses the zero energy state. In the
topological phase, the crossing through zero energy is
ing number calculation. For convenience, we’ll consider a direct consequence of the presence of the zero energy
the case where ∆B = −∆A = ∆0. If we compare mode at the end of the chain. Here we have two zero
Eq. (4) – with symmetric hybridization Vs,k – and Eqs. energy excitations at each end, thus it is natural that
(16) we have m1(k) = µ2 +∆2k +∆20,k −Vs2,k −(cid:15)2k and we have two level crossings (we notice that region I with
m (k) = −2(V ∆ −(cid:15) ∆ ). This suggests that the µ = 0 in Fig. 6 has a degenerate level crossing). When
2 s,k k k 0,k
symmetrichybridizationmayinduceatopologicalphase, µ(cid:54)=0 the level crossing modes do not need to be degen-
since we have non-vanishing m even to zero chemical erated,butwenoticethateventhoughwehavetwolevel
2
8
Figure 6. Results for the case of symmetric hybridization case as we vary the tunneling phase φ: i) First row shows the
excitation spectrum that preserves the parity of the superconductor. ii) Second row shows the Josephson current through the
Josephson junction. Here we have used L=150 and t(cid:48) =0.1.
crossings (and their particle-hole symemtric), the cross-
ings through zero always happen at the same φ point.
Second row of Fig. 6 shows the current flowing through
the junction, which is the derivative of the ground state
energy respective to the flux φ. We clearly see that the
current has a periodicity of 2π (one pump) in the trivial
phase, Fig. 6d. On the other hand, the periodicity of
the Josephson currents in Figs. 6e and 6f are 4π (two
pumps), characterizing the topological superconducting
phaseandprovidinganalternativeevidenceforthepres-
ence of Majorana states.
E. Anti-symmetric hybridization
d. Winding number: we proceed the analysis of the
topological properties with the winding number calcula-
tion. For convenience, and since the self-consistent re-
sults do not distinguish the SC in the bands, we’ll con-
sider the case where ∆ = ∆ = ∆ . Therefore, com-
A B 0
paring Eq. (4) – with anti-symmetric hybridization, Vas Figure 7. Here we show the energy spectrum of the self-
– and Eqs. (16) we have that m1(k)=µ2+∆2k−∆20,k− consistent results, for two fixed values of the chemical po-
V2 −(cid:15)2 and m (k) = −2µ∆ . As a result we notice tential and increasing anti-symmetric hybridization (µ = 0
as,k k 2 0,k
that only for a non-zero chemical potential and intra- on (a) and µ=−1.4 on (b)).
band superconductivity we have non-vanishing m and
2
the system may include a topological phase. Calculating
the winding number, as described in Eq. (14), one ob- property of phase III is hidden by particle-hole symme-
tains a trivial solution (W =0) for all self-consistent so- try. Moreover, Ref. [49] shows that in this phase local-
lutionsinparameterspace[49]. Eventhoughthewinding ized states are present in the edges of the chain (despite
numberseemstoindicateatrivialsolution,theresultsof having finite energy when µ (cid:54)= 0). As concerns phase I,
Ref. [49] for a system with no intra-band pairing show it is a topological phase that presents Weyl fermions [49]
that the phases corresponding to regions I and III of the whose topological character remains also undetected by
phase diagram in Fig. 1 are topological. The topological the winding number calculation. In that reference it is
9
Figure8. Resultsforanti-symmetrichybridizationaswevarythetunnelingphaseφ: i)Firstrowshowstheexcitationspectra
that preserve the parity of the superconductor. ii) Second row shows the Josephson current flowing through the Josephson
junction. Here we have used L=250 and t(cid:48) =0.1.
shown an alternative procedure to uncover the topologi- there are no level crossings in the excitation spectrum
cal nature of this phase. and the current is 2π periodic as we can see in Fig. 8a
for the case of region IVa.
e. Edge states – now we proceed with the analysis
In phase I, even though we have no gap in the bulk
of the zero-energy modes explicitly through the energy
spectrum of an infinite system, it is still possible to cal-
spectrum of a finite chain. We also have used L = 100
culatetheJosephsoncurrentinafiniteone. Thejunction
sites, which is large enough to prevent finite size effects.
itself opens up a small gap in the spectrum if L is not
We analyze the energy spectrum for two fixed values of
too large and t(cid:48) is not too strong. Of course, in the limit
chemical potential, µ = 0 and µ = −1.4, and increasing
L→∞thegapcloses, butifthetunnelingt(cid:48) istoolarge
thehybridizationaccordingtotheself-consistentsolution
(or the barrier too small) the junction just couples both
of Fig. 1. The results for anti-symmetric hybridization
ends analogously to a periodic boundary condition (i.e.,
are shown in Fig. 7. We immediately see that the zero-
infinite system). Thus, a typical excitation spectrum for
energy states for µ=0 are not robust, in the sense that
very small energies in the gap generated by the coupling
theydisappearwhenµ(cid:54)=0. Thisisthedifferenceofzero-
accross the junction (positive and negative excitation) is
energymodesinthesuperconductor(phaseIforsymmet-
shown in Figs. 8b and 8c.
richybridization)andzero-energymodesintheinsulator
(phase III for the anti-symmetric hybridization). The Even though Figs. 8b and 8c show no level crossings
chemical potential is not breaking any symmetry, but during the pumps, we may proceed with the same calcu-
the zero-energy modes in the superconductor are topo- lations as before and obtain the Josephson current. The
logically protected and survive after the introduction of result is shown in Figs. 8e and 8f for two values of the
afiniteµ,whileintheinsulatorthosezero-energymodes chemical potential. Clearly, both figures exhibit 4π pe-
are not protected and can be eliminated as you see in riodic Josephson current, even without zero energy level
this figure. crossings revealing in some sense the hidden topological
nature of this Weyl-phase.
f. 4π Josephson effect: we may also analyse the
topological properties of the system via Josephson junc-
tion scheme (Fig. 3). We start looking to the ex-
citation spectrum (bogoliubons) during two pumps for IV. CONCLUSIONS
each superconducting phase in the phase diagram. The
anti-symmetric case has three types of superconducting Inthispaperwehavestudiedamodelofap-wave,one
phases: intraband gapped SC, interband gapped SC and dimensional, multiband superconductor. This represents
interband gapless SC, as shown in Fig. 1e. Both gapped a generalization of the single band model for odd-parity
superconductingphases(IIandIVa)showsimilarexcita- superconductivity that gives rise to a much richer phase
tionspectraandtheirtypicalbogoliubonsthatkeepsthe diagram with a variety of quantum phase transitions.
ground state parity are shown in Fig. 8a. As expected, Theodd-paritysuperconductivityispreservedinthisex-
10
tension, but inter-band superconductivity is now present thenatureofthetopologicalphasesandtheirendstates,
in addition to the intra-band ones. The presence of two- we have analyzed the energy spectrum of a finite sys-
bands in our model allows us to include hybridization, tem. We have compared the energy spectrum between
increasing the space of parameters. We have considered the anti-symmetric and symmetric results, or the trivial
symmetric and anti-symmetric hybridizations. Both are andtopologicalresults,respectively. Wealsocheckedthe
permitted, depending on the parities we choose for the localization of the zero-energy states.
angular momenta of the two orbitals. In order to provide further evidence for the presence
Wehavecalculatedtheself-consistentsolutionsforthe ofedgeMajoranastateswehaveshownthatinthetopo-
inter- and intra-band superconducting order parameters logicalphasesonefindsa4π-periodic(fractional)Joseph-
as functions of the chemical potential and the strength son current as one changes the magnetic flux accross a
of the symmetric or anti-symmetric hybridization. The ring composed of the superconductor with an insulator
self-consistent calculation of the order parameters allow inserted between its ends. The result is consistent with
to obtain the T =0 phase diagram of the system. When the results for the winding number and edge states for
increasing anti-symmetric hybridization, both intra- and thetopologicalphaseinthecaseofsymmetrichybridiza-
inter-band superconductivity emerge in the phase dia- tion. In addition, we also found the same 4π-periodic
gram, buttheycompeteandexcludeoneanotherfordif- Josephsoncurrentinthehiddentopologicalphaseidenti-
ferent values of band-filling. On the other hand, when fiedpreviouslyasWeyl-typeinthecaseofanti-symmetric
increasingthesymmetrichybridization,bothtypesofsu- hybridization.
perconductivity are present and they coexist. An inter-
Asafinalnote,wehighlightthatsymmetrichybridiza-
estingresultisthatinter-bandsuperconductivityhasthe
tion in addition to odd-parity inter-band superconduc-
highest value of order parameter, indicating that it has
tivity stabilizes a topological non-trivial phase, which
the higher critical temperature and makes it responsible
presents localized states at the ends of the chain.
for the superconductivity appearing in the system.
ACKNOWLEDGMENTS
A general approach for obtaining the winding num-
ber of a system described by 4 × 4 matrices was pre-
sented. It may be applied whenever particle-hole sym- The authors would like to thank the CNPq and
metry and spinless time-reversal symmetry are present FAPERJ for financial support. They also are grate-
in a Bogoliubov-de Gennes (BdG) Hamiltonian, which is ful to Emilio Cobanera for discussion and calling at-
the case of the two-bands BCS superconductors studied tention to Ref. [47], and Griffith M.A.S. for useful
here. Accordingtothisapproach,adominantinter-band discussions. Partial support from FCT through grant
couplingwithsymmetrichybridizationbetweenbandsin- UID/CTM/04540/2013 is acknowledged.
duces a topological superconducting phase. The non-
trivial topological character of this phase was shown
through a calculation of the winding number, using the
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