Majorana modes in time-reversal invariant s-wave topological superconductors

1.7 MB·English

by  Shusa Deng

Majorana modes in time-reversal invariant s-wave topological superconductors
Shusa Deng,1 Lorenza Viola,1 and Gerardo Ortiz2
1Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, NH 03755, USA
2Department of Physics, University of Indiana, Bloomington, Indiana 47405, USA
(Dated: January 20, 2012)
We present a time-reversal invariant s-wave superconductor supporting Majorana edge modes.
Themulti-bandcharacterofthemodeltogetherwithspin-orbitcouplingarekeytorealizingsucha
2 topologicalsuperconductor. WecharacterizethetopologicalphasediagrambyusingapartialChern
1 numbersum,andshowthatthelatterisphysicallyrelatedtotheparityofthefermionnumberofthe
0 time-reversal invariant modes. By taking the self-consistency constraint on the s-wave pairing gap
2 into account, we also establish the possibility of a direct topological superconductor-to-topological
insulator quantumphase transition.
n
a
J PACSnumbers: 73.20.At,74.78.-w,71.10.Pm,03.67.Lx

9
1
Since Majoranasuggestedthe possibilityfora fermion netic insulator, or to a magnetic field in materials with

] tocoincidewithits ownantiparticlebackin1937[1],the strong spin-orbit (SO) coupling [13].
n searchfor the Majoranaparticlehascatalizedintenseef- Our motivation in this work is to explore whether a
o
fortacrossparticleandcondensed-matterphysicistsalike path to TSs exists based on conventional bulk s-wave
c
- [2]. Particles either constitute the building blocks of a spin-singlet pairing superconductivity. We answer this
r fundamental physical theory or may effectively emerge questionbyexplicitlyconstructingamodelwhich,tothe
p
u as the result of the interactions of a theory. A Majorana best of our knowledge, provides the first example of a
s fermion is no exception to this principle, with neutrinos 2DTS with s-wavepairingsymmetry, and supports Ma-
.
t potentially epitomizing the first view [3], and localized jorana edge modes without breaking TR symmetry [14].
a
quasiparticleexcitationsinmatterillustratingthesecond Thekeyphysicalinsightisthemulti-bandcharacterofthe
m
[4]. Remarkably, Majorana fermions can give rise to the model,inthesamespiritoftwo-gapsuperconductors[15],
-
d emergenceofnon-Abelianbraiding[5]. Thus,inaddition but with the SO coupling playing a crucial role in turn-
n to their significance for fundamental quantum physics, ing a topologicallytrivial two-gapsuperconductor into a
o interest in realizing and controlling Majorana fermions non-trivialone. Ourresultsadvanceexistingapproaches
c
has been fueled in recent yearsby the prospectof imple- inseveralways. First,multi-bandsystemsclearlyexpand
[
mentingfault-toleranttopologicalquantumcomputation the catalog of TI and TS materials. Following the dis-

2 [6, 7]. As a result, a race is underway to conclusively covery of s-wave two-band superconductivity in MgB
2
v
detect and characterize these elusive particles. in 2001, a number of two-gap superconductors ranging
3
from high-temperature cuprates to heavy-fermion and
8 A variety of condensed-matter systems hosting local-
6 iron-basedsuperconductorshavealreadybeencharacter-
ized Majorana elementary excitations have been pro-
4 ized in the laboratory [16], giving hope for a near-future
posed, notably certain quantum Hall states [5] and so-
8. called topological superconductors (TSs) [8, 9]. Unfortu- material implementation. Furthermore, from a theoreti-
0 cal standpoint, our TR-invariant model also supports a
nately, these exotic states of matter require the explicit
1 direct TI-to-TS (first-order) quantum phase transition
breakingoftime-reversal(TR)symmetryandtheirphys-
1
(QPT), allowing one to probe these novel topological
: ical realization seems to be at odds with existent mate-
v phases and their surface states by suitably tuning con-
rials. Such is the case, for instance, of superconductors
i trol parameters in the same physical system.
X withp +ip spin-tripletpairingsymmetry. Thishasnot
x y
Exact solution with periodic boundary conditions.—
r prevented researchers to pursue creative proposals that
a WeconsideraTR-invarianttwo-bandHamiltonianofthe
rely on a combination of carefully crafted materials and
form H =H +H +H +H.c., where
devices. Fu and Kane [10], in particular, suggested the cd so sw
use of a (topologically trivial) s-wave superconducting 1
film on top of a three-dimensional topological insulator Hcd = 2 (ucdψj†τxψj −µψj†ψj)−t ψi†τxψj,
(TI),whichbyproximityeffecttransformsthenon-trivial Xj hXi,ji
surface state of the TI into a localized Majorana excita- Hso = iλ ψj†τzσνψj+ν,
tion [4, 11] (see also [12] for related early contributions). j,νX=xˆ,yˆ
Whileexperimentalrealizationofthisideaawaitsfurther
progress in material science, alternative routes are be- Hsw = (∆cc†j, c†j, +∆dd†j, d†j, ), (1)
Xj ↑ ↓ ↑ ↓
ing actively sought, including schemes based on metallic
thin-filmmicrostructures,quantumnanowires,andsemi- represent the two-band (c and d) dynamics, the SO in-
conductor quantum wells coupled to either a ferromag- teraction, and s-wave superconducting fluctuations, re-
2
spectively. In the above equations, µ is the chemical
potential,ucd representsanonsitespin-independent“hy- 6 C+=0
bridization term” between the two bands, fermionic cre- 4 1 −1
ation operators at lattice site j (unit vectors xˆ,yˆ) and 2 −1 (two 0pairs) 1
spin σ =↑,↓ are specified as c†j,σ or d†j,σ, depending on ∆ 0 0 −2 2 0
the band, and (∆c, ∆d) denote the mean-field s-wave 0
pairing gaps. By letting ψ ≡ (c ,c ,d ,d )T, the −2 −1 (two pairs) 1
j j, j, j, j,
↑ ↓ ↑ ↓
Paulimatricesτν andσν actontheorbitalandspinpart, −4 1 −1
respectively. Noticethatwehaveimplicitlyassumedthat −6 0
theintrabandSOcouplingstrengthsobeyλc =−λd ≡λ. −6 −4 −2 0ucd 2 4 6
In this way, in the limit where µ = 0 = ∆ = ∆ , H re-
c d
duces to a known model for a TI [17]. 6 C+=0
For general parameter values and periodic bound-
4 1 −1
ary conditions (PBC), H can be block-diagonalized
by Fourier transformation in both x and y. That 2 −1 1
0 −2 0 2 0
is, we can rewrite H = 21 k(Aˆ†kHˆkAˆk − 4µ), with ∆ 0
Aˆ†k = (c†k, ,c†k, ,d†k, ,d†k, ,c kP, ,c k, ,d k, ,d k, ), and −2 (two pairs)
Hˆk an 8 ×↑ 8 m↓ atri↑x. A↓n−ana↑lyt−ica↓l so−lut↑ion−ex↓ists in −4 1 −1
the limit where the pairing gaps are π-shifted, ∆ = 0
c −6
−∆d ≡ ∆, since Hˆk decouples into two 4 × 4 matri- −6 −4 −2 0u 2 4 6
cd
ces. By introducing new canonical fermion operators,
FIG. 1: (Color online) Topological characterization of the
ak,σ = √12(ck,σ + dk,σ), bk,σ = √12(ck,σ − dk,σ), we phasestructureofHamiltonianH viathepartialCNsumC+
may rewrite H = 12 k(Bˆk†Hˆk′Bˆk − 4µ), with Bˆk† = as a function of ucd and ∆, with t = 1 and arbitrary λ 6= 0,
(a†k, ,b†k, ,a k, ,b k, ,Pa†k, ,b†k, ,ak, ,bk, ), and Hˆk′ = f(obrortetopmre)s.enTtahteivbelacchkem(dicaashlepdo)telninteiarlsepµre=se0nt(stoapn)ainnsdulµat=or−o1r
Hˆ1′,k↑⊕Hˆ↓2′,k−, w↑ith−Hˆ1↓′,k,−Hˆ2′↓,k −bei↑ng T↓R of↑one another, metal phase, depending on the filling, with ∆ = 0. CNs are
and calculated for (Nx,Ny) = (100,100) lattice sites. Note that
we may havetwo pairs of edge modes with C =0.
+
Hˆ1′,k=(cid:18)mkσz−−i∆µσ+yλk·~σ −mkσz +i∆µσy+λk·~σ∗ (cid:19).
overhalftheBrillouinzoneforalltheoccupiedbandswas
Here, λk = −2λ(sinkx,sinky), mk = ucd −2t(coskx + used. Here, we propose a different Z2 invariant which is
cosky), and ~σ ≡ (σx,σy). The excitation spectrum ob- guaranteed to workin the presence ofTR: taking advan-
tained from diagonalizing either Hˆ or Hˆ is tageofthedecoupledstructurebetweenTR-pairs,weuse
1′,k 2′,k
the CNs of the two occupied negative bands of Hˆ only
1′,k
(say,C andC )anddefinethefollowingparityinvariant:
1 2
ǫn,k =±rm2k+Ω2+|λk|2±2 m2kΩ2+µ2|λk|2, (2)
q P ≡(−1)mod2(C+), C ≡C +C . (3)
C + 1 2
swuhmereedtahnedoΩrd2er≡ǫµ1,2k+≤∆ǫ22.,kQ≤PT0s≤occǫu3,rkw≤heǫn4,ktheisgaasp- Let |ψn,ki denote the band-n eigenvector of Hˆ1′,k. Then
the required CNs C , n=1,2, can be computed as [21]
closes (ǫ2,k = 0, for general ∆ 6= 0), leading to the crit- n
ical lines determined by mkc = ±Ω, where the criti- 1 π π
cal modes kc ∈ {(0,0),(0,π),(π,0),(π,π)}. It is worth Cn = π Z dkxZ dkyImh∂kxψn,k|∂kyψn,ki. (4)
noticing that through a suitable unitary transformation π π
− −
(see Eq. (4) of Ref. [18]), the SO interaction in Eq. (1)
The resulting topological phase structure is shown in
is formally mapped into px+ipy and px−ipy intraband Fig. 1 by treating the pairing gap ∆ as a free control
interaction, hinting at the existence of non-trivial topo-
parameter. In an actual physical system, ∆ cannot be
logical phases, as we demonstrate next.
changed at will, but only be found self-consistently by
Topological response.— Since H preserves TR invari- minimizing the free energy (or ground-state energy at
ance, bands which form TR-pairs have opposite bulk zero temperature). While we shall return on this issue
Chern numbers (CNs) Cn, leading to n occupiedCn =0 later,wefirstfocusonunderstandingthe physicalmean-
(includingbothHˆ andHˆ ). Thus,Pint∈roducinganew ing of the above invariant and on establishing a bulk-
1′,k 2′,k
Z topological invariant is necessary in order to distin- boundary correspondence for our model.
2
guish between trivial and TS phases. In Ref. [19], the Interestingly, there is a direct connection between
parity of the sum of the positive CNs was considered, the invariant P and the fermion number parity of the
C
whereas in Ref. [20] an integral of the Berry curvature TR-invariant modes. Without loss of generality, let
3
µ = 0, and focus on the ground-state fermion number
3 3
(a) (b)
parity of the four TR-invariant points in the first
2 2
Brillouin zone, k . Since Hˆ and Hˆ are decoupled,
c 1′,k 2′,k 1 1
we need only concentrate on the ground-state parity
property of Hˆ . Let us introduce the new basis: εkx0 εkx0
1′,k
−1 −1
{sias†k,cHˆ,↑1′|,vkacbie,cbo†kmc,↓e|svHac1i,,k|cva=cim,ak†kccσ,↑zb⊕†k∆c,↓σ|vxa,cwi}it.heInigetnhvisalbuaes- −−32 −−32
±mkc,±∆, and anbidentical matrix for Hˆ2′,kc in the TR- −34 −2 0kx 2 4 −34 −2 0kx 2 4
(c) (d)
bWahsiesn |m{−ka|†kc,↓>|vac|i∆,b|†k,c,↑t|hvaeci,g|rvoaucni,d−as†ktca,t↓eb†kco,↑f|vaecaic}h. 2 2
c 1 1
mode k is in the sector with odd fermion parity,
Pk = ceiπ(a†kc,↑akc,↑+b†kc,↓bkc,↓) = −1, otherwise it is εkx 0 εkx0
c −1 −1
in the sector with even fermion parity, Pk = 1.
c −2 −2
By analyzing the relation between |mk | and |∆| for
each kc, we can see that the TS (triviacl) phases with −−34 −2 0 k 2 4 −−34 −2 0k 2 4
x x
P = −1(1) correspond to the ground state with
C
cQokinccPidkecs ≡withPFthe=fe−rm1i(o1n).numThbuesr,poaurirtyZo2f itnhvearfioaunrt FHIGon. 2a:c(yClionldoerrofnolrinµe)=E0x,ctit=ati1o,nλs=pe1ct.ruPmaneolf(Ha)a,mCil+ton=ia1n:
∆=2,u =3;Panel(b),C =−1: ∆=2.5,u =2;Panel
TR-invariant modes from one representative of each cd + cd
(c), C = 0: ∆ = 2, u = 1; Panel (d), C = 2: ∆ = 0.8,
Kramer’s pairs, consistent with the fact that only a + cd +
u = 1.5. Note that the bulk gap scales as min(λ,∆). The
cd
partial CN sum can detect TS phases in the presence of
numberof lattice sites (Nx,Ny)=(40,100).
TR symmetry. While the relation between non-trivial
topological signatures (such as the fractional Josephson
effect) and the local fermion parity of Majorana edge
states has been discussed in the literature [6, 22, 23], γ0 = γ0†, indicating that the edge states in our system
satisfy Majorana fermion statistics.
invoking the fermion number parity of the TR-invariant
modesinbulk periodic systemstocharacterizeTSphases Phase diagram with self-consistent pairing gap.—
has not, to the best of our knowledge. Within BCS mean-field theory, let V ≡ Vk,k′ > 0 de-
Open boundary conditions and edge states.— A hall- notetheeffectiveattractionstrengthineachband. Then
markofaTSisthepresenceofanoddnumberofpairsof the pairing gap ∆ = ∆c = −Vhck, c k, i = −∆d,
↑ − ↓
gapless helical edge states, satisfying Majorana fermion and the ground-state energy can be written as Eg =
statistics. Thus, in order to understand the relation be- 2NxNy(∆2/V) + k(ǫ1,k + ǫ2,k − 2µ). The first (con-
tween P (or P ) and the parity of the number of edge stant) term is thePcondensation energy, which was ne-
C F
states, i.e., a bulk-boundary correspondence, we study glected in H. By using Eq. (2) and minimizing Eg, we
the Hamiltonian H on a cylinder. That is, we retain obtainthe stable self-consistentpairing gap∆ as a func-
PBC only along x, and correspondingly obtain the exci- tionoftheremainingcontrolparameters[24]. Theresult-
tationspectrum,ǫ ,byapplyingaFouriertransforma- ing zero-temperature phase diagram is shown in Fig. 3.
n,kx
tion in the x-direction only. For simplicity, let us again For µ = 0 (top panel), the average fermion number is
focus on the case µ = 0. The resulting excitation spec- consistent with half-filling, and thus with an insulating
trum is depicted in Fig. 2 for representative parameter phase when ∆ = 0. In particular, when 0 < |ucd| < 4,
choices. Specifically, for odd P (C = 1 in panel (a) the ground state is known to correspond to a TI phase
C +
andC =−1inpanel(b),respectively),H supportsone [17]. Interestingly, without self-consistency, the TI can-
+
TR-pair of helical edge states on each boundary, corre- notbeturnedintoaTSdirectly,asshowninthetoppanel
sponding to the Dirac points k =0 (a) and k =π (b). of Fig. 1. However, after self-consistency is taken into
x x
Different possibilities arise for even PC. While C+ = 0 account, the topologically trivial phase with C+ = ±2
can clearly also indicate the absence of edge states, in disappears, and a first-order QPT can connect the two
panel (c) one TR-pair of helical edge states exists on phases. For µ=−1 (bottom panel),the averagefermion
each boundary for both Dirac points k =0,π (for a to- number is found to be less than half-filling, realizing a
x
taloftwo pairs,as alsoexplicitly indicated inFig.1). In metallic phase when ∆ = 0. Derivatives of the ground-
panel (d) (C =2), both TR-pairs of helical edge states stateenergyindicatethatallQPTs,excepttheTI-to-TS
+
correspond to the Dirac point k = 0 instead. Since, as phase transition, are continuous.
x
remarked,ourHamiltonianexhibitsparticle-holesymme- Discussion.—Anumberofremarksareinorder. First,
vtrayl,uethǫe eq,uawthioenreγγǫn,kx =isγt−†hǫen,aksxsohcoialdtsedfoqrueaasci-hpaerigtiecnle- wpahiirliengthgeapcshooibceeyionfgSλO=co−upλlinagndst∆reng=th−s∆andaffso-rwdasvae
n,kx ǫn,kx c d c d
annihilation operator. Thus, for zero-energyedge states, fully analyticaltreatment,relaxingthese conditions may
4
der to simplify calculations. Preliminaryresults indicate
12
that a qualitatively similar behavior (that is, the possi-
C =0 5
+ bility of even/odd numbers of pairs of gapless Majorana
11 1 −1 4.5
4 surface states) also exists for 3D systems obtained from
10 3.5 a natural extension of our 2D Hamiltonian. It is espe-
3 ciallysuggestiveto note thataπ phase shiftinthe order
V 9 2.5 parameteracrosstwobandsisalsobelievedtoplayakey
2 roleinironpnictidesuperconductors[26],hintingatpos-
8 TI 0 TI 1.5 sible relationshipsbetweenTSbehaviorandso-calleds
7 (two pairs) 1 pairing symmetry. Whileamoredetailedinvestigationi±s
µ=0 0.5 underway, it is our hope that multi-band superconduc-
6 0 tivity maypoint tonew experimentallyviable venuesfor
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
u
cd exploringtopologicalphasesandtheirexoticexcitations.
It is a pleasure to thank Charlie Kane for insightful
12
0 C+=−1 5 discussions. Support from the NSF through Grants No.
11 1 4.5 PHY-0903727(to L. V.) and 1066293(Aspen Center for
10 2 4 Physics) is gratefully acknowledged.
3.5
9
3
V 8
2.5
7 0 Metal 2 [1] E. Majorana, NuovoCimento 14, 171 (1937).
(two pairs) 1.5 [2] F. Wilczek, NaturePhys. 5, 614 (2009).
6
1 [3] F. T. Avignone, S. R. Elliott, and J. Engel, Rev. Mod.
5 µ=−1 0.5 Phys. 80, 481 (2008).
[4] M. Z.Hasan and C. L. Kane,Rev.Mod. Phys.82, 3045
4 0
0.5 1 1.5 2 u 2.5 3 3.5 (2010).
cd
FIG. 3: (Color online) Phase diagram as a function of u [5] G. Moore and N. Read,Nucl. Phys.B 360, 362 (1991).
cd
and V with the pairing gap ∆ calculated self-consistently. [6] A. Kitaev, Phys.Uspekhi44, 131 (2001).
The magnitude of ∆ is represented by a color, whose scale is [7] C. Nayak et al.,Rev.Mod. Phys.80, 1083 (2008).
indicatedontheside. Thenumberoflatticesites(Nx,Ny)= [8] N. Read and D.Green, Phys. Rev.B 61, 10267 (2000).
(80,80). [9] D. A. Ivanov,Phys. Rev.Lett. 86, 268 (2001).
[10] L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407
(2008).
[11] T.D.Stanescuetal.,Phys.Rev.B81,241310(R)(2010).
be necessary to make contact with real materials. Nu- [12] B. A. Volkov and O. A. Pankratov, JETP Lett. 42, 178
mericalresults ona cylinder showthat the levelcrossing
(1985); O. A. Pankratov, S. V. Pakhomov, and B. A.
of the Majorana edge states in the TS phase is robust Volkov,Solid State Commun. 61, 93 (1987).
against perturbations around λ = −λ , including the [13] A.C.PotterandP.A.Lee,Phys.Rev.Lett.105,227003
c d
possibility that the SO coupling vanishes in one of the (2010); J. D. Sau et al., arXiv:1103.2770; L. Mao et al.,
bands. TS behavior also persists if |∆ |−|∆ | 6= 0, as arXiv:1105.3483; J. D. Sau et al., Phys. Rev. Lett. 104,
c d 040502(2010);J.Alicea,Phys.Rev.B81,125318(2010).
long as the phase difference between pairing gaps is π.
[14] TR-invariantTSsin 2Dand3Dhavebeenalso analyzed
In the presence of a phase mismatch ε, edge modes are in X.-L. Qi et al., Phys. Rev. Lett. 102, 187001 (2009),
foundto become gapped,with a minimalgapthatscales
based, however,on p-wavepairing.
linearly with ε. Interestingly, however, preliminary re- [15] H. Suhl, B. T. Matthias, and L. R. Walker, Phys. Rev.
sults indicate that adding a suitable Zeeman field can Lett. 3, 552 (1959).
allow(atthe expense ofbreakingTR invariance)gapless [16] J.Nagamatsuetal.,Nature410,63(2001);R.Khasanov
Majorana excitations to be restored, with a precise tun- et al., Phys. Rev. Lett. 98, 057007 (2007); M. Jourdan
et al., ibid. 93, 097001 (2004); Y. Kamihara et al., J.
ingofthe phasedifferencebeing nolongerrequired. Itis
Am. Chem. Soc. 130, 3296 (2008); A.P. Petrovi´c et al.,
alsoworthnotingthatonecanreinterpretthebandindex Phys.Rev.Lett.106, 017003 (2011); T.Hanaguriet al.,
inH asalayerindex,andsoH maybethoughtofasde- Science 328, 474 (2010).
scribing a bilayer of superconductors with phase-shifted [17] G. Rosenberg and M. Franz, Phys. Rev. B 82, 035105
pairing gaps, and an interlayer coupling H . Beside es- (2010).
cd
tablishingaformalsimilaritywiththe scenariodiscussed [18] M.Sato,Y.Takahashi,andS.Fujimoto,Phys.Rev.Lett.
by Fu and Kane [10], such an interpretation may offer 103, 020401 (2009).
[19] R. Roy,Phys.Rev.B 79, 195322 (2009).
additional implementation flexibility, as the possibility
[20] L. Fu and C. L. Kane, Phys.Rev. B 74, 195312 (2006).
to control the superconducting and SO couplings by an
[21] In order to avoid the random phases emerging
appliedgatevoltagehasbeendemonstratedrecently[25].
from diagonalizing H, in numerical computations of
Second,wehavethusfarrestrictedto2Dsystemsinor- Cn we approximate the integrand in Eq. (4) as
5
Im[ln(hψn,k|ψn,kxihψn,kx|ψn,kyihψn,ky|ψn,ki)]/ǫ2,where [24] A.Kubasiak,P.Massignan,andM.Lewenstein,EPL92,
k ≡k+ǫkˆ ,kˆ areunitvectors,andǫ≪1.This 46004 (2010).
enxs(yu)res that Cxn(y)becxo(my)es numerically gauge-invariant. [25] M. Ben Shalom et al., Phys. Rev. Lett. 104, 126802
[22] L. Fu and C. L. Kane, Phys. Rev.B 79, 161408 (2009). (2010).
[23] K. T. Law and P. A. Lee, Phys. Rev. B 84, 081304(R) [26] I. I. Mazin et al., Phys.Rev.Lett. 101, 057003 (2008).
(2011).