Equivalence of topological mirror and chiral superconductivity in one dimension

1.1 MB·English

by  Eugene Dumitrescu

Equivalenceoftopologicalmirrorandchiralsuperconductivityinonedimension
Eugene Dumitrescu1, Girish Sharma1, Jay D. Sau2, and Sumanta Tewari1
1Department of Physics and Astronomy, Clemson University, Clemson, SC 29634
2 CondensedMatterTheoryCenter,DepartmentofPhysics,UniversityofMaryland,CollegePark,MD20742
Recentlyithasbeenproposedthataunitarytopologicalmirrorsymmetrycanstabilizemultiplezeroenergy
Majorana fermion modes in one dimensional (1D) time reversal (TR) invariant topological superconductors.
Hereweestablishanexactequivalencebetween1D“topologicalmirrorsuperconductivity”andchiraltopologi-
calsuperconductivityinBDIclasswhichcanalsostabilizemultipleMajorana-Kramerspairsin1DTR-invariant
topologicalsuperconductors. Theequivalenceprovesthattopologicalmirrorsuperconductivitycanbeunder-
stoodaschiralsuperconductivityintheBDIsymmetryclassco-existingwithtime-reversalsymmetry. Further-
5
more, weshowthatthemirrorBerryphasecoincideswiththechiralwindinginvariantoftheBDIsymmetry
1
class, which is independent of the presence of the time-reversal symmetry. Thus, the time-reversal invariant
0
topological mirror superconducting state may be viewed as a special case of the BDI symmetry class in the
2
well-knownAltland-Zirnbauerperiodictableoffreefermionicphases. Weillustratetheresultswiththeexam-

n plesof1Dspin-orbitcoupledquantumwiresinthepresenceofnodelesss superconductivityandtherecently
±
a discussedexperimentalsystemofferromagneticatom(Fe)chainsembeddedonalead(Pb)superconductor.
J

5
Introduction : Mirror symmetry, when coupled with time- symmetry. We then establish an exact equivalence between

] reversal(TR)andparticle-hole(PH)symmetries,hasrecently “mirror Berry phase”1 used to justify the presence of multi-
l
l beenproposed1tostabilizeaone-dimensionaltopologicalsu- ple MKPs in DIII systems and the integer winding number
a
perconductingphasewithanintegernumberofspatiallyover- invariant ofthe BDIclass25. However, the winding invariant
h
- lappingzeroenergyMajoranafermionmodes. Thenumberof fortheBDIclasscontinuestobewell-definedandunchanged
s
Majorana edge modes in a so-called “topological mirror su- evenwhenthetime-reversalsymmetryisbroken. Theequiva-
e
m perconductor” is indexed by a mirror Berry phase γm Z lenceprovesthat“topologicalmirrorsuperconductivity”1can
∈
which was defined in Ref. 1. Mirror symmetry and a mirror beviewedasbelongingtotheBDIsymmetryclassinthewell-
.
t Berry phase topological invariant have been used to theoret- known periodic table of Altland-Zirnbaur classification of
a
m ically explain the existence of multiple Majorana modes in freefermionicphases4–6andthattheco-existingtime-reversal
heterostructure based topological superconductors involving symmetrydoesnotqualitativelymodifythetopologicalphase.
-
d a proximity induced s±-wave pairing potential on spin-orbit Therefore, in one dimension the chiral BDI symmetry is in
n coupled semiconductor wires1. A mirror symmetry has also factmoregeneralthanmirrorsymmetryinthesensethatper-
o beenrecentlyinvokedtodiscussthetopologicalpropertiesin turbationswhichbreakboththesymplecticTRsymmetryand
c thecontextofferromagneticatomicchainsembeddedonthe topological mirror symmetry may keep the chiral symmetry
[
surfaceofaPbsuperconductor2,3. invariant,allowingmultipleMajoranafermionmodesevenin

1 Topological classification based on a unitary mirror sym- theabsenceofmirrorsymmetry. Incontrast,chiralsymmetry
v metry aims to extend the Z -invariant of the DIII symmetry breakingtermsalsonecessarilybreakeithermirrorortimere-
2
5 class to allow the existence of multiple Majorana Kramers versalsymmetry(undereitherofthesecircumstancesthemir-
8 pairs(MKPs)1. IntheAltland-Zirnbauerframeworkofclassi- rorBerryphasecannotbedefined1)andthereforetopological
9
0 ficationoffreefermionicphases4–6, Majoranafermionshave mirrorsymmetrycannotbethoughtofasanindependentsym-
0 beenpredictedtooccurasedgestatesinlowdimensionalsys- metryinone-dimension.
. temsbelongingtothetopologicalclassesD7–14,DIII15–25,and Mirror versus chiral symmetric systems: In this paper we
1
0 BDI26–33. Thisraisesthefundamentalquestionofwhetheror will consider a superconducting system with particle-hole
5 notmirrorsymmetryinonedimensionrefinesthetopological symmetryΞandtime-reversalsymmetryΘ. Superconducting
1 classification in the Altland-Zirnbauer framework, and if so, systems with mirror symmetry for which the “mirror Berry
: is mirror symmetry a critical ingredient for the protection of phase”isdefined1 haveanadditionalunitarysymmetryoper-
v
i MKPsintime-reversalinvariantDIIIsystems? Inonedimen- ator such that [ ,Θ] = [ ,Ξ] = [ ,H] = 0 (where
X sionitiswellknownthatthechiralBDIsymmetryclassalso H isMthe BdG HamMiltonian foMr which exMplicit examples are
r supports an arbitrary number of Majorana end modes due to discussed in later sections) and 2 = 1. On the other
a itsZinvariant26,27,32and,withconcurrentTRsymmetry,stabi- hand,systemsinthechiralBDIsyMmmetryc−lasshaveapseudo
lizesmultipleMajorana-Kramerspairs25 asedgemodes. Itis timereversaloperator (with 2 =1)whichincombination
thereforenaturaltoaskhowthesetwosymmetries(mirrorand withtheparticle-holesOymmetrOyΞdefinesachiralityoperator
chiral),whichbothproducemultipleMKPsinthepresenceof =Ξ withtheproperties 2 =1, ,H =0,[Θ, ]=0.
concurrent TR, are related andinvestigate the conditions un- CNoteth·aOtwehaveassumedheCrethatt{hCetwo}time-reverCsalop-
derwhichtheymaydiffer,ifatall. erators and Θ commute. From these definitions it is clear
O
In this work we formulate a general procedure illustrat- thatwhentheseBDIsymmetryoperatorsarepresentonecan
ing how the presence of mirror symmetry promotes a one- definea“mirror”operator = Θ,whichsatisfiesallthe
M O·
dimensionalclassDIIItopologicalsuperconductortothechi- properties mentioned for the Mirror operator. Alternatively
ral topological class BDI that co-exists with a time-reversal for systems with time-reversal and mirror symmetry one can
2
definethepseudo-timereversal =Θ ,whichisrequired connectionofthetwochiralHamiltoniansarerelatedby
O ·M
tocharacterizesystemsintheBDIsymmetryclass.Therefore,
(cid:88)
superconducting systems with TR and BDI symmetries have a(k,θ =0)= f n,k ∇ n,k (7)
n k
(cid:104) | | (cid:105)
thesameoperatorcontentassystemswithTRandmirrorsym- n
metries. =(cid:88)f n,k Z(k)†∇ [Z(k)n,k ] (8)
n k 0
EquivalenceofmirrorBerryphaseandBDIwindingnum- (cid:104) | | (cid:105)
n
ber: The topological invariant associated with the BDI sym- (cid:88)
=a (k,θ =0)+ f n,k [Z(k)†∇ Z(k)]n,k (9)
metryi.e. thewindingnumberfor coincideswiththemirror 0 n k 0
(cid:104) | | (cid:105)
BerryphasedefinedinRef.[1]arCealsorelated. Toseethis, n
wefollowRef.[1]todefinethemirrorBerryphaseanddefine =∂kArg[Det(U(k)V†(k))]=∂kArg[Det[Q(k)]]. (10)
afamilyofHamiltoniansparametrizedbyθas
Therefore the integral of the Berry connection, which is re-
H(k,θ)=H(k)cosθ+ sinθ. (1) lated to the mirror Berry phase, is the same as the winding
C numberinvariantoftheBDIclass.
Noteherethatwehaveintroducedaslightmodificationtothe Example 1: Spin-orbit coupled nanowire proximity cou-
definitionusedinRef.[1],whereforthechiralityoperatorwe pled to s± superconductor: As a concrete example, we il-
haveused =Π insteadofΠ=iΘ Ξ.Butsince com- lustrate the equivalence of topological mirror and chiral su-
muteswithCH thi·sMisessentiallyequival·entapartfromMthefact perconductivity for a system consisting of a spin-orbit cou-
thatwiththistransformation,themirrorBerryphaseinRef.[ pled semiconductor nanowire with proximity induced s±-
1] becomes the total Chern number of H(k,θ) . The Chern wavesuperconductivity1. Inprinciplethismaybeexperimen-
numberisgivenbyintegratingthecurvatureoftheBerrycon- tally achieved by depositing an InSb nanowire onto an Iron
nectionandusingStokestheoremisrelatedtotheintegralof based superconductor with a sign changing extended s-wave
theBerryconnectionwrittenas orderparameter. TheeffectiveBogoliubov-deGennes(BdG)
Hamiltonian for the nanowire with proximity induced super-
a(k,θ)=(cid:88)fn n,k,θ ∇θ,k n,k,θ (2) conductivityisH =(cid:80)kΨ†kH(k)Ψk where,
(cid:104) | | (cid:105)
n H(k)=( 2tcos(k) µ)σ τ +α sin(k)σ τ (11)
0 z R y z
− −
atθ = 0,where n,k,θ arewave-functionsparametrizedei- +∆scos(k)σ0τx.
| (cid:105)
theronthetopHemisphereπ/2>θ >0orthebottomHemi-
Here k k is the 1D crystal momentum and Ψ =
sphere−π/2<θ <0. (c ,c ,≡c† x, c† )T is a four component Nambu spkinor
TheHamiltonianH(k)isoff-diagonalwithanoff-diagonal k↑ k↓ −k↓ − −k↑
acting in the τ (particle-hole) and σ (spin) spaces. Addi-
matrixQ(k)as
tionally, t is the nearest neighbor hopping, µ is the chemi-
(cid:18) (cid:19) calpotential,α isthestrengthofRashbaspinorbitcoupling
0 Q(k) R
H(k)= Q(k)† 0 (3) (whichwehavenchosentobealongyˆwithoutlossofgener-
ality)and∆ istheproximityinducedpairpotential.
s
AswithanysuperconductingmeanfieldBdGHamiltonian
inthebasiswhere isdiagonalandhasawindingnumber
definedintermsofCthephaseArg(det(Q(k)))26. ThematrWix Eq. (11) is invariant under a particle-hole transformation de-
notedbytheoperatorΞ. ThePHconstraintforBlochHamil-
Q(k)canalsobedecomposed(singularvaluedecomposition)
tonians is ΞH(k)Ξ−1 = H( k) and the anti-unitary PH
as
− −
operator in our basis is given by Ξ = σ τ . In addi-
y y
K
Q(k)=U†Σ V , (4) tion to belonging in the time reversal symmetry class DIII
k k k with a Z invariant (by virtue of the time-reversal symme-
2
try ΘH(k)Θ−1 = H( k) where Θ = iσ τ with the
whereU ,V areunitaryandΣ isapositivediagonalmatrix. y 0
k k k − K K
complexconjugationoperator),theHamiltonianinEq.(11)is
The Hamiltonian H(k) can be transformed using a unitary
alsoinvariantunderamirrorsymmetry1 = iσ τ since
transformation y 0
H(k) −1 = H(k) (i.e. [ ,H(k)]M= 0). −Note that the
(cid:18) (cid:19) M M M
U 0 momentum has not changed sign under the mirror operation
Z(k)= k (5)
0 V since the mirror operator is unitary and the reflection as-
k M
sociatedwiththemirrorsymmetryistakenabouta1Dmirror
todefineanotherchiralHamiltonian line1.
Since the Hamiltonian in Eq. (11) is invariant under Θ
(cid:18) (cid:19)
H¯(k)=Z(k)H(k)Z(k)† = 0 Σ(k) . (6) and M as defined above, it is also clearly invariant under
Σ(k) 0 their composition which we define as = Θ . Ex-
O · M
plicitly, = σ τ which squares to +1 and the Hamilto-
0 0
SincethisHamiltonianH¯(k)haswave-functions n,k that nian in EOq. 11 tranKsforms under this operator according to
0
are easy to write down in terms of the diagonal m|atrix(cid:105)Σ(k) H(k) −1 = H( k), which is the pseudo time-reversal
O O −
one can easily check that this Hamiltonian has vanishing symmetry introduced above. The presence of a TR-operator
windingnumber0andalsozeroChernnumber0. TheBerry with 2 = 1, alongwithPHsymmetry, meansthatEq.(11)
O
3
also satisfies the requirements to be in the topological class Operator Symmetry Ex.1 Ex.2
BDIindexedbyanintegerwindinginvariant Z. Aswe
showedinthelastsection, themirrorBerrypWhas∈ecalculated Ξ PH σyτyK σyτyK
ninumthbeeprresetnhcaetcoafnMbeadnedfinΘediswiidthenthtiecahletlopothfethBeDpIsewuidnodiTnRg M Mirror iσyτ0 idˆ·στ0
operatorW = Θ (with 2 = 1) and the PH operator Θ DIIITR iσ τ K iσ τ K
O · M O y 0 y 0
Ξ. Belowwefirstreviewtheactionof andthepseudoTR
operator onthePaulimatricesσ ,andMthenconsiderpertur- O BDITR K (dˆ·yˆ+i(dˆ×yˆ)·σ)K
i
O
bationswhichremoveeachsymmetryindividually.
Π DIIIChiral σ τ σ τ
0 y 0 y
It is straightforward to observe the following relations in-
volvingtheactionofTR(conventionalaswellaspseudo)and C BDIChiral σ τ dˆ·στ
y y y
mirrorsymmetriesonthePaulimatrices:
Θσ Θ−1 = σ (12a) TABLEI.Summaryofsymmetryoperatorsandtheirexplicitforms
i i
− inExamples1,2.Particle-holeandtime-reversalsymmetriesareanti-
σ −1 =+η σ (12b)
i i i unitaryoperatorswhicharetheproductofaunitaryoperator,acting
M M
σ −1 = η σ (12c) inparticle-holeandspinspace,andthecomplexconjugationoperator
i i i
O O − K.
whereη = 1fori=x,z andη =+1fori=y(notethat
i i
−
inobtainingEq.12cwehaveusedthefactthat = Θ ).
O ·M
Thus,underthemirroroperation( )spincomponentsinthe
M z
x z planeacquireaphaseof 1whilethespincomponent
− −
normal to the mirror plane (see Fig. 1) is unchanged. Note
thattheSU(2)angularmomentumalgebraoftheσ operators
([σ ,σ ]=2i(cid:15) σ )ispreservedundertheactionof since
i j ijk k
two(one)operatorsareeven(odd)under 34. O
O
Consider case (i) – A perturbation which potentially can
break the mirror symmetry of Eq. (11) is a Zeeman term
HZ =V·σwhen|Vxz|=(cid:54) 0where|Vxz|=(cid:112)Vx2+Vz2,and Mxz
V = (V ,V ,V ). Such a perturbation also breaks the con-
x y z y
ventionalTRsymmetryΘbutkeepsthepseudoTRsymmetry αˆ
( Θ)andhencethechiralsymmetryofthetotalHamilto-
x
M·
nianintactsolongasV =0. Thus,inthiscase,eventhough
y
themirrorBerryphasecannolongerbedefined,theBDIchi-
ralinvariant remainswelldefinedandthesystemcanhost FIG.1. Mirrorsymmetryforspin-orbitcoupledwires(Example1).
W
an integer number of protected Majorana fermion modes at Inthiscasethe1Dnanowire(red)liesalongthex-axis.Thebluear-
theedges. Nowconsidercase(ii)–whenV = (0,V ,0)the rowdenotesthedirectionofthespin-orbitfieldinspinspacewhich
y
Hamiltonianismirrorsymmetric( isunbroken)butchiral definesthemirrorplane(lightblue).WhenaZeemanfieldliesinthe
and TR symmetries both break doMwn (i.e. the Hamiltonian mirrorplanethemirrorsymmetryisbrokenbutchiralsymmetryis
is not invariant under Θ and, consequently, also not under preserved. IfaZeemanfieldisnormaltothemirrorplanethemir-
rorsymmetryispreservedwhiletimereversalandchiralsymmetries
= Θ). In this case, even though the mirror symme-
O M· are removed. However, in this case, there is no mirror topological
trypersists,themirrorBerryphaseprocedureisnolongerap-
invariantsincethemirrorBerryphaseisonlydefinedinthepresence
plicable due to the lack of time reversal symmetry (note that
oftime-reversalsymmetry(classDIII).Figureformirrorsymmetry
themirrorBerryphaseisonlydefinedinthepresenceofΘ1).
forferromagneticatomchainembeddedonPbsuperconductor(Ex-
Thus we find that the mirror symmetry helps us find a suit- ample2)shouldbeanalogous.
ablepseudoTRoperator whichinturnhelpsusdefinethe
O
chiral operator = Ξ. This procedure is valid as long
C O ·
astheTRoperatorΘisasymmetryorasincase(i)wherein
physicalsystem.
both Θ and are broken by the same Zeeman field. Such
M
a procedure of defining a chiral symmetry does not work in Example 2: Ferromagnetic atom (Fe) chain embedded
case (ii) and also in case (iii) where the mirror symmetry is in Pb superconductor: Motivated by recent experimental
broken(say,byanadditionalnext-nearest-neighborspin-orbit findings2,3, as a second illustrative example we consider a
couplingHSO = α(cid:48)sin(2k)(c σ)τ withc yˆ25)butΘis simple effective model for a ferromagnetic (Fe) atom chain
k · z ⊥
unbroken. Note that in both case (ii) and (iii) neither mirror embedded in a Pb superconductor which can support Ma-
Chernnorthechiralwindingnumberinvariantcanbedefined. jorana modes and topological superconductivity. The effec-
However,aswehaveestablishedinthiswork,whenbothin- tive mean-field BdG Hamiltonian, which gives a sufficient
variantsaredefined(i.e.,inthepresenceofΘ),theyareiden- descriptionoftopologicalsuperconductivityoftheferromag-
tical,andthetopologicalmirrorsuperconductivityandchiral neticnanowireassumingthattheunderlyingdegreesoffree-
BDIsuperconductivityareequivalentdescriptionsofthesame dominthePbsubstratehavebeenintegratedout,canbewrit-
4
tenas33,35,36H =(cid:80) Ψ†H(k)Ψ where, it breaks = Θ) and thus the system belongs to class
k k k O M·
D where even number of localized Majorana bound states
H(k)=( 2tcos(k) µ)σ τ (13)
0 z hybridize into finite-energy quasiparticles. Thus the mirror
− −
+∆ sin(k)d στ +V στ . symmetryalonedoesnotprotectspatiallylocalizedMajorana
p x 0
· ·
multiplets at the sample edges. Let us now introduce a Zee-
In Eq. (13) ∆ is the magnitude of the induced spin-triplet
p man field which only lies in the mirror plane which is per-
pairing potential in the ferromagnetic nanowire. The Zee- pendiculartothedˆ-vector. Nowthemirroroperatornolonger
mantermduetoaconstantinternalmagnetizationinthefer-
commuteswiththeHamiltonianH andthusmirrorsymmetry
romagnet, V = (V ,V ,V ) (which is assumed to be V =
x y z isbroken. HoweverinthiscasesinceΘisalsobrokenbythe
(0,0,J)).
Zeeman field the pseudo TR operator = Θ remains
Agenericd-vectorpointinginanarbitrarydirectioninspin O M·
unbroken. Thus chiral symmetry is unbroken and this is a
spaceiswrittenasdˆ=(sinθcosφ,sinθsinφ,cosθ)inpolar classBDIsystemasdiscussedearlCier33,35,36.
coordinates. In order to understand the role of mirror sym-
Conclusion: In this work we show that topological mir-
metry let us first consider Eq. (13) with no intrinsic magne-
ror superconductivity that results from the co-existence of a
tization (i.e V = 0). In this case, the only spin space Pauli
mirrorsymmetryandsymplectictime-reversalsymmetrymay
matrix (σ) appears in the superconducting term and the ob-
also be viewed as a co-existence of symplectic time-reversal
vious choice of would be = idˆ στ . Thus the mir-
0 andchiraltopologicalsuperconductivityintheBDIsymmetry
ror plane is the pMlane perpendiMcular to th·e dˆ-vector. Follow- classinone dimension. ThemirrorBerryphase1 isfound to
ingtheprocedurediscussedpreviouslywedefineanoperator
coincidewiththewindingnumberinvariantthatcharacterizes
=Θ =(iσ sinθcosφ+σ sinθsinφ iσ cosθ) =
z 0 x theBDIsymmetryclassinonedimension. Thewindingnum-
O ·M − K
(dˆ yˆ+i(dˆ yˆ) σ) . Clearly 2 =+1and H(k) −1 = berandotherqualitativeaspectsofthephasesuchasnumber
· × · K O O O
+H( k) and thus fulfills the requirements of a class BDI of edge Majorana modes, continue to survive even when the
−
or pseudo time reversal operator. The chiral operator is symplectictime-reversalΘandmirror-symmetry arebro-
= Ξ = dˆ στ forourHamiltonian. Indeed, thefCorm kenweakly.ThusthetopologicalmirrorphaseisadMiabatically
y
C O· ·
oftheoperatorsinExample1canbeunderstoodbytheabove connected to a BDI phase by infinitesimal perturbations that
relationdescribingthestructureofthe operator.Inthespin- breakthemirrorandtime-reversalsymmetrywhilepreserving
O
orbit coupled system the direction of the spin-orbit field (re- theBDIchiralsymmetry .Suchaperturbationdoesnothave
C
member the Rashba term involved a σ ) plays the role of an aqualitativeeffectsuchassplittingedgeMajoranamodes.We
y
effectived-vector forthe effectivep-wavepairing createdby illustrate our point with two examples, namely, a spin-orbit
thecombinationofspin-orbitcouplingandsingletsupercon- coupledsemiconductornanowirewithproximityinducedex-
ductivity. tended s-wave pairing potential1, and the recently discussed
NowletusturnontheeffectiveZeemanfieldVandexam- experimental system of chains of ferromagnetic atoms on a
inewhathappenstothemirrorsymmetry. Aslongastheonly spin-orbitcoupledsubstrateofPbsuperconductor2,3.
non-zerocomponentofV isalongthedˆvector, mirrorsym- Acknowledgment: This work is supported by AFOSR
metry remains intact because the mirror operator still com- (FA9550-13-1-0045). J.D.S. would like to acknowledge the
muteswiththeHamiltonianinEq.13. Howeverthisdirection UniversityofMaryland,CondensedMattertheorycenter,and
oftheZeemanfieldbreaksthechiralsymmetry = Ξ(as theJointQuantuminstituteforstartupsupport.
C O·
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