Interplay between electronic topology and crystal symmetry: Dislocation-line modes in topological band-insulators

2.1 MB·

by  Robert-Jan Slager

Interplaybetweenelectronictopologyandcrystalsymmetry: Dislocation-linemodesintopological
band-insulators
Robert-Jan Slager,1 Andrej Mesaros,2 Vladimir Juricˇic´,3 and Jan Zaanen1
1Instituut-LorentzforTheoreticalPhysics,UniversiteitLeiden,P.O.Box9506,2300RALeiden,TheNetherlands
2Department of Physics, Boston College, Chestnut Hill, MA 02467, USA
3InstituteforTheoreticalPhysics,UtrechtUniversity,Leuvenlaan4,3584CEUtrecht,TheNetherlands
We elucidate the general rule governing the response of dislocation lines in three-dimensional topological
bandinsulators. AccordingtothisK-b-trule, thelatticetopology, representedbydislocationlinesoriented
indirectiontwithBurgersvectorb,combineswiththeelectronic-bandtopology,characterizedbytheband-
4 inversionmomentumKinv,toproducegaplesspropagatingmodeswhentheplaneorthogonaltothedislocation
1 linefeaturesabandinversionwithanontrivialensuingfluxΦ=Kinv·b (mod 2π). Althoughithasalready
0 beendiscoveredbyY.Ranetal., NaturePhys. 5, 298(2009), thatdislocationlineshostpropagatingmodes,
2 the exactmechanism of their appearancein conjunction withthe crystal symmetries ofa topological stateis
providedbytheK-b-trule. Finally,wediscusspossibleexperimentallyconsequentialexamplesinwhichthe
c
modesareobliviousforthedirectionofpropagation,suchastherecentlyproposedtopologically-insulatingstate
e
D inelectron-dopedBaBiO3.

2 PACSnumbers:71.10.Pm72.10.Fk73.20.-r73.43.-f

]
l Topologicalband-insulators(TBIs)representanewclassof linesl(τ)(parametrizedbyτ ∈[0,1]),characterizedbyatan-
l
a quantum materials that, due to the presence of time-reversal gent vector t ≡ dl/dτ, with the discontinuity introduced to
h
symmetry(TRS),featureaninsulatingbulkbandgaptogether the crystalline order described by a Burgers vector b. Both
-
s with metallic edge or surface states protected by a Z topo- thesevectorscanonlybeorientedalongtheprincipalaxesof
2
e logical invariant[1–4]. This Z classification of topological the crystal, and a screw (edge) dislocation is obtained when
m 2
insulatorsisapartoftheclassificationoffreegappedfermion b (cid:107) t (b ⊥ t), see Fig.1. Moreover, the Burgers vectors
.
t matter in the presence of the fundamental antiunitary time- areadditive, andinfullgeneralitywecanthusconsideronly
a
m reversal and particle-hole symmetries, the so-called tenfold dislocationswiththeBurgersvectorsequaltoBravaislattice
way[5–7]. Ontheotherhand, topologically-insulatingcrys- vectors. The crucial observation is that translational lattice
-
d talsbreakcontinuoustranslationalandrotationalsymmetries symmetry is preserved along the dislocation line. Therefore,
n downtodiscretesymmetriesmathematicallycharacterizedby the full lattice Hamiltonian in the presence of a dislocation
o the space groups. By considering the crystal symmetries, an orientedalongthez-axis(t=e )canbewrittenas
z
c extralayerinthisZ classificationofTBIshasbeenrecently
[ 2 (cid:88)
uncovered[8]. ThisspacegroupclassificationofTBIsresults H (x,y,z)= eikzzH2D(x,y,k ). (1)
3D eff z
2
v intheenrichmentofthetenfoldwaywithextraphases,suchas kz
crystalline (or “valley”) phases [9] and translationally-active
4 Notice that the above 2D lattice Hamiltonian possesses the
phases,thelatterfeaturinganoddnumberofband-inversions
4
(wallpapergroup)symmetryofthecrystallographicplaneor-
0 at non-Γ points in the Brillouin zone (BZ) [8, 10]. Disloca-
thogonaltothedislocationline,becausetheBurgersvectoris
4 tions are of central interest in this endeavor, being the topo-
1. logical defects exclusively related to the lattice translations. aBravaislatticevector.
On the other hand, the electronic topology of a TBI is
0 In two dimensions (2D), the role of these lattice defects has
4 beenrecentlyelucidatedinTBIs[10,11],aswellasintopo- characterized by the band-inversions at time-reversal invari-
1 ant(TRI)momentaK intheBZ[8]. Adislocationdisturbs
logical superconductors [12, 13] and interacting topological inv
: the crystalline order only microscopically close to its core.
v states [14–16]. In particular, it has been shown that these
i lattice defects in two-dimensional TBIs act as probes of dis- Wecanthereforeuseelasticcontinuumtheorytodescribeits
X
effectatlowenergies. Theelasticdeformationofthecontin-
tincttopologicalstatesthroughbindingofthelocalizedzero-
r uousmediumisencodedbyadistortionfieldε oftheglobal
a energymodesintheircore[10]. i
Cartesian reference frame e , eα = δα, with i,α = 1,2,3
i i i
Although early on it was identified that in three- [21,22]. Themomentumnearthetransitionfromatopologi-
dimensional TBIs dislocation lines support propagating heli- callytrivialtoanontrivialphasewiththebandgapclosingat
calmodes[17],thepreciseroleofdislocationshavenotbeen the momentum K is k = (e + ε ) · (K − q), with
inv i i i inv
explored thoroughly [18–20]. In particular, the relation be- q (cid:28) K ∼ 1/a as the momentum of the low-energy elec-
inv
tween the lattice symmetry and the electronic topology, as tronic excitations, ε ∼ a/r, a is the lattice constant, and r
wellasthecharacterizationofthesetopologicalstatesthrough is the distance from the defect core. Therefore the disloca-
the response of the dislocation lines has not been addressed. tion gives rise to a U(1) gauge field A = −ε · K that
i i inv
Dislocationsinthreedimensions(3D)aredefectswithricher minimallycouplestotheelectronicexcitations,q → q+A.
structurethantheirtwo-dimensionalcounterparts. Theyform The translational symmetry then implies that the gauge field
2
FIG.1: (Coloronline)IllustrationoftheK-b-trulerelatingtheelectronictopologyinthemomentumspace,andtheeffectof
dislocationsinrealspace. PanelsAtoDshowtheelectronic-bandtopologyoftheR[T-p3(4) ]andM [T-p3(4) ]phaseson
R M
asimplecubiclatticeandtheM −R(p4 )andX(cid:48)−R(p4 )weakphasesontetragonallattices. Adislocationwith
M,R X(cid:48),R
Burgersvectorb=e actsontheencircledTRImomentaintheplanesorthogonaltothedislocationline. Asaresult,the
x
coloredplaneshostaneffectiveπflux. TheresultingnumberofKramerspairsofhelicalmodesalongtheedgeandscrewparts
oftheloopisindicatedwiththebluenumber. (A)ThesymmetricRphasehasatopologicallynontrivialplanehostingaπflux
orthogonaltoanyofthethreecrystallographicdirectionsandhenceanydislocationloopbindsmodesalongtheentirecore,as
shownforaloopinthexˆ−zˆplane,panelE.(B)IntheM phase,translationallyactivephasesintheTRIplanesorthogonalto
k andacrystallinephaseink =πplanehostπfluxes. Hencethedislocationloopbindstwopairsofmodes,asdisplayedin
z x
panelF.Thesemodesaresymmetry-protectedagainstmixing. (C)IntheM −Rphase,onlytheTRIplanesnormaltok host
z
aneffectiveπ-fluxandhencethesamedislocationloopbindsmodesonlytotheedge-dislocationparts,asdisplayedinpanelG.
Thesemodesarenotprotectedagainstmixing. (D)IntheX(cid:48)−RphaseallTRIplanesorthogonaltothedislocationslineshave
atrivialflux,and,accordingtotheK-b-trule,neithertheedgenorthescrewdislocationoftheloopbindsmodes,asillustrated
inpanelH.
hasnontrivialcomponentsonlyintheplaneorthogonaltothe of the propagating helical modes along the dislocation line.
dislocationline,consistentwithEq.(1),carryinganeffective For k ·ˆt = K ·ˆt, the system develops a Kramers pair
inv
fluxΦ=K ·b,aswedemonstratebelow. of true zero modes Ψ ≡ (ψ ,Tψ )(cid:62), with T as the time-
inv 0 0 0
Consider an edge and a screw dislocation both oriented reversal operator and T2 = −1 [10]. This is a consequence
along the z-axis. We use that for the edge dislocation with ofthefactthatbydefinitiontheK pointishostingaband
inv
Burgers vector b = ae , the dual basis in the tangent space inversion. Henceforth, the physics is essentially captured by
x
at the point r is Ex = (cid:0)1− ay (cid:1)ex + ax ey, Ey = aDiracHamiltonianwithnegativemass. Deviatingfromthis
2πr2 2πr2
ey, Ez = ez, whileforthescrewdislocationwithb = ae , ’parent’momentumbyq·ˆt,theeffectivelow-energyHamilto-
z
wehaveEx =ex, Ey =ey, Ez = b (−yex+xey)+ez; nianforthepropagatingmodesthengenerallydevelopsalin-
2πr2
r2 ≡ x2+y2 [21]. Thecorrespondingdistortionforbothan eargapH ∼q·ˆt,tolowestorder.ThegappedKramerspair
eff
edgeandascrewdislocationisthenreadilyobtainedtolead- ofdescendantstatesisthenpresentaslongasH (q)remains
eff
ingorderina/rtobeε = y b, ε = − x b(seeSup- topologicallynontrivial,andmaythenbecapturedbyH =
x 2πr2 y 2πr2 eff
plementalMaterialin[23]fordetails),andthecorresponding v Σ (q·ˆt)+O(q2).Here,thePaulimatrixΣ actsinthetwo-
t 3 3
gaugepotentialA(r)hasnontrivialcomponentsintheplane dimensionalHilbertspaceofthedislocationmodes,whichare
orthogonaltothedislocationline,A·t=0,and oftheformΨ ≡ (ψ ei(q·ˆt)(r·ˆt),(Tψ )e−i(q·ˆt)(r·ˆt))(cid:62),and
qt 0 0
v isthecharacteristicvelocity,whichissetbythesymmetries
t
−ye +xe −ye +xe
A= x y(K ·b)≡ x yΦ. (2) anddetailsofthebandstructure.
2πr2 inv 2πr2
We thus see how the dislocation line t, the Burgers vec-
WhenthisfluxΦ(mod2π)isnonzero,thedislocationscarry tor b, and the TRI momenta K , through the K-b-t rule,
inv
propagatinghelicalmodesprovidedthatthe2DHamiltonian conspire into a precise condition determining the existence
in a TRI plane orthogonal to the dislocation line, ˆt ≡ t/|t|, of the dislocation propagating modes in certain directions in
hosts a band inversion. The lattice symmetry that relates the a topologically-insulating phase, consistent with the space
band-inversion momenta then protects these modes. This is groupclassification[8](Fig. 1). Byvaryingtandbinalldi-
what we refer to as the K-b-t rule. This rule and the fol- rections,thenumberof”parent”zeromodesinanyprojection
lowingdescendantconstructiontogetherimplythatthebound plane is in one-to-one correspondence with the space group
statesforagivenk·ˆtmomentumcombineintothespectrum classificationintermsoftheK momenta. Specifically,for
inv
3
a(translationally-active)phasewithasingleK momentum, arejoinedinaloopinthex−z-planewithb=e ,weexpect
inv x
edgeandscrewdislocationsbindasingleKramerspairofhe- propagatingmodes,whichareindeedfoundinournumerical
licalmodesifK ·b(cid:54)=0.Incaseofa(translationally-active computations. Clearly,TRStogetherwiththecrystalsymme-
inv
or a crystalline) phase with multiple K momenta, a pro- tryprotectsthemodesfrombackscattering.
inv
jected2Dsystemmayalsoresultinadoublepairofmodesif Next we consider the system in the three-dimensional M-
the effective system entails a 2D crystalline phase. The two phase [T − p3(4) ], characterized by band-inversions lo-
M
pairs are then protected by the symmetries relating the K cated at the momenta M ≡ (π,π,0), X(cid:48) ≡ (π,0,π), Y(cid:48) ≡
inv
momenta. Mostinterestingly,thereisthepossibilitythatboth (0,π,π),whicharerelatedbythethreefoldrotationalsymme-
edgeandscrewdislocationsbindmodesinanycrystaldirec- try (Figs. 1B, and S5 and S10 in the Supplemental Material
tionresultinginpropagatingmodesalongthefulldislocation [23]). According to the K-b-t rule, for the edge-segment of
loops. Inparticular,inacompletelyisotropiclatticewithO the dislocation loop, k = 0 and k = π planes both host
h z z
crystal symmetry a strong variant of this effect can be real- a π flux, originating from the M and X(cid:48) points. Addition-
ized: thegaplessstatesinthedislocationloopthatpropagate ally, the k = π plane hosts a two-dimensional crystalline
x
in the way completely oblivious to the lattice directions. In phase,andthusthescrew-dislocationpartsalsohosttwopairs
contrast,inaweakphasecharacterizedbytheweaktopolog- ofmodes. Therefore,thereisatotaloftwoKramerspairsof
ical invariant M, protected helical modes are obtained only gaplessdislocationmodesalongtheloop,whichareprotected
when the product M · b (mod2π) is nontrivial [3, 17–19]. by symmetry. Note that their existence crucially depends on
This is consistent with the fact that double pairs of modes the fact that the M and X(cid:48) momenta are symmetry-related.
originating from two K not related by symmetry may be Were this not the case, the weak index of this T −p3(4)
inv M
gappedout. However,accordingtotheK-b-truleiftheK phase,M = (0,0,0),wouldpredictnodislocationmodesat
inv i
are related by symmetry such pairs are protected from gap- all.
pingout. Weexpectthatthestabilityofthedislocationmodes Incontrast,letusnowbreakthecubicsymmetrybyconsid-
in3DTBIsagainstatime-reversalinvariantdisorderisinone- ering the M −R-phase (p4 ) on a tetragonal lattice with
M,R
to-onecorrespondencewiththestabilityofthe2Dtopological a =a =a,a =a/α,wherea isthelatticeconstantinthe
x y z i
phase yielding the helical modes. Therefore, the most sta- direction e , and α (cid:54)= 1 is the lattice deformation parameter
i
ble dislocation modes should originate from a 2D Γ−phase, (Figs. 1Cand2). Noticeherealsoasubtledifferencebetween
followed by the ones arising from translationally-active and suchaweakphaseandacrystalline(valley)phase,whichhas
crystallinephases[8]. an even number of band-inversions protected by a 3D space
These general statements can be illustrated by a simple groupsymmetry. Theusualstrongandweakindices[3,4,17]
tight-bindingmodelwithtwoorbitalswithdifferentparityand inthisphaseare(ν;M ) = (0;0,0,1), andthusM·b = 0.
i
two spin states [24] on the simple cubic lattice with space Nonetheless, the k = 0 and k = απ planes are topolog-
z z
grouppm¯3m ically nontrivial therein (Fig. 1 and [23]). As a result, for a
dislocationloopwithb=e ,accordingtotheK-b-trule,we
H =A(γ sink +γ sink +γ sink )+M γ , (3) x
nn 1 x 2 y 3 z nn 0 findmodesboundonlytotheedge-dislocationparts. Asboth
withkastheelectronmomentum,theγ-matricesactinginthe theseplanescontributethemidgapstates,weexpectadouble
orbitalandthespinspaces,andweusenaturalunits((cid:126)=c= Kramerspairofthepropagatingmetallicstates,whichournu-
a=1).Topologicalphasesofthemodel(3)canbetunedwith mericalcomputationsindeedconfirm. However,thesemodes
themasstermM =m−2B(3−cosk −cosk −cosk ). canmixinthedislocationloop,sincenosymmetryrelatesthe
nn x y z
The parameters A(B) are related to the inter-(intra-)orbital momentaM andRgivingrisetothem.
hoppings, andmisrelatedtothedifferenceintheonsiteen- Finally,weconsiderthetetragonalX(cid:48)−R(Y(cid:48)−R)phase
ergies for different orbitals [23]. Let us assume that the sys- obtained by deforming the cubic lattice in the e (e ) di-
y x
tem is in an R-phase (labeled T-p3(4) in Ref. 8), which is rection with the corresponding band-inversions at (π,0,π)
R
characterizedbyaband-inversionattheRpoint[momentum [(0,π,π)]and(π,απ,π)[(απ,π,π)]momenta(Fig.1D).No
(π,π,π)]intheBZ,(Figs. 1Aand2). Ifb=e andt=e , dislocationmodesappearintheX(cid:48)−Rphase,whereasinthe
x z
the effective Hamiltonian in the k = π plane reduces to Y(cid:48)−Rphaseonlytheband-inversionatmomentum(απ,π,π)
z
theπ fluxprobleminthetwo-dimensionalM-phase(labeled contributes a π flux, thus yielding the modes for both types
T −p4 inRef.8),whichisdefinedbytheband-inversionat of dislocations. These results are therefore similar as in the
M
theM point,andhencepossesseszeromodes. Forthefamily R-phase on the cubic lattice, as also confirmed by numeri-
of Hamiltonians (1) then any momentum k = π + q in- calcomputationsshowingmodespropagatingalongtheentire
z z
finitesimallyclosetok =πyieldsthewould-bezeromodes loop in the x−z plane. However, the difference is that not
z
in the absence of the term ∼ γ . When included, this term alldirectionsareequivalenthere, andhencevelocitiesofthe
3
givesrisetothelinearlydispersingpropagatingmodesalong modesalongtheloopareanisotropic.
thedislocationline, accordingtotheaboveK-b-trule. Fur- TheresponseofdislocationsinTBIsisexperimentallycon-
thermore, the same rule implies that an edge or a screw dis- sequential as these defects are ubiquitous in any crystals.
location along any Bravais lattice vector in the simple cubic The electron-doped mixed-valent perovskite oxide BaBiO
3
latticeeffectivelyactsasaπ flux. Hence,whenthesedefects with a simple cubic crystal symmetry has been recently pre-
4
A M/B=10, z=0, R phase B M/B=10, z=0, R phase
1.5 (0,π+,π)" (π,π-,π)"
(0,0,π) 2π
1.0 + +
(π,0,π)"
"
0.5 " (π,π,0)"
+ "+
En 0.0 (0,+0,0) (0,π,0)("π,0+,0)"1" 0 1
-0.5
En
-1.0 0
0
-1.5
-1
-2.0 0 π k 2π
0 50 n-8230 100 150
C M/B=7, z=0, α=0.5, M-R phase D M/B=7, z=0, α=0.5, M-R phase
1.5 (0,π+,απ)" (π,π,-απ)"
1.0 (0,+0,απ) (""π,0,απ+)" 2π
0.5
+(0,π,0) ("π,π-,0)"
En0.0 (0,+0,0) ("π,+0,0)"" 1 0 1
-0.5
En
-1.0 0
-1.5 0
-1
0 π k 2π
-2.0
0 50 n-8230 100 150
FIG.2: (Coloronline)Theeffectofdislocationlinesin3Dtopologicalbandinsulators. Edgeandscrewdislocationswith
Burgersvectorb=e aretreatedsimultaneouslybyconsideringdislocationloopsof8×8siteswithinthetight-bindingmodel
x
(3)onthelatticewith16×16×16sitesincaseofperiodicboundaryconditions,see[23]fordetails. PanelsAandCshowthe
spectrumofthedislocationmodestraversingthegapincaseofthecubicRphase[T-p3(4) ]andtetragonalM −R(p4 )
R M,R
phases,withthecorrespondingelectronictopologicalconfigurationsshownintheinsetsontheupperleft. Thecirclesindicate
theTRImomentahostingband-inversionsthatyieldeffectiveπ-fluxes. Additionally,thespectraldensityasfunctionofthe
momentumkalongthedislocationlineisdisplayedintheinsetsonthelowerright. WefindasingleconefortheRphase,and
adoubleconefortheM −Rphase,consistentwiththeK-b-trule. Theenergylevelsarenoweightfolddegenerateasboth
k =0andk =πplanesaretopologicallynontrivial,andthuseachyieldspropagatingmodesduetotheeffectiveπflux
z z
introducedbythedislocations. Thezeromodesareoffsetduetoafinitesystemsize(seealsoFig.S11inRef.23). PanelsBand
Ddisplaytherealspacelocalizationofthemodes,wheretheweightofthewavefunctionisindicatedbythesizeofthecircles
andthecolorcodingindicatesthecorrespondingphase. Mostimportantly,thecubicRphasefeaturestopologically-protected
propagatingdislocationmodesalongthecompleteloop. Incontrast,theM −Rphasebindsmodesonlyalongthe
edge-dislocationparts. Inparticular,wefindeightenergylevelsincorrespondencewiththenumberofallowedmomentaalong
thedislocation.
dicted to be a TBI [25]. More interestingly, the topological these states, since the surface irregularities should not affect
phase is believed to be the symmetric R-phase [T-p3(4) ], thisprobe.
R
whichshouldhostpropagatinggaplessmodesalonganydis- We hope that our findings will motivate further studies of
location loop, as we have found here. The effective tight- theroleofdislocationsinTBIswithdifferentcrystalsymme-
binding model (3) describes this phase with the parameters tries,andespeciallytheirknottingwhichshouldplaytherole
A = 2.5eV·A˚,B = 9.0eV·A˚2, M = 5.08eV,andthelattice ofbraidingforthesetopologicaldefectsandmightberelevant
constant a = 4.35A˚ [25], implying that the velocity of the for quantum computation. Finally, our results based on the
dislocationmodesisv = A/(cid:126) = 4.3×105m/s,whilethelo- “intuitive”classificationofthedislocationsthroughtheBurg-
(cid:112)
calization length λ ∼ B/(12Ba−2−M) (cid:39) 4.8A˚ [26]. ers vectors ultimately call for a mathematically precise char-
The enhanced density of states near the dislocation line in acterizationofthesedefectsintermsofthehomotopyclasses
a non-Γ TBI [27–30] should be observable by local probes, forthediscretespacegroups.
suchasscanningtunnellingmicroscopy,wherethedislocation Note added. Recently, we became aware of the work by
reachesthecrystalsurface. Thesurfacestateshybridizewith ShiozakiandSato[31],wheretheresponseofdislocationsin
the dislocation modes, and the precise local redistribution of 3DTBIswasstudiedusingK−theory.
theincreaseddensityofstatescanbemodeledforthespecific ThisworkispartoftheD-ITPconsortium,aprogramofthe
experimentalsituationusingourtheory. Angle-resolvedpho- NetherlandsOrganisationforScientificResearch(NWO)that
toemissionspectroscopymayalsobeusefulformappingout is funded by the Dutch Ministry of Education, Culture and
5
Science (OCW). This work is supported by the Dutch Foun-
dationforFundamentalResearchonMatter(FOM).V.J.ac-
knowledges the support of NWO. The authors acknowledge
fruitfuldiscussionswithXiao-LiangQiandZhi-XunShen.
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6
Supplementary Material:
Interplay between electronic topology and crystal symmetry:
Dislocation-line modes in topological band-insulators
Robert-JanSlager,AndrejMesaros,VladimirJuricˇic´ andJanZaanen
A.Modeldetails
Weherepresentthedetailsofthemodelthatweusetoobtaintopologicalstatesoncubicandtetragonallattices. Thetight-
bindingmodelcontainstwospindegenerateorbitalswithaHamiltonian
(cid:88) (cid:88)
H =(cid:15)(k)1+ d (k)γ + d γ , (S1)
α α αβ αβ
α αβ
where γ are the five Dirac matrices obeying the Clifford algebra {γ ,γ } = 2δ , γ are the ten commutators γ =
α α β αβ αβ αβ
1[γ ,γ ]. Specifically,wetakethefollowingbasisoftheγ-matrices
2i α β
γ =σ ⊗τ γ =σ ⊗τ , γ =σ ⊗τ , γ =σ ⊗τ , γ ≡−γ γ γ γ =σ ⊗τ ,
0 0 3 1 1 1 2 2 1 3 3 1 5 0 1 2 3 0 2
withσandτ beingthestandardPaulimatricesactinginthespinandorbitalspace,respectively;1=σ ⊗τ withσ ,τ asthe
0 0 0 0
2×2unitymatrices.
TheexplicitformoftheHamiltonian(S1)isthendeterminedbyexploitingthesymmetries. Forthetight-bindingmodelwe
assume a well-defined parity, in addition to time-reversal symmetry. Time-reversal symmetry is represented by the operator
T = ϑK, with ϑ = iσ ⊗ τ and K as complex conjugation, while the parity operator is P = γ . As the commutators
2 0 0
γ transform under time-reversal and parity with the opposite sign, the assumption of having both these symmetries thus
αβ
results in d = 0 in Eq. (S1). The remaining functions d (k) of the effective model can then be obtained from the theory
αβ α
of invariants. Let us consider the space group pm¯3m with the point group O . This point group has two one-dimensional,
h
onetwo-dimensionalandtwothree-dimensionalirreduciblerepresentations. Thematrixγ anticommuteswiththeγ-matrices
0
{γ ,γ ,γ } and therefore represents the mass term, which is by construction even under the parity. The corresponding mass
1 2 3
term has to be rotationally-symmetric and therefore an even polynomial in the momentum k. Using also that the γ-matrices
{γ ,γ ,γ } form a three-dimensional representation under rotations in O , we arrive at the following minimal continuum
1 2 3 h
Hamiltonianthatcapturestopologicallynontrivialphases
H =A(k γ +k γ +k γ )+M γ +O(k3), (S2)
eff x 1 y 2 z 3 nn 0
withM =m−2B(k2+k2+k2)andwedroppedtheterm∼1notrelevantforthetopologicalanalysishere.ThisHamiltonian
nn x y z
generalizes the 2D Bernevig-Hughes-Zhang (BHZ) to three dimensions (3D) and its different incarnations have already been
usedtodescribetopologicalstatesinBismuth-basedcompounds[S1].
Letussubsequentlydeterminethelattice-regularizedversionoftheeffectivemodel(S2). Consideringasimplecubiclattice
withnearest-neighbor(nn)hoppings,weobtainthefollowingtight-bindingHamiltonian
H =A(γ sink +γ sink +γ sink )+M γ , (S3)
nn 1 x 2 y 3 z nn 0
withtheeffectivemassparameter
M =m−2B(3−cosk −cosk −cosk ).
nn x y z
We set the lattice constant a = 1, and the parameters A (B) represent hopping amplitudes between different (same) orbitals,
whilemisthedifferenceoftheonsiteenergiesbetweenthetwoorbitals, similarasinthetwo-dimensional(2D)BHZmodel.
The hoppings in the Hamiltonian (S3) are equal in the three orthogonal directions due to the rotational symmetry. The above
HamiltonianreducestothecontinuumHamiltonian(S2)whenexpandedaroundtheΓortheR-pointlocatedatthemomentum
(π,π,π)intheBrillouinzone(BZ).Additionally,inordertoenrichthephasediagram,weaddnext-nearest-neighborhopping
termsintheexactsamefashionasthenearest-neighborones. Noticethatonacubeeachsitehasfournext-nearestineachofthe
7
threemutuallyorthogonalcrystallographicplanesresultingintheHamiltonian
A˜
H = [sin(k +k )(γ +γ )+sin(−k +k )(−γ +γ )]
nnn 2 x y 1 2 x y 1 2
A˜
+ [sin(k +k )(γ +γ )+sin(−k +k )(−γ +γ )]
2 x z 1 3 x z 1 3
A˜
+ [sin(k +k )(γ +γ )+sin(−k +k )(−γ +γ )]
2 y z 2 3 y z 2 3
−4B˜[3−cos(k )cos(k )−cos(k )cos(k )−cos(k )cos(k )]γ , (S5)
x y x z y z 5
sothetotalHamiltonianofthetight-bindingmodelonthecubiclatticeis
H =H +H (S6)
TB nn nnn
Different topological phases are obtained by varying the mass term multiplying γ matrix. The topological phase transitions
5
occur when the mass term vanishes at the time-reversal invariant momenta in the BZ and the corresponding phases can be
characterizedbythemasstermatthesespecialmomenta[S2]. Asaresultthephasediagram,withA=1,isreadilyobtainedas
afunctionofM/BandB˜/B,seeFig. S1.
ByloweringthefullrotationalsymmetryoftheHamiltonian(S3),oneobtainsaminimaltight-bindingHamiltoniandescribing
topologicalphasesonatetragonallattice
Htetragonal =A γ sink +A γ sink +A γ sink
nn x 1 x y 2 y z 3 z
+[m−2B (1−cosk )−2B (1−cosk )−2B (1−cosk )]γ , (S7)
x x y y z z 5
whichwewilluselater.
FIG.3: FigureS1: Thephasediagramofthetight-bindingmodel(S6),withspacegrouppm¯3mandO point-groupsymmetry.
h
AsfunctionoftheparameterM/BandthenearestneighborhoppingB˜/Bthemassparameteratthetime-reversalinvariant
momentacanbetunedtoproducethedifferentelectronictopologicalconfigurationsasshowninthefigure.
8
B.Analyticalandnumericaldescriptionofthedislocationmodesinthree-dimensionaltopologicalband-insulators
Letusnowconsidertheeffectofdislocationsinthecoarse-grainedcontinuumtheoryobtainedfromthetight-bindingmodel
introduced in the previous section. Such lattice defects are described within the elastic continuum theory using vielbeins,
encodingthemapfromtheperfectlatticetothedistortedlattice[S3]. ThetorsionTi andcurvatureRi arethenrelatedtothe
j
vielbeinsEi andthespinconnectionωi bytheEinstein-Cartanstructureequations
α j
Ti =dEi+ωi ∧Ej
j
Ri =dωi +ωi ∧ωk. (S8)
j j k j
Foradislocationdefect,thecurvaturevanisheswhilethetorsionissingular,Ti = biδ(l)withbastheBurgersvectorandlas
thepositionofthedislocationline. Sincecurvaturetensorvanishesforadislocation,thecorrespondingspin-connectioncanbe
settozero.
B.1.Edgedislocations
Incaseofanedgedislocationorientedalongthezˆ-direction,thevielbeintakestheform
1− by bx 0
2πr2 2πr2
Eˆ ≡Eαi = 0 1 0, (S8)
0 0 1
wherebisthemagnitudeoftheBurgersvectorb,whichisassumedtobealongthexˆ-directionb=be ,andr2 ≡x2+y2. The
x
inverseoftheabovevielbeinisreadilyfoundtobe
(cid:16) (cid:17)−1 (cid:16) (cid:17)−1 
1− by − bx 1− by 0
Eˆ−1 ≡Eα = 2πr2 2πr2 2πr2 . (S9)
i  0 1 0
0 0 1
Sinceforanelementarydislocationb=awithaasthelatticeconstant,thedistortionfieldE =e +ε totheleadingorderin
i i i
a/risthenreadilyobtained
ay y ax x
ε = e = b ε =− e =− b.
x 2πr2 x 2πr2 y 2πr2 x 2πr2
ThecorrespondinggaugepotentialA=−ε ·K ,withK astheband-inversionmomentum,canbethenstraightforwardly
i inv inv
obtainedandisgivenbyEq.(2)inthemaintext.
We can now consider the effect of an edge dislocation in a 3D topological insulator. We note that along the core of the
dislocation translational symmetry is preserved. Henceforth, k is a good quantum number and we obtain a family of two-
z
dimensionalHamiltonians
(cid:88)
H(x,y,z)= eikzzH (x,y,k ).
eff z
kz
Thedimensionally-reducedHamiltonianH (x,y,k ))canthenbetreatedusingwell-knownmethods[S2,S4].Hence,wecan
eff z
determinethespectrumofdislocationmodesdirectly.
WenowmakethismoreconcreteforthemodelwiththeHamiltonian(S6). WeassumethatthesystemisintheT-p3(4) or
R
Rphase[S6]asshowninFig. S2. Usingk→k+A=k+ 1 e ,straightforwardcalculationyields
2r ϕ
∂ 1 i 1
H (x,y,k ))=iγ ∂ +iγ ( r + )+γ [M −2B((cid:52)+ ∂ϕ− )]+sin(k )γ , (S10)
eff z r r ϕ r 2r 0 r2 4r2 z 3
where
M =m−8B−2B(1−cosk )=Mˆ −2B(1−cosk ).
z z
Here, the Laplacian (cid:52) = ∂2 + ∂r + 1 ∂2, and (r,ϕ) are the usual polar coordinates related to the Cartesian ones by x =
r r r2 ϕ
rcosϕ, y = rsinϕ. To analyze the propagating midgap modes bound to the dislocation, let us first neglect the last term in
9
theHamiltonian∼sink . IntheremainingHamiltonianthetermM playstheroleoftheeffectivemassinthetwo-dimensional
z
BHZmodel. Therefore,thisHamiltonianhostszeromodesforthevaluesoftheparametersm,B,k obeying0 < M/B < 4.
z
Now, take k = π. Then, as 8 < m/B < 12, it follows that 4 < Mˆ/B < 8, which is precisely the condition for a zero
z
modesolutionboundtothedislocation. ThesolutionofEq.(S10)withoutthelastterm∼ γ hasbeenderivedanddiscussed
3
indetailinRefs. [S2,S4]. Morespecifically,inthisparameterrangethesystementailstheT-p4orM phaseinthereduced2D
Hamiltonian.Similarly,thenextk valueinafinitesystemresultsinadislocationmodeiftheeffectivemassMˆ(k )stillsatisfies
z z
the topological condition 4 < Mˆ/B < 8. Moreover, we can exactly determine the corresponding spectrum by reintroducing
thelasttermintheHamiltonian(S10),∆ ≡ sin(k )γ . Sincek commuteswiththeHamiltonianandγ anticommuteswith
3 z 3 z 3
theotherγ-matricesintheHamiltonian,theterm∆ actsasamasstermforthemodescomprisingzero-energysubspaceinits
3
absence.Thecorrespondinggapisthus±|sin(k )|withrespecttothezeroenergy,andinthethermodynamiclimitthesemidgap
z
modesarelinearlydispersingalongthedislocationlinewiththevelocitysetbythehoppinginthisdirection,inagreementwith
the K-b-t rule. These conclusions are schematically shown in Fig. S2. By displaying the allowed values of k in a finite
z
system,k = 2πn/L,withL ∈ Zandn = 0,1,2...L−1,ontheunitcircleintheimaginaryplane,therealpartcharacterizes
z
the effective Mˆ/B, while the imaginary part signals the anticipated energy of the dislocation mode. We note that the shift in
Mˆ indeed ensures that the T-p3(4) phase hosts no dislocation modes with k = 0 and in the thermodynamic limit all the
R z
propagatingmodesindeeddescendfromthezeromodesattheband-inversionmomentum(π,π,π),asexpectedfromtheK-b-t
rule.
A B
(0,π,π)" (π,π,π)"
ΔE
+ -
+
(0,0,π) + (π,0,π)"
"
"
ΔM
4 2 0
+ (π,π,0)"
+"
(0,π,0)
"
+
+
(0,0,0)
(π,0,0)"
"
FIG.4: FigureS2: TheleftpanelshowstheschematicconfigurationoftheT-p3(4) orRphaseintheBrillouinzonewithan
R
edgedislocationwithBurgersvectorb=e andisorientedalongthezˆdirectionintherealspace. Therightpaneldisplaysthe
x
modesk ontheunitcircleinthecomplexplane. Theimaginarypartthenencodesforthemassgapofthedislocationmode
z
withrespecttothezeromode,whereastherealpartrelatestotheshift∆M(k )=−M/B+Mˆ/B =2(1−cos(k ))ofthe
z z
effectivemassparameterintheassociated2DBHZmodel.
Theaboveanalyticalresultscaneasilybeverifiedbynumericalcomputations,andthetwoshowanexcellentagreement. In
Fig. S3weshowthespectrumofa12×12×12systemonatorus(periodicboundaryconditions)andtherealspacelocalization
ofthedislocationmodes. WenotethatthemodesareneatlylocalizedandcomeastwoKramerspairs,onefromthedislocation
andonefromtheanti-dislocation. Moreover,sinceexp(ik )takesvaluesinthesetcomprisingofthetwelfthrootsofunity,we
z
cancomparetheenergygapswiththeanticipatedsin(k )dependenceandfindagreementuptotwodecimals. Inaddition,we
z
changethemassparametermtoconfirmthatmodescanbeaddedorremovedfromthespectrumconsistentwiththecondition
inEq.(S10),seeFig. S4. Theseresultsreproduceforvarioussizesofthesystem(rangingupto16sitesinlength),confirming
the presence of the zero mode together with the descendant propagating states, as predicted by the K-b-t rule. Furthermore,
whenthenext-nearestneighborhoppingsareincluded,thephaseT −p4 protectedbybothTRSandcrystalsymmetrieswith
M
thebandinversionsatthemomenta(0,π,π),(π,0,π),and(π,π,0)canberealized,seeFig.S1. Inthatcase,adislocationwith
Burgers vector b = ae and oriented along the z axis,ˆt = e , in both planes k = 0 and k = π acts as π-flux, and thus a
x z z z
doubleKramerspairisexpected. Thisisindeedwhatwefindnumerically,seeFig. S5.
Itisstraightforwardtogeneralizetheexplainedreasoningtootherphases. Letusillustratethiswiththeprimitivetetragonal
systemtakingA =A = 1A andB =B = 1B inEq.(S7). Fortheseparametersweobtaintheweakp4 phasewith
x y 2 z x y 2 z M,R
band-inversionsatM andRpoints. ThisphaseisdisplayedinFig. S6,andisnotprotectedbyeithertime-reversalsymmetry
10
FIG.5: FigureS3: Thespectrumofthe12×12×12system(M/B =10)intheT-p3(4) phasewiththeedgedislocation
R
withperiodicboundaryconditions. ThedislocationmodescomeindegenerateKramerspairs,originatingfromthedislocation
andanti-dislocation(panelA).Theinsetdisplaysthecompletespectrum. Wenotethattheenergylevelsofthesemodesinthe
gaparelocatedatE =0,±E =0.50andat±E =0.86,inagreementwiththetheory. PanelBshowstherealspace
n N N
localizationofthedislocationmodesforafixede plane. Theweightofthewavefunctionisrepresentedherebytheradiusof
z
thecircles,whereasthecolorindicatesthephasefollowingthesameconventionsasinFig2inthemaintext. Duetothe
translationalsymmetry,thee planesareidentical. Finally,panelCshowsthethespectraldensityasfunctionofthemomentum
z
k =k alongthedislocationlinefora12×12×28system. Thisspectraldensityisalsoplottedascircles,wheretheradius
z
indicatestheweight,inordertofurtheremphasizetheexcellentagreementwithnumerics.
or3Dspacegroupsymmetry, asopposedtoavalleyphase. Byinsertinganedgedislocationwithb = e andˆt = e wesee
x z
that both time-reversal invariant planes orthogonal to ˆt are topologically nontrivial and thus yield zero modes, which in turn
producedescendantpropagatingmodesalongthedislocation. Wefindthepropagatingmodesnumericallyforthesystemswith
10×10×10anda12×12×12sites,seeFig.S6,andwiththeplotofthespectralweightofthedislocationmodesshowninFig.
S7. NoticethedoublingoftheDiracconesinceinthisphasetherearetwonontrivialtopologicallynontrivialplanesorthogonal
to the dislocation line each yielding a π flux. In turn, this nontrivial flux produces a double Kramers pair of the zero modes,
whichyieldsdescendentpropagatingstatesalongthedislocationdefect.