A topological look at the quantum spin Hall state

2.7 MB·English

by  Huichao Li

A topological look at the quantum spin Hall state
Huichao Li, L. Sheng, and D. Y. Xing
∗
National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
We propose a topological understanding of the quantumspin Hall state without considering any
symmetries,anditfollowsfromthegaugeinvariancethateithertheenergygaporthespinspectrum
gapneedstocloseonthesystemedges,theformerscenariogenerallyresultingincounterpropagating
gaplessedgestates. BasedupontheKane-Melemodelwithauniformexchangefieldandasublattice
staggered confining potential near the sample boundaries, we demonstrate the existence of such
2
gapless edge states and their robust properties in the presence of impurities. These gapless edge
1
states are protected bythe band topology alone, rather than any symmetries.
0
2
PACSnumbers: 72.25.-b,73.20.At,73.22.-f,73.43.-f

n
a
J Since the remarkable discovery of the quantum Hall TRS [15], are present. When the TRS is broken, it was

9 effect (QHE) [1], the study of edge state physics has found [10] that a small gap appears in the spectrum of
attracted much attention on both theoretical and ex- the edge states, which was obtained for a ribbon geome-
]
l perimental sides. Recently, a new class of topological tryunderidealboundaries,i.e.,boundariescreatedbyan
l
a states of matter has emerged, called the quantum spin infinitehard-wallconfiningpotential. However,sincethe
h Hall (QSH) states [2, 3]. A QSH state of matter has a edge states are localized around the sample boundaries,
-
bulk energy gap separating the valence and conduction theycanbesensitivetothevariationofon-sitepotentials
s
e bands and a pair of spin-filtered gapless edge states on near the boundaries [16].
m
theboundary. TheQSHeffectwasfirstpredictedintwo- Inthis Letter,inorderto revealthegeneralcharacter-
. dimensional (2D) models [2, 3], and was experimentally istics of the edge states and their connection to the bulk
t
a confirmed soon after in mercury telluride quantum wells topologicalinvariantinaQSHsystem,wepresentatopo-
m
[4]. TheQSHsystemsare2Dtopologicalinsulators[5,6] logicalargument similar to the Laughlin’s Gedanken ex-
- protected by the time-reversal symmetry (TRS), whose periment without considering any symmetries. We show
d
n edgestatesarerobustagainstperturbationssuchasnon- that, as required by the nontrivial band topology and
o magnetic disorder. gauge invariance,either the energy gapor the spin spec-
c A simple model of the QSH systems is the Kane-Mele trum gap (the gap in the spectrum of Pσ P) needs to
z
[
model [2], defined on a honeycomb lattice, first intro- closeontheedgesofaQSHsystem. Thesetwoscenarios

1 duced for graphene with spin-orbit couplings (SOCs). It will lead to gapless or gapped edge modes, respectively.
v was suggested [2] that the intrinsic SOC in graphene In particular,it is demonstratedthatgapless edge states
0
would open a band gap in the bulk and also establish can appear in a TRS-broken QSH system by tuning the
9
spin-filtered edge states that traverse the band gap, giv- confining potential at the boundaries. They are associ-
6
1 ingrisetotheQSHeffect. EventhoughtheintrinsicSOC ated with the bulk topological invariant, and are robust
. strengthin pure grapheneis too smallto produce anob- against relatively smooth impurity scattering potential.
1
servable effect under realistic conditions [7], the Kane- Ourresultoffersaninterestingexampleforcounterpropa-
0
2 Mele model captures the essential physics of the QSH gatinggaplessedgestatesthatarenotprotectedbysym-
1 state with nontrivial band topology [8, 9]. In the pres- metries, which sheds light on the underlying mechanism
: enceoftheRashbaSOCandanexchangefield,theKane- of the QSH effect in a broad sense.
v
i MelemodelentersaTRS-brokenQSHphase[10]charac- Let us first look back on a looped ribbon of the QHE
X
terized by nonzero spin Chern numbers [11, 12]. Prodan system,withamagnetic fluxφ (inunits offlux quantum
r proved [12] that the spin-Chern numbers are topological hc/e)threadingtheringadiabatically[17–19]. TheFermi
a
invariants, as long as the energy gap and the spectrum energy E is assumed to lie in an energy gap. In the
F
gapoftheprojectedspinoperatorPσzP stayopeninthe spirit of the Laughlin’s argument, increasing φ from 0
bulk, where P is the projection operator onto the sub- to 1 effectively pumps one occupied state from one edge
space of the occupied bands and σz the Pauli matrix for to the other, giving rise to the transfer of one charge
theelectronspin. UnliketheZ2invariant[13],therobust betweenthe edges,essentiallybecause there is a nonzero
properties of the spin-Chern numbers remainunchanged Chern number (Hall conductivity) in the bulk. On the
when the TRS is broken [10, 12]. other hand, the system Hamiltonian is gauge invariant
The existence of counterpropagating edge states with under integer flux changes, i.e., if φ is increased from 0
oppositespinpolarizationsisanimportantcharacteristic to 1, the system will reproduce the same eigenstates as
of the QSH state. It is believed that the edge states can at φ = 0. To assure this gauge invariance, there must
be gapless only if the TRS [2] or other symmetries, such be gaplessedge modes onthe edges (when the perimeter
as the inversion symmetry [14] or charge-conjugation of the ring is large), so that the spectral flow can form
2
a closed loop, as illustrated in Fig. 1a, along which the the valence (or conduction) band, as shown in Fig. 1c.
electron states can continuously move with changing φ. In this case, the spin spectrum gap must vanish on the
edges, but the energy gap may remain open in the edge
state spectrum.
(a)
EF (a) (b)
(b) s
e abrupt potential
g
Conduction bands d smooth potential
e
air
Valence bands h
c
m
(c) ar
Conduction bands
Valence bands zigzag edges
Left edge Right edge
FIG. 2: (a) A schematic of an armchair honeycomb lattice
ribbon with atom sites in two sublattices being labeled by
FIG. 1: (a) A schematic of the flow of electron states in a
solid dots and hollow dots, where a is the distance between
loopedribbonoftheQHEsystem,withadiabaticallyincreas-
ingthemagnetic fluxφthatthreadsthering. Thebulkelec- nearest-neighbor sites. (b) Profiles of |Vi| as functions of xi,
tron states below Fermi energy EF drift from left to right, foranabruptconfiningpotential(dashedline)andarelatively
gapless edge modes ascend through EF on the right edge, smooth confining potential (solid line).
states above EF drift from right to left, and gapless edge
modesdescendthroughEF ontheleftedge,formingaclosed TheabovetopologicaldiscussionontheQSHsystemis
loop. (b) In the first scenario for the QSH system, gapless verygeneral,independentofanysymmetries. Todemon-
edge modes appear on theedges of thering, so that electron
stratethetwoscenariosinFigs.1band1c,inwhatfollows
statesineachspinsectorbehavelikeintheQHEsystem,but
wetaketheKane-Melemodel[2]forahoneycomblattice
those in two spin sectors move in opposite directions. (c) In
ribbon as an example, by taking into account different
the second scenario for the QSH system, the edge states are
gapped, whereas the spin spectrum gap closes on the edges. confining potentials near the edges of the ribbon. It was
In the bulk, electron states in the two spin sectors drift in shownthatinasuitableparameterrange,theKane-Mele
the same way as in (b), but on the edges they join together systemisintheQSHphaseprotectedbytheTRS,andit
within thevalence (conduction) band. canbecome a TRS-brokenQSH phase [10], when a spin-
splitting exchange field is applied. Consider an armchair
We now propose a topological understanding of the ribbon along the y direction, as shown in Fig. 2a, in-
QSH system in terms of the same looped ribbon geom- cluding Nx dimer lines across the ribbon. (Results for a
etry. The occupied valence band can be decomposed zigzagribbonaresimilar.) The boundariesareatx1 =0
into two spin sectors by using the projected spin oper- and x = √3(N 1), where the distance between
Nx 2 x −
ator [10, 12]. (The unoccupied conduction band can be nearest-neighborsitesischosenastheunitoflength. The
dividedsimilarly.) Thetwospinsectorsareseparatedby Hamiltonian can be written as H = H +H +H
KM E C
anonzerospinspectrumgapinthebulk. Theycarryop- with
positespinChernnumbers,sothatincreasingφpumpsa
2i
stateofthespinupsectorintheoccupiedbandfromone HKM = −t c†icj + √3VSO c†iσ·(dkj ×dik)cj
edge to the other, and pumps another state of the spin Xij Xij
h i hh ii
dsyoswtenmsetcotroercoinvetrhteheopinpiotisailteeigdeirnescttaitoens.asInφochrdaenrgefosrfrtohme + iVR c†ieˆz·(σ×dij)cj, (1)
Xij
0to 1,the spectralflowneedsto formclosedloops,simi- h i
larly to the QHE system. However,for the QSH system, as the Hamiltonian of the Kane-Mele model. Here, the
if not enforced by any symmetry, two different scenarios firsttermisthenearest-neighborhoppingtermwithc†i =
can occur on the edges. One is that gapless edge modes (c†i ,c†i ) as the electron creation operator on site i and
appearontheedges,sothatstatescanmovebetweenthe the↑ang↓ularbracketin i,j standingfornearest-neighbor
h i
conduction and valence bands with changing φ to form sites. ThesecondtermistheintrinsicSOCwithcoupling
closed loops in the spin-up and spin-down sectors sepa- strengthV ,whereσ arethePaulimatrices,iandj are
SO
rately,asshowninFig.1b. Inthiscase,thestatesinthe twonextnearestneighborsites,kistheiruniquecommon
two loops cannot evolve into each other due to the non- nearestneighbor,andvector d points fromk to i. The
ik
vanishingspinspectrumgapbothinthe bulkandonthe third term is the Rashba SOC with coupling strength
edges. The other scenario is that a closed loop of spec- VR. HE = g ic†iσzci stands for a uniform exchange
tral flow is formed between the two spin sectors within field of strengtPh g. HC represents a sublattice staggered
3
confining potential, which is given by HC = iVic†ici fixed and two different decay lengths, is plotted in Fig.
with P 3a and Fig. 3b. One can see easily that the edge states
appear as thin lines in the middle bulk band gap of the
Vi = V0 e−xξi +e−xNxξ−xi , (2) energy spectrum. In Figs. 3a and 3b, the spin polariza-
± (cid:18) (cid:19)
tionofthe edgestatesis labeledwith and ,indicating
↑ ↓
where is taken to be positive for sites on sublattice A that two spin-filtered channels on each edge flow along
(solid d±ots) and negative on sublattice B (hollow dots), opposite directions. For the nearly hard-wall confining
asshownintheFig.2a. InEq.(2),Viisstronglyxdepen- potentialofξ =0.1,thesublatticepotentialVi isnonzero
dentacrosstheribbon,equalto V0 attheedges(xi =0 only on the outermost armchair lines, similar to the as-
and x = x ). It decays expo±nentially away from the sumption in Ref. [16]. In this case, two small energy
i Nx
edges, with a characteristic length ξ, as shown in Fig. gaps are observed in the edge state spectrum shown in
2b. When the ribbon width is much greater than ξ, V Fig. 3a, in agreementwith the previousobservation[10],
i
essentially vanishes in the middle region of the ribbon. asaconsequenceofthebrokenTRS.Withincreasingthe
Here, we note that in the case of a uniform staggered decaylengthξ oftheconfiningpotential,theenergygaps
potential Vi = V0 (Vi being independent ofxi), itwas of the edge states become smaller and smaller. As ξ is
shown[10] that±with|inc|reasing V , there is a transition large enough, e.g., ξ = 4 in Fig. 3b, interestingly, the
i
from the TRS-broken QSH pha|se|to an ordinary insu- edge states become gapless.
lator state, where the middle band gap closes and then We now calculate the ky-dependent spectrum of pro-
reopens. Therefore, for the confining potential Vi given jected spin operator PσzP, whose matrix elements are
by Eq. (2) with large V0, the ribbon in Fig. 2 can be givenby ϕm(ky) σz ϕn(ky) withmandnrunningover
h | | i
regarded as a TRS-broken QSH ribbon sandwiched in alltheoccupiedstates. Bydiagonalizingthismatrix,the
between two trivia l band insulators. spectrum of the projected spin PσzP can be obtained.
For the Kane-Mele model Eq. (1), if V = 0, σ com-
R z
mutes with the Hamiltonian H. Therefore, σ is a good
0.2 (a) (b) z
y (t)0.0 R↓ L↑ R↑ L↓ R↓ L↑ R↑ L↓ qcounasnitsutsmonfujumsbtetrw.oItvfaolluloewss t1h,awththicehspareecthruigmhlyofdPegσeznP-
nerg-0.2 x=0.1 x=4.0 erate. When the Rashba te±rm is turned on, σz and H
E
no longer commute, and the degeneracy is lifted. In this
-0.4
case, the spectra of Pσ P between +1 and 1 spread
z
−
1.0 towards the origin, but a gap remains for a bulk sample
m
ctru0.5 (c) (d) itfhrtehsehoalmdp[1li2tu].de of the Rashba term dose not exceed a
pe0.0 x=0.1 x=4.0
s Fortheribbongeometry,thesituationismorecompli-
n -0.5
pi cated due to the existence of the edges, and numerical
S
-1.0 calculations are performed to obtain the spin spectrum.
-0.4 -0.2 0.0 0.2 0.4 -0.2 0.0 0.2 0.4 The calculated spectrum of Pσ P for the same param-
3k (π) 3k (π) z
y y eters as those in Figs. 3a and 3b is shown in Figs. 3c
and 3d, which exhibits very interesting behavior. For
FIG. 3: (color online) The energy spectrum (a, b) and the the hard-wall confining potential with ξ =0.1, while the
spectrum of the projected spin PσzP (c, d) of an armchair energy spectrum of edge states are slightly gapped, with
ribbon for ξ = 0.1 (a, c) and ξ = 4 (b, d), in which L (R)
increasingk thespectrumofPσ P continuouslychange
standsforstatesontheleft(right)edge,and↑(↓)fortheup y z
between+1to 1withoutshowinganygap,correspond-
(down) spin polarization. The horizontal arrows in (a) point
−
tothesmall energy gaps intheenergyspectrum. Atξ=0.1, ing to the second scenario shown in Fig. 1c. On the
while the energy spectrum (a) is gapped, the spin spectrum other hand, for a relatively smooth confining potential
(c) is gapless. At ξ = 4, the energy spectrum (b) is gapless, with ξ =4.0, the energy spectrum of the edge states are
but thespin spectrum (d) is gapped. gapless, but the spectrum of Pσ P displays a big gap,
z
and the sudden changes happen at the cross points in
In order to assure the system in the TRS-brokenQSH the energyspectrumofthe edge states,correspondingto
state, we set the parameters V = 0.1t, V = 0.1t, thefirstscenarioshowninFig.1b. FromFig.3,itfollows
so R
and g = 0.1t. The length of the armchair ribbon N is that as long as the system is in the QSH state, a gapless
y
taken to be infinite. The energy spectrum of the ribbon, characteristic always appears either in the energy spec-
together with the corresponding eigenfunctions ϕm(ky), trum of edge states or in the spectrum of PσzP, leading
canbe numericallyobtainedbydiagonalizingthe Hamil- to the two types of closed loops for the continuous flow
tonian for each momentum k in the y direction. The of the electron states illustrated in Figs. (1b, 1c).
y
calculated energy spectrum of the armchair ribbon with The result shown in Figs. 3b and 3d, for the relatively
widthNx =240,fortheconfiningpotentialwithV0 =12t smooth confining potential, is of particular interest. It
4
indicates that gapless edge states can exist in the TRS- dentofl0. Figure4showstheevolutionofthecalculated
broken QSH system, accompanied with a gapped spec- eigenenergiesof a 120 60 systemin the band gapupon
×
trum of Pσ P. Such an interesting behavior can be fur- adiabaticinsertionofa magneticflux φinto the ring,for
z
ther understood by the following argument. As long as three different impurity scattering potentials. The num-
thebulkenergygapdoesnotclose,theprojectedspinop- ber concentration of the impurities is fixed at 1%. For
eratorPσzP is exponentiallylocalizedinrealspacewith a very short correlation length l0 = 1, for which the im-
a characteristic length about λ ¯hv /∆ , where v is purity potential is nearly uncorrelated from one site to
F E F
∼
theFermivelocityand∆E isthemagnitudeoftheenergy another, we see from Fig. 4a that at U0 = t, the energy
gap. [12] For the parameter set used in Fig. 3, λ is esti- levelsofthe edge statesavoidto crosseachotherasthey
mated to be between 1 to 2 lattice constants. When the move close, resulting in small energy gaps in the spec-
confiningpotentialV isvaryingrelativelyslowlyinspace, trum, as indicated by the arrows. This level repulsion
i
i.e., ξ λ, one can find that Pσ P roughly commutes behavior is a signature of the onset of backwardscatter-
z
≫
with the confining potential. In this case, the confining ing [11]. When the characteristic length l0 is increased
potential is of no influence on the spectrum of PσzP. to l0 =3 with U0 =t fixed, correspondingto a relatively
Since the spin spectrum has a gap in the bulk [10], this smoothimpurityscatteringpotential,alltheenergygaps
gap remains to open on the smooth edges, as seen from vanish, as shown in Fig. 4b. The energy levels move in
Fig. 3d. As a result, the energy gap has to close due straight lines and continue to cross each other, a clear
to the topological requirement, resulting in gapless edge indication of quenching of the backward scattering [11].
modes, as obs erved in Fig. 3b . SuchalevelcrossingfeatureisintactwhenU0isincreased
up to 3t for fixed l0 = 3, as shown in Fig. 4c. We thus
(a) conclude that the edge states remain to be robust in the
(b) (c)
presence of relatively smooth impurity scattering poten-
0.1 0.2
tial of intermediate strength. This result can be under-
l= 1.0 stoodbaseduponanargumentsimilartothatinthepure
0 l= 3.0
y (t) U0= t l0= 3.0 U00= 3t ocafsteh.eWprhoejneclt0edissgprienatoepretrhaatonrtPheσchPa,rtahceteirmisptiucrlietyngstchatλ-
erg0.0 U0= t 0.1 tering potential nearly commutes wz ith Pσ P, and hence
n z
E does not affect much the spin spectrum gap, so that the
energy gap needs to close on the edges, which explains
the level crossing behavior of the edge modes.
-0.1 0.0
In summary, based upon a general topological argu-
ment without relying on the TRS or other symmetries,
0.0 0.5 0.0 0.5 1.0 0.0 0.5 1.0 we show that in a QSH system either the energy gap or
Magnetic flux φ
the gap in the spectrum of PσzP needs to close on the
edges. WefindthataTRS-brokenQSHsystemcanhave
FIG. 4: (color online) Eigenenergies of the edge states as a either gapless or gapped edge states, depending on the
functionofthemagneticfluxφthreadingtheloopedgeometry propertiesoftheconfiningpotentialneartheboundaries.
withsize120×60for(a)l0 =1.0,U0 =t,(b)l0 =3.0,U0 =t, The gapless edge states are protected by the bulk topo-
and (c) l0 = 3.0, U0 = 3t, where the impurity concentration
logical invariant rather than any symmetries, which can
is fixed at 1%. The other parameters are taken to be VSO =
remain to be robust in the presence of impurities.
VR = g = 0.1t, V0 = 12t, and ξ = 4. Arrows in (a) indicate
some of therelatively large energy gaps. Acknowledgment This work is supported by the
State Key Program for Basic Researches of China un-
Finally, we wish to discuss the robustness of the gap- der GrantNos. 2009CB929504(LS), 2011CB922103and
less edge states found in the present TRS-broken QSH 2010CB923400(DYX), by the National Natural Science
system. WeconsideraN N sampleformingalooped Foundation of China under Grant Nos. 11074110 (LS),
x y
×
geometry as that shown in Fig. 1. N nonmagnetic im- 11174125, 11074109, and 91021003 (DYX), and by a
I
purities are assumed to be randomly distributed in the project funded by the PAPD of Jiangsu Higher Educa-
sample at positions R with α = 1, N . An extra tion Institutions.
α I
···
term HI = iwic†ici is added to the total Hamiltonian
H todescribPetheeffectoftheimpurityscattering,where
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