Topological Polaritons in a Quantum Spin Hall Cavity

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by  Alexander Janot

Topological Polaritons in a Quantum Spin Hall Cavity
Alexander Janot,1 Bernd Rosenow,1 and Gil Refael2
1Institut für Theoretische Physik, Universität Leipzig, 04009 Leipzig, Germany
2Institute of Quantum Information and Matter, Department of Physics,
California Institute of Technology, Pasadena, CA 91125, USA
(Dated: April 14, 2016)
Westudythetopologicalstructureofmatter-lightexcitations,socalledpolaritons,inaquantum
spin Hall insulator coupled to photonic cavity modes. We identify a topological invariant in the
presence of time reversal (TR) symmetry, and demonstrate the existence of a TR-invariant topo-
6
logical phase. We find protected helical edge states with energies below the lower polariton branch
1
and characteristic uncoupled excitonic states, both detectable by optical techniques. Applying a
0
ZeemanfieldallowsustorelatethetopologicalindextothedoublecoverageoftheBlochsphereby
2
the polaritonic pseudospin.

r
p
A
Recently,topologicallynontrivialstatesofmatterwith a)
protected edge or surface states have attracted much at- ǫ
3 UP LP
1 tention [1–3]. While the first realizations were found in
electronic systems [4–7], topologically nontrivial phases excitonic

l] ofperiodicallydrivensystems[8–11]andphotons[12–17] zz yy
al have been discovered within the last years. A time re- helical xx
edgestate
h versal (TR) invariant topological phase can exists when
q
- abandinversionoccurs, asafunctionofmomentum, be- Γ q0 Q
s b) σ=l
me towrbeietnalosrtbaittaeslsatraetecsouwpitlehddbiffyesrpenint-poarbriittyi,natnedrawcthioenn.these ǫ σ=r∼BZ
z
. In electronic systems, the time reversal operator
t
a squarestominusone,T2 = 1,implyingtheexistenceof LP
m − pola-
degenerateKramerspairs,suchthatthecrossingoftopo- σ=r rization
- logical edge states is protected. In contrast, for bosonic σ=l vector x y
d systems with T2 =+1, in general there is no TR invari- Γ Q q
n
o ant topologically nontrivial phase [18–20]. The photonic
Figure 1. (color online) Defining properties of a topologi-
c topological insulators [12–17] either break TR or have a
callynontrivialpolariton: a)theLPandUPbranches(orange
[ built-in degeneracy, which protects edge states in a way
lines)aresplitbyspin-orbitcoupling,exceptatTRinvariant
2 similartotheKramer’sdegeneracyinfermionicsystems. momenta (Γ,Q) (thick black dots). For topological polari-
v We consider strongly coupled light-matter systems tons purely excitonic states (gray dot) emerge along a line
2 in two-dimensions, so-called polaritons, in which the (cid:126)q=(cid:126)q0, and helical edge states (thick purple line) are present
9 bosonic polariton can inherit its topological properties below the LP. b) with a TR breaking Zeeman field BZ (cid:54)= 0,
0 thepolarizationvector(cid:126)nofeachLP-dispersionbranchcanbe
from the electronic part. Building on such an example,
4 trackedoverthewholeBrillouinzone. Thepolaritonistopo-
0 realizedbyquantumspinHall(QSH)electrons[4,6]cou-
logicallynontrivialif(cid:126)ncoverstheentireBlochsphere(inset).
. pled to cavity photons, we develop a framework which
1 Here,thenorth(south)pole(blue(red)color)standsforright
0 allowstocharacterizetopologicalstatesofpolaritons(cf. (left)circularpolarization,whilethex-y planerepresentslin-
5 Fig.1). Incontrasttorecentproposalsoftopologicalpo- early polarized light.
1 laritons[21–23]whereTRbrokensystemswerediscussed,
v: we focus on TR invariant topological polaritons. Here,
The two polarization directions of the photonic compo-
i wegoconsiderablybeyondpreviousworksandi)definea
X nent can be identified with a pseudospin [26], which can
topological invariant for TR symmetric bosonic systems,
r ii) explain that contrary to [18–20] a topologically non- be described by an effective Hamiltonian, and whose di-
a
rection can be observed by detecting photonic emission.
trivialphaseispossibleduetoavortex-likesingularityin
For the LP branch it takes the form
the exciton-photon coupling, and iii) describe how TR-
invariant topological polaritons can be detected experi- H ((cid:126)q)=(cid:15) ((cid:126)q) + (cid:126)σ (cid:126)h((cid:126)q) , (1)
LP LP
mentally by looking for dark excitonic states and edge ·
states, and by studying the polarization in the presence where(cid:126)h((cid:126)q) is a momentum dependent effective magnetic
of an external Zeeman field. field. With (cid:15) ((cid:126)q) we denote the lower polariton disper-
LP
Main results – A polariton consists of an exciton cou- sion in the absence of pseudospin coupling, and (cid:126)σ is the
pled to a cavity photon, such that lower (LP) and upper vector of Pauli matrices. TR invariance TH((cid:126)q)T 1 =
−
polariton (UP) dispersion branches are formed [24, 25]. H( (cid:126)q) with TR operator T = σ requires that the
x
− − K
2
x- and y-component of the effective magnetic field(cid:126)h are Refs. [29, 30] is analogous to defining spin-eigenstates
even functions and the z-component is an odd function with nontrivial spin-Chern number in a QSH insulator
ofmomentum. Thepseudospinpolarizationoftheeigen- with broken spin-symmetry [31, 32].
state χ is given by (cid:126)n = χ (cid:126)σ χ = hˆ with If TR-symmetry is broken by a small Zeeman field,
1,2 1,2 1,2 1,2
| (cid:105) (cid:104) | | (cid:105) ∓
hˆ (cid:126)h/(cid:126)h. If (cid:126)n points north (south) the emitted light BZ = 0, the pseudospin degeneracy at the TR invariant
is ≡right|(l|eft) polarized, while the x-y plane represents mom(cid:54) enta is lifted, and two completely non-degenerate
linearlypolarizedlight,seeinsetFig.1b. Inthepresence dispersionrelationsexist,seeFig.1b. Fortopologicalpo-
of an additional parity symmetry, H ( (cid:126)q) = H ((cid:126)q), laritonswithC =0,wethenfindthatforeachbranchof
one finds h 0, and the emitted lighLtPis−always linLeParly thedispersiont±he(cid:54) experimentallymeasurablepseudospin
z
polarized. Ph≡ysicalmechanismswhichleadtoanon-zero polarization(cid:126)n((cid:126)q)coverstheBlochsphere. Thus,inaTR
effective magnetic field are, for example, a longitudinal- broken setup, the experimentally measurable pseudospin
transverse splitting of the electromagnetic field [26] or a polarization(cid:126)n((cid:126)q)andthepseudospinmodelEq.(1)allow
spin-orbit coupling of the electronic building blocks as todistinguishbetweentopologicallytrivialandnontrivial
discussed below. In both cases, it turns out that (cid:126)n polaritons.
1,2
winds twice around the z-axis if (cid:126)q encircles the Γ-point. Microscopic model – The Hamiltonian of the two-
Duetocontinuityof(cid:126)h((cid:126)q),wefindthat(cid:126)h((cid:126)Γ)=0,implying dimensional QSH insulator in the basis of orbital states
thattheLP(UP)dispersionisdegenerateattheΓ-point, +1/2 , +3/2 , 1/2 , 3/2 is [6]
{| (cid:105) | (cid:105) |− (cid:105) |− (cid:105)}
see Fig. 1a; in our QSH-polariton model,(cid:126)h((cid:126)Γ)=0 at all (cid:32) (cid:33)
other TR invariant momenta as well. H ((cid:126)k)= He+((cid:126)k) 0 , H+((cid:126)k)=d(cid:126)((cid:126)k) (cid:126)σ , (2)
The Chern number [27, 28] for the eigenstate χ1,2 e 0 He−((cid:126)k) e ·
| (cid:105)
countshowmanytimes(cid:126)n wrapsaroundtheunitsphere
1,2
if(cid:126)qcoverstheBrillouinzone,C (cid:82) hˆ (∂ hˆ ∂ hˆ). with wavevector (cid:126)k and spin-orbit field d(cid:126). Because of
Clearly, in the presence of pari1t,y2 ∼sym∓mq(cid:126)etr·y, Cqx1,2×=q0y as TR-symmetry He−((cid:126)k) = He+(−(cid:126)k)∗, where α = {+,−}
h 0. However, even without parity, C = 0: The labels a pseudospin. In the following we will use the
z 1,2
≡
sic=alaxr,ytr,izpleexpacrotldyuoctncceo,natnaidnstheuaschiscaonmopdodnefnutnchˆtiiownitohf pcoasr(akmx)etrizcaotsi(okny)d)xw/yith=AA=sin(cid:126)(vkFx//ay,),Bdz>=0, Mand+MB(2R−,
− ∈
momentum, because hx,y is even and hz is odd. Since wherevF istheFermivelocity, athelatticespacing, and
the integral over the Brillouin zone is invariant under (cid:126)kismeasuredina−1. ForM <0andB > M /2thenor-
| |
momentum inversion, the Chern number has to vanish, malizedspin-orbitfielddˆ d(cid:126)/d(cid:126) coverstheBlochsphere
≡ | |
in agreement with the general classification [18–20]. and the QSH insulator is topologically nontrivial [6].
For polaritons in a QSH cavity the LP eigenstates are AnelectromagneticfieldA(cid:126) iscoupledminimallytothe
superpositionsofexcitonic b andphotonicwavefunc- semi-conductor via p(cid:126) p(cid:126)+eA(cid:126) with elementary charge
tions a : Φ = β b| 1,2(cid:105)+α a with real co- e > 0. We work in →the Coulomb gauge, linearize the
1,2 1,2 1,2 1,2 1,2 1,2
efficien|ts β(cid:105) |,α (cid:105), see b|elow(cid:105) for de|tails(cid:105). An explicit Hamiltonian in A(cid:126) and expand the photon field in plane
1,2 1,2
calculation shows T Φ1,2((cid:126)q) = Φ1,2( (cid:126)q) . By con- waves with amplitudes A(cid:126)q(cid:126)σ where (cid:126)q is the photon mo-
struction, the effectiv|e mode(cid:105)l is t∓he| proj−ecti(cid:105)on onto the mentum and σ labels the polarization [33]. We find for
photonic sector, such that χ = a . Both mod- the optical transition matrix elements,
1,2 1,2
| (cid:105) | (cid:105)
els have the same polarization vector(cid:126)n , which implies
thaInt Cor1d,2er=t0ofroervethaletmheicrnoosnc-otpriicvimalotdoep1l,,o2ltoogoy. of the mi- gµσν((cid:126)k(cid:48),(cid:126)k,(cid:126)q)=δ(cid:126)k(cid:48)(cid:126)k+q(cid:126) eA(cid:126)qσ·(cid:104)ψµk(cid:48)| ∂H∂e(cid:126)k((cid:126)k) |ψνk(cid:105) , (3)
croscopic model, we define TR partners Φ = (Φ2 where µ,ν label the pseudospin and band index of the
Φ1 )/√2 which transform according t|o ±T(cid:105)Φ (|(cid:126)q) (cid:105)±= eigenstates of Eq. (2). In order to evaluate Eq. (3) we
|Φ (cid:105)( (cid:126)q) . Remarkably, the corresponding C|he±rn n(cid:105)um- usetheWigner-Eckarttheorem[34]andthespecificform
b| e∓rs−are(cid:105)non-zero, C = 2, an interesting result given of the basis states of Eq. (2) [6]. We find that optical
thatthegeneralclass±ificat∓ion[18–20]rulesoutnon-trivial
transitions do not change the pseudospin α.
Chern numbers for eigenstates of the Hamiltonian. The
A particle-hole transformation of Eq. (2) yields the
robustnessofC isduetoavortexlikenon-analyticityof hole Hamiltonian Hα((cid:126)k) = Hα( (cid:126)k)∗, with wave func-
the exciton-pho±ton Bloch Hamiltonian Eqs. (6,7), which h − e −
tions ψh =(ψe ) , and energies (cid:15) ((cid:126)k)= (cid:15) ( (cid:126)k)>0
canonlyberemovedwhenthesplittingoftheLPbranch α(cid:126)k α (cid:126)k ∗ h − e −
vanishes. As signatures of the topological phase we find (the chemical po−tential is zero). Accounting for the at-
an odd number of lines of momenta (cid:126)q encircling the tractive Coulomb interaction of electron and hole the
0
Γ-point for which α ( (cid:126)q ) = 0, exp{eri}mentally acces- wave function of optically active excitons is
2 0
{ }
saidbdleitiboyn,lowoekipnrgedfoicrtapodlaarrkitoenxiccitoedngicessttaatteesa.t O{(cid:126)qu0r}.prIon- |ψαxq(cid:126)(cid:105)=(cid:88)φC((cid:126)k) |ψαhq(cid:126)/2 (cid:126)k(cid:105)⊗|ψαeq(cid:126)/2+(cid:126)k(cid:105) , (4)
cedure of defining pairs of TR partners as suggested in (cid:126)k −
3
where φ ((cid:126)k) denotes the Fourier transform of the LP-splitting vanishes and the dispersion is twofold de-
C
electron-hole wave function with respect to the relative generate. Hence, the Chern numbers Cγ are protected
coordinate. Semi-conductors typically have a large di- by the splitting of LP and UP branches.±
electric constant, which screens the Coulomb interac- It is enlightening to analyze the photonic component
tion and results in an exciton Bohr radius much larger
aLP aLP aLP aLP
as the lattice constant. Thus, the binding function aLP = 2 ∓ 1 eiϕ ar + 2 ± 1 e−iϕ al , (9)
φ ((cid:126)k) is strongly peaked around k = 0 and ψx | ± (cid:105) 2 | (cid:105) 2 | (cid:105)
C | αq(cid:126)(cid:105) ≈
ψh ψe with energy (cid:15) ((cid:126)q) (cid:15) ((cid:126)q/2)+(cid:15) ((cid:126)q/2). of ΦLP . The coefficients aLP, aLP R are continuous
|Suαchq(cid:126)/2e(cid:105)xc⊗it|onαsq(cid:126)a/r2e(cid:105) described by txhe H≈amhiltonian e in t|he±e(cid:105)ntire Brillouin zone,1with2aLP∈=0 (cid:126)q, aLP(Γ)=
1 (cid:54) ∀ 2
aLP(Γ), and aLP(Q) = aLP(Q), such that aLP has to
Hα((cid:126)q)=Hα((cid:126)q/2) 1α+1α Hα((cid:126)q/2) , (5) −hav1eanoddnum2beroflines1ofzeroswhengoing2fromthe
x h ⊗ e h ⊗ e
Γ-point to the Q-points. Then, the polarization vector,
acting on a four dimensional Hilbert space of electron- (cid:126)n aLP (cid:126)σ aLP , points north (south) at the Γ-point
hole pairs. (Q+-p∼oin(cid:104)ts+)a|n|d+wi(cid:105)ndstwicearoundthez-axisinbetween.
Diagonalizing Eq. (5) and projecting onto excitons Thus, (cid:126)n covers the Bloch sphere twice. We find that
givestheeigenstates bα withenergy(cid:15)x =2d(cid:126). Thecav- the pola±rization vector of the excitonic component bLP
ityphotondispersion|is(cid:105)ω =ω (cid:112)1+(D/ω|)|(cid:126)q2 withω does not cover the Bloch sphere and cannot contri|b±ute(cid:105)
q(cid:126) 0 0 0
set by the cavity thickness, D ≡(cid:126)2c2ph/ω0a2, photon ve- to Eq. (8). For all wavevectors (cid:126)q0 at which aL2P = 0,
locity c , and momentum (cid:126)q measured in a 1. Coupling a dark exciton eigenstate, ΦLP = bLP , exists. This
ph − | 2 (cid:105) | 2 (cid:105)
excitons and right (r), left (l) circularly polarized pho- constitutes a clear signature for topological polaritons.
tons yields polaritons. In the basis b+ , b , ar , al Breaking TR-symmetry with a Zeeman field BZ, we
the Hamiltonian takes the form {| (cid:105) | −(cid:105) | (cid:105) | (cid:105)} find that the LP-eigenstates ΦLP for B = 0 have
| 1,2(cid:105) Z (cid:54)
(cid:18) (cid:19) the same topological structure as the states ΦLP for
HP = (cid:15)Gx1† ωG1 , (6) BtinZuo=usl0y,di.eef.otrhmeed|ΦiL±nPto(cid:105)tohbetaeiingeendsftoarteBsZΦ=LP0 cfoa|rnBb±e(cid:105)c=on0-.
| 1,2(cid:105) Z (cid:54)
Thus, if (cid:126)n covers the Bloch sphere in the case B =0,
and the exciton-photon coupling is obtained via Eq. (3), Z
then in th±e case B =0 the polarization vector(cid:126)n also
Z 1,2
(cid:54)
G= g0 (cid:18)(1−dˆz)e−2iϕ −(1+dˆz) (cid:19) , (7) covEedrgsethsteaBtelsoc–hWspehefirres.t evaluate the electron Hamilto-
4 (1+dˆ ) (1 dˆ )e2iϕ
z − − z nian Eq. (2) on a cylindrical geometry with boundaries
along the x-direction and then couple the correspond-
with dˆ = d /d(cid:126) and e iϕ (d id )/d id . The
z z ± x y x y ingexcitonstophotoneigenmodes[35]. Thenumerically
| | ≡ ± | ± |
coupling g (cid:15) ((cid:126)q), and is proportional to the photon
0 x obtained spectrum is shown in Fig. 2a. At each bound-
∝
amplitude times a numerical constant from the transi-
ary we find one pair of edge states with energies below
tion matrix element Eq. (3). However, for the study of
the LP branch. How can these edge states be related
topological properties we can safely neglect the continu-
to the polaritonic Chern numbers discussed above, and
ous (cid:126)q-dependency and treat g as constant.
0 why extend the edge states all the way down to zero
Topological invariant – The Hamiltonian Eq. (6) has
energy? Since the excitons are a direct product of topo-
two eigenstates for both LP and UP branch: Φγ =
baγ1n,d2|bp1,h2o(cid:105)t+onaγ1p,2s|eau1d,2o(cid:105)-swpiinthorγs |=aγ1,{2L(cid:105).P,UWPe},coenxsctitr|ounc1t,|2bn(cid:105)γ1e,2w(cid:105) lfidonogsdipcainnlolynd-evngaoennnitesrhriavinciayglCoefhletechrtneronenxucmaintbodenrhsdoCilse±px,e=irtsi∓oisn2e.ixnTpheacdetdepidsteioutno-
ebiagseinsssttaatteess,:bu|Φtγ±o(cid:105)be=y T(|ΦΦγ2γ(cid:105)(±(cid:126)q)|Φ=γ1(cid:105))Φ/γ√(2,(cid:126)qw)h.icWheardeefinnoet twoitThRthaelloewxcsittooncHonasmtriultcotnaiannoapnedratboerhawvheischexcaocmtlmyultikees
a Chern numbers as | ± (cid:105) | ∓ − (cid:105) a fermionic TR operator, for details consult our Supp.
(cid:90) Mat. [35]. Due to this symmetry, the excitonic system is
i
C±γ =−2π εij(cid:104)∂kiΦγ±|∂kjΦγ±(cid:105) , (8) aislsgoiviennsbyymνm=etr1yCclxassCAxII [a2n9d].aHZer2e,inthdeexprfoefracetxocrit1o/n4s
(cid:126)k∈BZ was introduced4b|ec+au−se o−f|the Chern number doubling
where ε is the Levi-Civita symbol. An explicit evalua- duetothetensorproductofholeandelectronspace[35].
ij
tionofEq.(8)yieldsCLP = 2andCUP =0forpositive Wenotethatthecoupledsystemofexcitonsandphotons
detuning, and vice ver±sa for∓negative±detuning. The ex- does not have any (pseudo) fermionic TR symmetry and
istence of CLP = 2 is due to the vortex structure on issolelyinsymmetryclassAI.Thenon-analyticityofthe
the diagonal±of th∓e coupling Eq. (7). For a transition polaritonHamiltonian(6)doesnotallowtoobtainalat-
to a phase with different Chern numbers to occur, this tice representation and to study edge states directly. A
vortex structure has to be removed, implying vanishing latticeHamiltonianisobtainedbyembeddingthepolari-
diagonal elements in Eq. (7). At the transition point the ton space into a larger Hilbert space which contains all
4
a) |Φ |1p.0h 0.4 qx=0.03
1 ×10−2 0.3 ǫ˜/2|M|=0.07 g00.12
/
2|M| ˜ψph0.2 ˜g 0.1
ǫ/0.5 0.5 0.1 M|0.5
2|
/
0 ˜∆ 0
−00 .3 −0.2 −0.1 q0x 0.1 0.2 0.3 0.0 0 20 y 40 60 −0.2 −0.1 q0x 0.1 0.2
b) Figure 3. (color online) Comparison of analytical (lines) and
C =0
± numerical (symbols) results for the polaritonic edge state.
UP
Leftpanel: Thephotonwavefunction(cf.Eq.(11))isshown.
ǫ LP C±=∓2 Rightpanel: Thecouplingstrengthg˜(upperplot)anddetun-
unbpahnydsiscal C±=0 ing ∆˜ (lower plot) of the effective model are depicted. Same
C = 2 parameters as in Fig. 2a are used.
± ±
Q~ ~Γ Q~
−
Figure2. (coloronline)Spectrumoftopologicalpolaritonsfor very good agreement with numerics as shown in Fig. 3.
a system with cylindrical geometry: a) The colored dots de- Finally, the one-dimensional effective Hamiltonian (for
pictpolaritonswithphotonicfractions|Φ|ph. Thesolidblack one edge) takes the form
linesshowtheLPandUPbranches(LPandUPsplittingnot
(cid:88)(cid:8) (cid:9)
visible on this scale). As parameters A = 5|M|, B = 25|M|, H = (cid:15)˜ b b +ω˜ a a +(g˜ b a +h.c.) .
E q †ρq ρq q †ρq ρq q †ρq ρq
ω = 3|M|/4, D = 35|M|, g = |M|/24, and N = 2000 lat-
0 0 qρ
tice points are used. b) The polariton branches (orange) and (12)
the unphysical electron-hole bands (light gray) are sketched
schematically. BecauseLP-branchandnegativeenergybands The operator a creates a right, ρ = R, (left, ρ = L)
†ρq
carryoppositeChernnumbersC±apairofhelicaledgestates moving photon with momentum q and dispersion ω˜ . In
q
(the physical part is marked purple) connects both.
one dimension the elementary excitations are collective
modes (plasmons) instead of excitons. Then, b creates
†ρq
right and left moving plasmons with dispersion (cid:15)˜ . For a
possibleelectron-hole(e-h)states[35],includingunphys- q
Luttinger liquid interactions renormalize the bare Fermi
icale-hpairswhichhaveeitherzeroornegativeenergies,
velocity [36], and the plasmon dispersion remains linear.
see Fig. 2b. Since the e-h bands at negative energy have
Experimental signatures – The signatures of topolog-
Chernnumbers 2, andsincethesumofallChernnum-
± ical polaritons are dark excitonic states along a line
bers in the Hilbert space of polaritons and artificial e-h
(cid:126)q in momentum space, and edge states below the LP
pairs has to be zero, the polariton Chern numbers are { 0}
branch, cf. Fig. 1a. Both are detectable via optical tech-
compensatedbythenegativeenergye-hChernnumbers,
niques [24]. Applying a Zeeman field allows to deter-
and edge states below the LP branch emerge, cf. Fig. 2.
mine the Chern number from analyzing the polarization
Effective edge model – The polaritonic edge states are
of the bulk polaritons, see Fig. 1b. Realizing topological
well described by coupling excitonic edge states to pho-
polaritons in a nontrivial QSH insulator is challenging.
toniceigenmodes[35]. Toleadingorderperturbationthe-
However, recent developments of engineering spin-orbit
ory we find
coupling in polaritonic systems [37] and accomplishing
(cid:18) (cid:19)2 slow photons in photonic crystals [38] may pave the way
g˜(q )
(cid:15)˜(q ) (cid:15)˜ (q ) x ∆˜(q ) , (10) for it.
x ≈ x x − ∆˜(q ) x
x Conclusion – We have considered a TR-invariant
Φ˜ ψ˜x g˜(qx) ψ˜ph , (11) model of polaritons in a QSH cavity, and introduced a
| ρqx(cid:105)≈| ρqx(cid:105)− ∆˜(q )| ρqx(cid:105) topological invariant for TR partners which is stabilized
x
bythepseudospinsplittingofpolaritons. Inthetopolog-
whereρ= R,L labelsrightandleftmover, ∆˜ ω˜ (cid:15)˜ ical phase, polaritonic edge states below the LP branch
x
{ } ≡ −
isthedetuningbetweenthephotonω˜ andexcitonenergy exist,aswellaslinesinmomentumspacewithuncoupled
(cid:15)˜ =(cid:126)v q , and g˜the coupling strength. The excitonic excitons, both detectable via optical techniques.
x F x
| |
and photonic edge-state wave functions are denoted by We thank T. Karzig and H.-G. Zirnstein for discus-
ψ˜x and ψ˜ph , respectively. The right (left) moving sions. AJ is supported by the Leipzig School of Natu-
| ρqx(cid:105) | ρqx(cid:105)
excitoncarriespseudospin+( ),whereasbothrightand ral Sciences BuildMoNa. BR would like to acknowledge
−
left moving photon are linearly polarized with electric DFG grants RO 2247/7-1 and RO 2247/8-1, and GR ac-
fieldparalleltotheplaneofincident. Wefindthatahigh knowledgesNSFgrantDMR1410435,aswellastheInsti-
photonicfractionoftheedgestatereliesonv c and tuteofQuantumInformationandMatter,anNSFcenter
F ph
∼
g ω . For (cid:15)˜ 2M our perturbative results are in supported by the Gordon and Betty Moore Foundation.
0 0 x
∼ (cid:28) | |
5
[32] E. Prodan, Phys. Rev. B. 80, 125327 (2009).
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6
SUPPLEMENTAL MATERIAL
In our main work we presented a symmetry argument that the Chern number of energy eigen-
states has to be zero for the effective model. Here, we provide an argument that this result
holds for the microscopic model, too. Furthermore, we verify that the Chern numbers of time
reversedpartnershastovanishfortheeffectivemodel. Then,weshowhowapseudofermionic
time-reversaloperatorforexcitonscanbeconstructedandpresenttheembeddingofthepolari-
ton Hilbert space into an enlarged Hilbert space of artificial electron-hole pair combinations.
Finally, we consider a system with boundaries, evaluate analytically the polaritonic edge state
wave function, and present details of the effective edge-state model.
Chern numbers the Γ-point. Then, we can remove the Γ-point from the
Brillouin zone integral, Eq. (S.2) is applicable, and our
Microscopic model – We found that the eigenstates of argument for C1,2 =0 remains valid.
polaritonsinaQSHcavitytransformundertimereversal The Chern numbers C of the TR-partners χ =
(TR) according to T Φ((cid:126)q) = Φ( (cid:126)q) . By symmetry (χ χ )/√2 are defin±ed according to Eq. (S.|2)±w(cid:105)ith
2 1
| (cid:105) ±| − (cid:105) | (cid:105)±| (cid:105)
argumentswewillshowthatforanystatewhichsatisfies polarization vector (cid:126)n = χ (cid:126)σ χ . Now, we will
this behavior the corresponding Chern number defined show that C = 0 fo±r the(cid:104)eff±e|ct|iv±e(cid:105)model (S.3). To
via this end we c±alculate explicitly (cid:126)n . For convenience we
parametrize the effective magneti±c field of Eq. (S.3) by
(cid:90)
i
C = εij ∂iΦ∂jΦ , (S.1) (cid:126)h = (h cos2ϕ,h sin2ϕ,hz)T with h ( (cid:126)q) = h ((cid:126)q),
−2π (cid:104) | (cid:105) ϕ( (cid:126)q) =⊥ ϕ((cid:126)q) + π⊥ and h ( (cid:126)q) = h⊥((cid:126)q−). Then⊥, the
q(cid:126)∈BZ ene−rgy eigenstates take thezf−orm − z
has to vanish. We decompose the integrand into
b((cid:104)e∂ciaΦu(s(cid:126)qe)|∂thjeΦ(i(cid:126)qn)t(cid:105)e+gr(cid:104)a∂tiioΦn(−o(cid:126)qv)e|r∂jtΦh(e−B(cid:126)q)r(cid:105)i)ll/o2u,inwhziochneisisvailnid- χ1,2 = √12(cid:32) (cid:112)(cid:112)11∓hˆhˆz ee−iϕiϕ(cid:33) , (S.4)
variant under inversion of momentum. Then, the z
∓ ±
second summand is recast: ∂ Φ( (cid:126)q)∂ Φ( (cid:126)q) =
i j
(cid:104) − | − (cid:105)
∂iTΦ((cid:126)q)∂jTΦ((cid:126)q) = ∂jΦ((cid:126)q)∂iΦ((cid:126)q) . We used that with normalized effective magnetic field hˆ =(cid:126)h/(cid:126)h. Us-
(cid:104)the TR-o|perator (cid:105)and p(cid:104)artial d|erivati(cid:105)ve commute, and ing Eq. (S.4), (cid:126)n can be calculated in a straight|fo|rward
that T is anti-unitary. Now, Eq. (S.1) takes the form manner, ±
C ε ( ∂ Φ((cid:126)q)∂ Φ((cid:126)q) + ∂ Φ((cid:126)q)∂ Φ((cid:126)q) )=0.
ij i j j i
∼ (cid:104) | (cid:105) (cid:104) | (cid:105)
Effective model – In our main work we have used sym-  hˆ cos2ϕ
z
metry arguments to show that the Chern number −
(cid:126)n =  hˆzsin2ϕ . (S.5)
(cid:90) ± ±−(cid:113) 
C = 41π d2q (cid:126)n·(∂qx(cid:126)n×∂qy(cid:126)n) . (S.2) 1−hˆ2z
BZ
It cannot cover the Bloch sphere, since the z-component
oftheenergyeigenstate χ oftheTRinvariantHamil-
1,2
| (cid:105) isalwayspositive(negative). Therefore, theChernnum-
tonian
bers of the TR-partners have to vanish, C =0.
±
H =(cid:15) +(cid:126)σ (cid:126)h , (S.3)
LP LP
·
has to be zero. In Eq. (S.2) the polarization vector of
the eigenstate χ is (cid:126)n = hˆ with hˆ = (cid:126)h/(cid:126)h. We Pseudo fermionic TR operator for excitons
1,2 1,2
| (cid:105) ∓ | |
note that Eq. (S.2) and Eq. (S.1) are equivalent for 2 2
×
Hamiltonians. For the QSH model studied in the main Theexcitonspectrumisdegeneratewithrespecttothe
manuscript, (cid:126)n winds twice around the z-axis if (cid:126)q en- pseudospin α +, , such that [H ,σ ]=0. Further-
1,2 x z
∈{ −}
circles the Γ-point, and the effective magnetic field has more, H is invariant under TR. In a basis b , b
x +
to vanish at the Γ-point in order to obtain a continuous with σ b = b the excitonic TR-opera{t|or(cid:105)is|T−(cid:105)=}
z
Hamiltonian (S.3), which results in an ill-defined polar- σ , wh|i±ch(cid:105) squ±a|re±s(cid:105)to one as required by the bosonic
x
K
ization vector(cid:126)n . Thus, the Chern number Eq. (S.2) is statistics of excitons. The product of T and σ com-
1,2 z
formally not well defined. This raises the question of the mutes with H as well, and hence is a symmetry of H ,
x x
validity of our previously presented argument. Although too. Wedefinetheanti-unitaryoperatorT =σ T. This
F z
thelimit(cid:126)q (cid:126)Γisnotunique,wehaveverifiedthatthere is a fermionic TR-operator in the sense that T = iσ
F y
→ K
are no singular contributions from a neighbourhood of with T2 = 1.
F −
7
Enlarged polariton Hilbert space G ((cid:126)q) = G ( (cid:126)q)σ . We emphasize that Eq. (S.9) is
co−ntinuousl−y d∗+efin−ed oxver the entire Brillouin zone for a
The BHZ Hamiltonian [6] in the basis +1/2 , topologically nontrivial QSH insulator, and couples only
+3/2 , 1/2 , 3/2 has the form {| (cid:105) the (physical) excitonic states to photons.
| (cid:105) |− (cid:105) |− (cid:105)}
(cid:32) (cid:33)
H+((cid:126)k) 0
H ((cid:126)k)= e , (S.6)
e 0 H ((cid:126)k) Polariton system on a cylindrical geometry
e−
where H+((cid:126)k) = d(cid:126)((cid:126)k) (cid:126)σ and H ((cid:126)k) = H+( (cid:126)k)∗ with First,weanalyzetheQSHinsulatoronacylindricalge-
pseudospein α +, · , two-dime−ensional weav−evector (cid:126)k ometry with periodic boundary conditions in x-direction
and spin-orbit∈fie{ld d−(cid:126).} We found that optical transi- and hard wall boundary conditions in y-direction. We
compute the spectrum and eigenstates by numerical di-
tions do not change the pseudospin α. Then, excitons
agonalizationofthelatticeHamiltonianofEq.(S.6). The
can be characterized by a quantum number α, too. In
eigenfunctions are plane waves in x-direction with quan-
order to study the topology of these excitons we em-
tumnumbersk ,andvanishaty =0,Lwithsystemsize
bed the exciton Hilbert space for each pseudospin α x
L = L . For given pseudospin α and wavevector k we
into an enlarged Hilbert space; a four dimensional space y x
find l = 1,...,N conduction (valence) band eigenstates
spannedbythetensor-productstates: α1/2 α1/2 ,
h e
{| (cid:105) ⊗| (cid:105) whereL=aN withN latticesitesandlatticeconstanta.
α1/2 α3/2 , α3/2 α1/2 , α3/2 α3/2 .
Ar|eppprerso(cid:105)exhni⊗mta|attioinng(cid:105)Hexα|≈H(cid:105)hαh⊗⊗1|αe +1(cid:105)eαh ⊗| Heα(cid:105)yhi⊗eld|s in t(cid:105)he}is |Tψhαheksxel(cid:105)p.roFvoirdaetaopbaosloisgisceatllfyornoelnetcrtirvoianlsQ|ψSαeHkxinl(cid:105)sualnadtohrotlhees
statewithl=1isanedgestatelocatednearaboundary.
 
0 dx idy dx+idy 0 From now on we will label any edge state by a tilde.
−
H+ =dx+idy −2dz 0 dx+idy . (S.7) Aslongasmomentumwasagoodquantumnumberan
x dx idy 0 2dz dx idy approximation for the exciton wave function was given
− −
0 dx idy dx+idy 0 bythedirectproductofholeandelectronwavefunction:
−
ψx ψh ψe with exciton momentum (cid:126)q.
Because of TR-symmetry Hx−((cid:126)q) = (Hx+(−(cid:126)q))∗. The T| hαisq(cid:126)(cid:105)ap≈pr|oxαimq(cid:126)/a2t(cid:105)io⊗n|faiαlsq(cid:126)/o2n(cid:105) a cylindrical geometry, since
HamiltonianEq.(S.7)describesanexcitonstatewithen-
k is no longer a good quantum number. Nonetheless,
ergy (cid:15) ((cid:126)q) = 2d(cid:126)((cid:126)q/2), and three unphysical pairs with y
x the exciton state can be expanded in product states of
| |
holes in the conduction band and/or electrons in the va-
electron and hole: ψx =1 ψx with
lenceband. ThefourbandsofEq.(S.7)haveChernnum- | αqxn(cid:105) b| αqxn(cid:105)
berswhichresultfromaddingthenontrivialChernnum- (cid:88)
1 = ψh ψe ψe ψh , (S.10)
bers of electronic conduction and/or valence band. This b | αkxl αkxl(cid:48)(cid:105)(cid:104) αkxl(cid:48) αkxl|
ll(cid:48)
resultsinadoublingoftheexciton(hole-electron)Chern
nfourmabrteirfi.ciWalehofilne-delCec±tro=n∓pa2irfsowrietxhcniteognastiavnedenCer±gy= ±(cid:15)2. where |ψαe,hkxl(cid:105) are the electron and hole eigenstates in-
− x troduced above. For sufficiently large systems we can
The two bands with electron and hole in the same band
approximate the excitonic wave function on a cylindri-
carry vanishing Chern numbers.
calgeometrybyprojectingtheexcitoniceigenstateswith
TheHamiltonianofpolaritonsinaQSHcavityembed-
periodicboundaryconditionsontotheelectron-holebasis
ded into the extended exciton space takes the form
states with boundaries, namely
H+ 0 G 
x + (cid:88)
HP = 0 Hx− G  , (S.8) |ψαxqxn(cid:105)≈ cαll(cid:48)n(qx) |ψαhqx/2l ψαeqx/2l(cid:48)(cid:105) , (S.11)
G+† G† Hp−h ll(cid:48)
− cαn(q )= ψh ψe 1 (ψx ψx ) .
where the exciton Hamiltonian is given in Eq. (S.7) and ll(cid:48) x (cid:104) αqx/2l αqx/2l(cid:48)|√2 | αqxqn(cid:105)±| αqx−qn(cid:105)
the photon Hamiltonian for right and left circularly po-
larized modes is of form H = ω 1 with dispersion ω. Above, the exciton state ψx is the plane wave so-
ph | αqxqn(cid:105)
The coupling matrix is lution for periodic boundary conditions with wavevec-
tor q = 2k , and q = 2(2π/L)n, where n runs from
(1 dˆz)dˆ e−iϕ (1+dˆz)dˆ eiϕ n = xN/2,.x..,N/2 n 1. We use the even superposition
G+ = g80  −(d1ˆ2−e−dˆ2z⊥i)ϕ2 − −(1dˆ⊥+2ed2ˆizϕ⊥)2  , (S.9) o(pnleus(−msiignnu)sfsoirgnn)≤for0−n(c>os0in(esienigeeenifguenncftuinocnt)ioannsd).the odd
(1 ⊥dˆ )dˆ e iϕ (−1+dˆ )dˆ eiϕ We note that: i) The electron-hole Hilbert space is
z − z
− ⊥ − ⊥ differentforhardwallandperiodicboundaryconditions,
with g0 being a constant, dˆz = dz/|d(cid:126)|, and so that 1b Eq. (S.10) is the identity in the former and a
dˆ eiϕ (d + id )/d(cid:126). TR-symmetry demands that projector in the latter space. ii) In the presence of edge
x y
⊥ ≡ | |
8
states we replace the n=0 state by the edge state wave with wavevector k , spinor φα (eigenvector of the σ
x 1 x
function Paulimatrixwitheigenvalue ±1),andreal-spacefunction
±
ψ˜x ψ˜h ψ˜e , (S.12) (cid:112)λ(λ2 ν2)
| αqx(cid:105)≈| αqx/2(cid:105)⊗| αqx/2(cid:105) ηkx(y)=2 ν− e−λysinh(νy) , (S.18)
i.e. cαll(cid:48)0 = δ1lδ1l(cid:48). iii) The wave function Eq. (S.11) has satisfying the boundary condition η (y = 0) = 0. The
an energy kx
two parameters λ,ν (measured in inverse lattice units)
(cid:88) are defined as
(cid:15) (q ,n)= cαn 2((cid:15) (q /2,l)+(cid:15) (q /2,l)) , (S.13)
x x | ll(cid:48) | h x e x (cid:48) (cid:114)
ll(cid:48) A 2M
λ , ν(k ) λ2 | | +k2 (S.19)
where(cid:15) ((cid:15) )istheeigenvalueoftheelectron(hole)eigen- ≡ B x ≡ − B x
e h
function of the QSH insulator on a cylindrical geometry. respectively, where A,B,M are the BHZ-parameters.
iv) We note that the expansion Eq. (S.11) is somewhat Above, all lengths (wavevectors) are measured in lat-
analogousofprojectingsineandcosinewaves(freeparti- tice units (inverse lattice units). The edge state exists
cles)ontoastandingwavebasis(particlesinabox). v)In if k <(cid:112)2M /B, and has an energy
x
thelimitofinfinitesystemsize, wavefunctionEq.(S.11) | | | |
and its spectrum Eq. (S.13) converge to the solutions for (cid:15)˜α(k )=α(cid:126)v k . (S.20)
e x F x
periodic boundary conditions.
Onacylindricalgeometrythephotoniceigenmodesare Edge states located near y = L have φα1 spinors and
energies (cid:15)˜α(k ) = α(cid:126)v k . In the fol−lowing we will
(cid:114) (cid:114) e x − F x
1 2 focus on the boundary y =0.
A(cid:126)σqxm(x,y)= Leiqxx Lsin(qmy)(cid:126)eσ (S.14) Now, the coupling Eq. (S.16) is evaluated using the
approximation Eq. (S.12) and the result Eq. (S.17). We
with wavevector q , q =π/Lm, m=1,2,..., polariza- find that only p-polarized light (linearly polarized light
x m
tion vector(cid:126)e , and energy withelectricfieldparalleltotheplaneofincidentwhichis
σ
perpendiculartothey-direction)couples: gασ=p gα =
(cid:113) m ≡ m (cid:54)
ω(q ,q )= ω2+(cid:126)2c2 (q2+q2 ) , (S.15) 0, whereas s-polarized light (electric field perpendicular
x m 0 ph x m
to the boundary) does not: gασ=s =0. We find
m
with cph as photon velocity and ω0 determined by the (cid:114) (cid:90)
i g 2 πm
thickness of the cavity. The coupling to excitons is g (k )= 0 η (y)2sin( y) , (S.21)
m± x −√2 2 L y kx L
(cid:88)
gασ(q )= (cαn(q )) (S.16)
nm x ll(cid:48) x ∗ for both pseudospins α= .
ll(cid:48) ±
As long as the exciton bulk states are energetically
ψc i[Hˆ ,xˆ] eA(cid:126)σqxm ψv , much higher as the edge state, (cid:15)˜x =(cid:126)vF qx 2M , we
×(cid:104) αqx/2l(cid:48)| e · (cid:126) | α−qx/2l(cid:105) can neglect those and use an effective de|scr|ip(cid:28)tion|, |
wherec,v labeltheconductionandvalenceband,respec-  (cid:15)˜ (q ) 0 g+(q )
x x x
tively.
HE(cid:48)(qx)= 0 (cid:15)˜x(qx) g−(qx) , (S.22)
The polaritonic modes are obtained by evaluating the (g+(q )) (g (q )) ω(q )
x † − x † x
coupling Eq. (S.16) numerically and diagonalizing the
corresponding polariton Hamiltonian with exciton en- where gα is a row vector in the photon-space, see
ergy Eq. (S.13) and photon energy Eq. (S.15) for given Eq. (S.21), and ω(qx) = ω(qx,qm)δmm(cid:48) a diagonal ma-
wavevector qx and pseudospin α. We find pairs of po- trix, see Eq. (S.15). In leading order degenerate pertur-
laritonic edge states lying energetically below the lower bation theory we find
polariton branch.
(cid:15)˜(q ) (cid:15)˜ (q )+(cid:15)˜ (q )
x (0) x (2) x
≈
Polaritonic edge state model =(cid:15)˜x(qx) (cid:88)|gm±(qx/2)|2 , (S.23)
− ∆ (q )
m x
m
Replacingky →−i∂y inEq.(S.6)allowsustocalculate |Φ˜ρqx(cid:105)≈|Φ˜(ρ0q)x(cid:105)+|Φ˜(ρ1q)x(cid:105)
analytically the edge-state wave function for hard wall = ψ˜x (cid:88)(gm±(qx/2))∗ A(cid:126)σ=p . (S.24)
boundary conditions in y-direction, see Ref. [2, 39, 40]. | ρqx(cid:105)− ∆ (q ) | qxm(cid:105)
m x
For wave functions located near y =0 [41] we find m
Above, we have defined the detuning ∆ (q )
ψ˜e (y)=η (y)φα , (S.17) m x ≡
αkx kx +1 ω(qx,qm) (cid:15)˜x(qx), and an index ρ which labels right (R)
−
9
and left (L) moving edge states. The excitonic compo- ergy of the localized photon wave function and g˜ an ef-
nent of the right (left) mover has pseudospin α=+( ), fectivecoupling. DiagonalizingEq.(S.26)andexpanding
whereas the photonic component is p-polarized indep−en- in powers of g˜/∆˜ yields
dently of the direction of motion. Since g is equal for
m±
(cid:18) (cid:19)2
bothpseudospins,theedge-stateenergyisdegeneratefor g˜(q )
(cid:15)˜(q ) (cid:15)˜ (q ) x ∆˜(q ) , (S.27)
ρ = {R,L}. Evaluating the photonic part of Φ˜ρqx(y) x ≈ x x − ∆˜(qx) x
Eq. (S.24) results in a wave function which is exponen-
g˜(q )
tially localized near the(cid:113)boundary, too. We define the |Φ˜ρqx(cid:105)≈|ψ˜ρxqx(cid:105)− ∆˜(qx)|ψ˜ρphqx(cid:105) , (S.28)
x
photonfractionasF P Φ˜ 2 whereP projects
≡ | ph ρqx| ph
onto the photonic part, so that Eq. (S.24) yields with detuning ∆˜ ω˜ (cid:15)˜ . This allows us to extract
x
≡ −
F(qx)≈(cid:115)(cid:88)m |g∆m±m(q(xq/x2))2|2 . (S.25) g˜(qx)= F(1qx)(cid:88)m |gm±∆(mqx(q/x2))|2 , (S.29)
A simple effective model for right or left moving po- ∆˜(qx)= F(q1 )2 (cid:88)|gm±∆(qx(q/2))|2 , (S.30)
x m x
laritonic edge states is m
(cid:18) (cid:19) ψ˜ph = 1 (cid:88)(gm±(qx/2))∗ A(cid:126)σ=p , (S.31)
HE(qx)= (g˜(cid:15)˜x(q(qx))) ωg˜˜((qqx)) , (S.26) | ρqx(cid:105) F(qx) m ∆m(qx) | qxm(cid:105)
x ∗ x
by comparing with Eq. (S.23) and Eq. (S.24).
where (cid:15)˜ (q ) = (cid:126)v q is the exciton energy, ω˜ the en-
x x F x
| |