Chiral topological excitonic insulator in semiconductor quantum wells
Ningning Hao, Ping Zhang, Jian Li, Zhigang Wang, Wei Zhang, Yupeng Wang
aa r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Chiral topological excitonic insulator in semiconductor quantumwells
Ningning Hao,
1, 2
Ping Zhang,
2, 3, ∗ Jian Li, Zhigang Wang, Wei Zhang, and Yupeng Wang Institute of Physics, Chinese Academy of Sciences,Beijing 100190, People’s Republic of China LCP, Institute of Applied Physics and Computational Mathematics,P.O. Box 8009, Beijing 100088, People’s Republic of China Center for Applied Physics and Technology,Peking University, Beijing 100871, People’s Republic of China
Abstract
We present a scheme to realize the chiral topological excitonic insulator in semiconductor het-erostructures which can be experimentally fabricated with a coupled quantum well adjacent to twoferromagnetic insulating films. The different mean-field chiral topological orders, which are due tothe change in the directions of the magnetization of the ferromagnetic films, can be characterizedby the TKNN numbers in the bulk system as well as by the winding numbers of the gapless statesin the edged system. Furthermore, we propose an experimental scheme to detect the emergenceof the chiral gapless edge state and distinguish different chiral topological orders by measuring thethermal conductance.
PACS numbers: 03.65.Vf, 73.21.Fg, 73.43.Lp ∗ Corresponding author; zhang [email protected] . INTRODUCTION The search for new phases of quantum matter is one of the essential topics in condensed-matter physics. Chiral topological band insulators (TBIs) are such a type that has beenattracting a lot of interest both theoretically and experimentally. Although like trivialinsulators in the sense that TBIs have a band gap in the bulk, they are fundamentallydistinguished from trivial ones by their having gapless modes on the boundaries. Thesegapless modes are robust under perturbations and cannot be gapped without going througha quantum phase transition. In the case of time reversal symmetry (TRS) breaking, a well-known TBI system is the Haldane’s model which is a minimal model to illustrate quantumanomalous Hall effect (QAHE) [1]. The QAHE topological phase is characterized by the(Thouless, Kohmoto, Nightingale, and Nijs) TKNN number [2] of the first Chern class ofa U(1) principal fiber bundle on a torus in the bulk system or the winding number ofHalperin’s edge-state theory [3, 4] on the boundary of the system. The coherence of thetwo different kinds of numbers is guaranteed by the bulk-edge correspondence. Since therigorously prerequisite magnetic field in Haldane’s model is difficult to realize in experiment,recently, there are some new proposals [5] to realize QAHE based on single-particle picture.In analogy with QAHE in single-particle picture, the superconductors in TRS-broken( p x + ip y ) weak pairing state in two dimensions with a fully stable bulk gap opened byelectron-electron interaction can also have chiral topological order [6]. The edge states ofthe chiral superconductor have half of the degrees of freedom compared to QAHE states dueto the particle-hole symmetry (PHS) and are called Majorana edge sates. In the spirit ofanalogy with superconductor, a natural and important issue is how to get chiral topologicalexcitonic insulator (TEI), which is addressed in this paper.In this paper, we consider an independently gated double-quantum-well structure sepa-rated by a spacer as shown in Fig. 1(a). The ferromagnetic insulating films are introduced tobreak TRS by inducing an effective Zeeman splitting in the two dimensional electron (hole)gas [2DE(H)G]. The magnetization is perpendicular to the two-dimensional layer. Note thatthe orbital effect of the ferromagnetic films to the 2DE(H)G can be neglected due to thelocal exchange interaction on the interface. An electron-hole fluid is created by modulatingthe voltages so that the Zeeman-split upper branch of the heavy-hole bands in 2DHG layercan move above the Zeeman-split lower branch of the electron bands in 2DEG layer. This2 ge FMFM2DEG2DHGspacerV gh (a) yz -202 -1.5 1.50E k(b) E e+ E e - E h+ E h - FIG. 1: (Color online) (a) Schematic structure of semiconductor quantum-wells system that holdschiral TEI. The external gates (V ge ( h ) ) can independently tune the chemical potential µ e ( h ) toobtain the electron and heavy-hole layer. The ferromagnetic insulating films support effectiveexchange fields ( V e , V h ). (b) The energy spectrum of the electron/hole bilayer system near theFermi energy E F ≈
0. Here the solid lines denote non-interacting single-particle energy spectrum E e ( h ) ± , while the dashed lines denote the exciton energy spectrum with an obvious mean-field gapopened. We take t e = t h = 1, µ e = µ h = −
4, ( V e , V h ) = (1 , α = 0 .
5, and β = 0 . procedure results in spatially separated but strongly interacting electron and hole fluids ifthe two layers are close enough. The external electric field produced by the bias voltagesand the intrinsic electric field due to doping in the process of fabricating quantum wells canenhance the structural inversion asymmetry and induce the tunable Rashba spin-orbit (SO)interaction [7–10]. Recall that due to the strong SO coupling and non-centrosymmeyticproperty [11], the broken parity of the order parameter is the prerequisite condition of thechiral superconductor. In analogy with chiral superconductor, we demonstrate that the chi-ral TEI can occur when the Rashba SO interaction is strong enough with respect to theamplitude of the excitonic order parameter (EOP). Generally the Rashba SO interactionstrength can be influenced by carrier density, gated voltages, material of quantum wells, etc[12–14]. In InAs heterostructures, for instance, this quantity can be electrically tuned to beas large as α ≈
50 meV ˚A [15]. Due to the missing PHS, the edge states of chiral TEI in thepresent system are not Majorana fermions. The implication of these gapless edge states forexperimental observations is also discussed in this paper.3
I. MODEL AND HAMILTONIAN
We start with an effective electron/hole semiconductor bilayer system confined in the x - y plane. Here for the hole layer only the heavy-hole bands are occupied as in typicalexperiments, while the light-hole bands are empty and are therefore not taken into accountin our model. The resultant tight-binding Hamiltonian for the Rashba spin-orbit coupledsemiconductor bilayer system is H = P p ( H ( p ) kin + H ( p ) R )+ H ( e − h ) int ≡H + H ( e − h ) int : H ( p ) kin = X ,σ ( − t p − µ p δ ij ) p † i,σ p j,σ + X j,σ,σ ′ V p τ p ( s z ) σ,σ ′ p † j,σ p j,σ ′ , H ( e ) R = 12 α [ X j ( e † j, ↑ e j + δx, ↓ − e † j, ↑ e j − δx, ↓ ) − i X j ( e † j, ↑ e j + δy, ↓ − e † j, ↑ e j − δy, ↓ )] + H.c. , H ( h ) R = i β [ X j ( h † j, ⇑ h j +2 δx, ⇓ − h † j, ⇑ h j − δx, ⇓ )+ i X j ( h † j, ⇑ h j +2 δy, ⇓ − h † j, ⇑ h j − δy, ⇓ )+ 3(1 − i ) X j ( h † j, ⇑ h j − δx + δy, ⇓ − h † j, ⇑ h j + δx − δy, ⇓ ) (1)+ 3(1 + i ) X j ( h † j, ⇑ h j − δx − δy, ⇓ − h † j, ⇑ h j + δx + δy, ⇓ )+ 4 X j ( h † j, ⇑ h j + δx, ⇓ − h † j, ⇑ h j − δx, ⇓ )+ 4 i X j ( h † j, ⇑ h j + δy, ⇓ − h † j, ⇑ h j − δy, ⇓ )] + H.c. , H ( e − h ) int = − X i,j,σ,σ ′ U ( eh ) i,j ( d ) e † iσ h † jσ ′ h jσ ′ e iσ .Here t p denotes the nearest-neighbor hopping amplitude while µ p represents the chemicalpotential in electron ( p = e ) or heavy-hole ( p = h ) layer. s z is the z -component of the Paulimatrices and V p τ p is the effective Zeeman splitting ( τ e =1 for electron layer and τ h = − p j,σ is the fermion annihilation operator at lattice site j with spin ± ↑ , ↓ )for p = e and spin ± ⇑ , ⇓ ) for p = h . α ( β ) is the Rashba SO interaction strength in theelectron (heavy-hole) layer. δx ( δy ) is the square lattice spacing along the x ( y ) direction.In the interaction term, U ( eh ) i,j ( d )= e /ε r(cid:12)(cid:12)(cid:12) ~R i,e − ~R j,h (cid:12)(cid:12)(cid:12) + d , where ε is the dielectric constantof the spacer and d is the interlayer distance. We only consider the interaction correla-4ive to exciton formation and ignore the electron-hole exchange interaction. The latticeHamiltonian can be transformed into the momentum space with the Fourier transformation( e ~k,σ , h ~k,σ )=1 / √ Ω P i e i~k · ~R i ( e i,σ , h i,σ ). The result reads H ( p ) kin ( ~k ) = X ~k,σ,σ ′ ,p [( ζ ( p ) ~k − µ p ) δ σ,σ ′ + V p τ p ( s z ) σ,σ ′ ] p † ~k,σ p ~k,σ ′ , H ( e ) R ( ~k ) = X ~k iα (sin k x − i sin k y ) e † ~k ↑ e ~k ↓ + H.c. , H ( h ) R ( ~k ) = X ~k iβ ( a k − ib k ) h † ~k ⇑ h ~k ⇓ + H.c. , (2) H ( e − h ) int = − X ~k,~k ′ ,~q,σ,σ ′ U ( eh ) ( q ) e † ~k + ~qσ h † ~k ′ − ~qσ ′ h ~k ′ σ ′ e ~kσ , where U ( eh ) ( q )= πe ǫq e − qd , ζ ( p ) ~k = − t p (cos k x +cos k y ), a k =2(3 cos k y sin k x − sin k x cos k x − k x ),and b k =2( − k x sin k y +sin k y cos k y +2 sin k y ). In the above Hamiltonian, the interlayertunneling is neglected, because the insulating spacer can supply a high barrier to stop thedirect interlayer hopping. We also neglect the intralayer electron-electron and hole-holeinteractions, since they are expected to renormalize the single-particle energy of eachlayer and have no essential influence on the topological properties of the system. In themean-field approximation, the above Hamiltonian can be written as H MF = X ~k,σ,σ ′ ,p [( ζ ( p ) ~k − µ p ) δ σ,σ ′ + V p τ p ( s z ) σ,σ ′ ] p † ~k,σ p ~k,σ ′ + X p H ( p ) R − X ~kσσ ′ (∆ σσ ′ ( ~k ) e † ~kσ h †− ~kσ ′ + H.c.) (3)+ 12Ω X ~k~qσσ ′ ∆ σσ ′ ( ~k )∆ ∗ σσ ′ ( ~k − ~q ) U ( eh ) ( q ) , where the EOPs are defined as∆ σσ ′ ( ~k ) = 1Ω X ~q U ( eh ) ( q ) D h − ~k + ~qσ ′ e ~k − ~qσ E . (4)In the Nambu notation with combined e - h field operator basis ψ =[ e ~k ↑ e ~k ↓ h †− ~k ⇑ h †− ~k ⇓ ] T , themean-field Hamiltonian is expressed as H MF = ψ † H MF ψ + const with H MF = Σ ( e ) ~k − µ e + V e s z ∆ ( ~k ) ∆ † ( ~k ) Σ ( h ) − ~k + µ h + V h s z , (5)5here Σ ( p ) ± ~k = ± ζ ( p ) ± ~k I ±H ( p ) R ( ± ~k ) and ∆ ( ~k ) = − ∆ ↑⇑ ( ~k ) ∆ ↑⇓ ( ~k )∆ ↓⇑ ( ~k ) ∆ ↓⇓ ( ~k ) (6)with ∆ σσ ′ ( ~k ) defined in Eq. (4).The complex EOPs ∆ σσ ′ ( ~k ) can be self-consistently obtained from exact numerical cal-culation of Eqs. (3) and (4) with respect to minimizing the ground state energy. In ournumerical calculation of ∆ σσ ′ ( ~k ), we set the lattice size 81 ×
81 and take t e = t h =1, µ e = µ h = − α =0.5, and β =0.5. There are four different kinds of choices for the perpendicular magnetiza-tion in the two magnetic films adjacent to the bilayer system. For the parallel configurations,in our numerical simulations we choose ( V e , V h )=(1 ,
1) and ( V e , V h )=( − , − V e , V h )=( − ,
1) and ( V e , V h )=(1 , − V e , V h ). Furthermore, we find that the EOPs will obtain k -dependent phases due to the Rashba SO coupling. For convenience of discussion, we define χ k =arctan(sin k y / sin k x ) and τ k =arctan( b k /a k ). As a typical example, the numerical resultsof EOPs for ( V e , V h )=(1 ,
1) are shown in Fig. 2. In this case, one can find from Fig. 2that the component ∆ ↓⇑ ( ~k ) in EOP matrix Eq. (6) is dominant, while the amplitudes ofthe other three components (∆ ↑⇑ , ∆ ↑⇓ , ∆ ↓⇓ ) are negligibly small. With keeping in mind thatthe k -dependent phases of EOPs are obviously due to the Rashba SO interaction, we haveanalytically constructed various possible SO interaction-induced phases in EOPs and turnedto compare these analytic approximate expressions with our exact numerical results. TableI summarizes the most optimal approximate phases for the four magnetic configurations.As an illustration, we plot in Fig. 3 our derived approximate condensate phases for the caseof ( V e , V h )=(1 , TABLE I. Dominant spin channel and approximate analytical EOP phase factors for different magnetization configurations ( V e , V h ) dominant EOP component phases of (∆ ↑⇑ , ∆ ↑⇓ , ∆ ↓⇑ , ∆ ↓⇓ )(1 ,
1) ∆ ↓⇑ ( ~k ) (1 , − ie iτ k , ie iχ k , e i ( χ k + τ k ) )( − ,
1) ∆ ↑⇑ ( ~k ) ( ie − iχ k , e − iχ k + iτ k , − , ie iτ k )( − , −
1) ∆ ↑⇓ ( ~k ) ( e − i ( χ k + τ k ) , − ie − iχ k , ie − iτ k , , −
1) ∆ ↓⇓ ( ~k ) ( − ie iτ k , − , e iχ k − iτ k , − ie iχ k )6 IG. 2: (Color online) The left (middle, right) panels respectively show our calculated magnitudes(real parts, imaginary parts) of the EOPs ∆ ↑⇑ ( ~k ), ∆ ↑⇓ ( ~k ), ∆ ↓⇑ ( ~k ), and ∆ ↓⇓ ( ~k ), at a typical setupof magnetization parameters ( V e , V h )=(1 , ↑⇓ ( ~k ), ∆ ↓⇑ ( ~k ), and ∆ ↓⇓ ( ~k )that are listed in Table I with ( V e , V h )=(1 , ↑⇑ ( ~k ) isnegligibly small (see Fig. 2(a2)). Here the upper and lower panels respectively plot the real andimaginary parts of these three phase factors. The black cirques denote Fermi surface. k -dependent phases in the EOPs maylead to the nontrivially chiral topological orders. For instance, let us consider the case of( V e , V h )=(1 , e iχ k ∼ k x + ik y k , and thus ∆ ↓⇑ ( ~k ) ∼ i | ∆ ↓⇑ ( ~k ) | k x + ik y k . Thatmeans the ( p x + ip y )-like pairing emerges.Moreover, an explicit picture of chiral TEI can be well understood in the two-bandapproximation. To reveal this fact, first the non-interacting part in the total Hamiltonianis rewritten in the single-particle eigenstate space as H = X ~k,s E es ( ~k ) ψ † es ( ~k ) ψ es ( ~k ) + E hs ( ~k ) ψ hs ( − ~k ) ψ † hs ( − ~k ) , (7)where E es = ζ ( e ) ~k − µ e + s p α (sin k x + sin k y )+ V e and E hs = − ζ ( h ) − ~k + µ h + s p β ( a k + b k )+ V h ( s =+ , − ) are respectively electron and heavy-hole band energies, and ψ ps denotes the rele-vant annihilation field operators. Here the single-particle eigenstates are given by ϕ e − ( ~k ) = e iθ k (cid:2) − if + ( k ) e − iχ k , f − ( k ) , , (cid:3) T ,ϕ e + ( ~k ) = e iθ k (cid:2) f − ( k ) , − if + ( k ) e iχ k , , (cid:3) T ,ϕ h − ( ~k ) = e iϑ k (cid:2) , , ig + ( k ) e iτ k , g − ( k ) (cid:3) T ,ϕ h + ( ~k ) = e iϑ k (cid:2) , , g − ( k ) , ig + ( k ) e − iτ k (cid:3) T , (8)where f ± ( k )= α √ sin k x +sin k y q α (sin k x +sin k y )+( √ α (sin k x +sin k y )+ V e ± V e ) and g ± ( k )= β √ a k + b k q β ( a k + b k )+( √ β ( a k + b k )+ V h ± V h ) . Note that θ k and ϑ k are k -dependent phasesand are in principal determined, during exciton formation, by exactly solving the groundstate of the system through our above self-consistent calculation. The single-particle bands E p ± ( ~k ) are shown in Fig. 1(b) (solid curves), from which it is easy to find that the excitonsare preferably formed between the lower electron band E e − and the upper hole band E h + .Moreover, the pairing relates to the Fermi surface of the bilayer system. With the valuesof the tunable parameters shown in the caption of the Fig. 1, the band E e − and band E h + have the nearly perfect nesting Fermi surface with the Fermi energy E F being nearlyzero, (namely, µ p = − t p ). Hence, we can deal with pairing in BCS picture in this situation.Now, we consider the electron-hole interaction part in Eq. (2) in terms of the filled electronband E e − and hole band E h + . In order to obtain an explicit picture, we use a roughapproximation by assuming a short-range interaction potential U ( eh ) ( q )= U δ ( q ). Then, after8ean-field treatment, the resultant two-band Hamiltonian for our exciton system is givenby ¯ H MF ≈ X ~k E e − ( ~k ) ψ † e − ( ~k ) ψ e − ( ~k ) + X ~k E h + ( ~k ) ψ h + ( − ~k ) ψ † h + ( − ~k ) − X ~k ( ¯∆( ~k ) ψ † e − ( ~k ) ψ † h + ( − ~k ) + H.c.) + 12 X ~k (cid:12)(cid:12)(cid:12) ¯∆( ~k ) (cid:12)(cid:12)(cid:12) U , (9)where ¯∆( ~k )= X s,s ′ U f s ( k ) g s ′ ( k ) D ψ h + ( − ~k ) ψ e − ( ~k ) E ( s, s ′ = ± ). The straightforward calculationcan prove X s,s ′ f s ( k ) g s ′ ( k ) ∼ k F . So ¯∆( ~k ) ≈ ∆ is almost k -independent and only nonzero around k F in BCS-type picture. In practice we can introducea factor γ ( ~k )= e − c ( k − k F ) e iω ( c and ω are real constants) to fit our exact multi-band self-consistent numerical results (say, Fig. 2) in the whole BZ. The gaped energy spectrum of¯ H MF is shown in Fig. 1(b) (dashed lines). Now, in the two-band approximation, the EOPsin Eq. (6) have the expressions as follows: ∆ ( ~k ) = ∆ e − c ( k − k F ) e i ( θ k − ϑ k + ω ) if + ( k ) g − ( k ) e − iχ k f + ( k ) g + ( k ) e − iχ k + iτ k − f − ( k ) g − ( k ) if − ( k ) g + ( k ) e iτ k , (10)where the phases e i ( θ k − ϑ k + ω ) are confirmed through our self-consistent calculation. It turnsout that Equation (10) gives a nice description of the numerical results. III. CHIRAL TOPOLOGICAL ORDER
In the presence of exchange fields ( V e , V h ) induced by the ferromagnetic films, the TRS ofthe system is broken. No less than the AQHE, the nonzero TKNN number can undoubtedlycharacterize the topological nature of the system if a stable bulk gap separates the groundstate and excited states. That means the topological property of the system will not bechanged without bulk gap closing in spite of adiabatically deforming | ∆ σσ ′ ( ~k ) | at the givenexchange fields. Hence, γ ( ~k ) in Eq. (10) is inessential for the system’s topological property.Moreover, only the dominant component of EOPs decides the system’s topological propertyat the given ( V e , V h , ). The straightforward calculation of I T KNN in Eq. (11) can prove theabove two arguments. 9n the following discussion, we use Eq. (10) to consider the system’s topological prop-erties. In general, in the spin-dependent Nambu space ( e ~k ↑ e ~k ↓ h †− ~k ⇑ h †− ~k ⇓ ), the EOPs indifferent spin channels are affected by the effective exchange fields. Additionally, the strongRashba SO interaction flaws the spin polarization of the carries along the z direction. Thetotal effect leads the factors f ± ( k ) g ± ( k ) to emerge in different spin channels of EOPs, andwhich decide the dominant one at given ( V e , V h , ). For convenience of the following discussion,we use (∆ uu , ∆ ud , ∆ du , ∆ dd ) to denote ∆ ( f + ( k ) g − ( k ) , f + ( k ) g + ( k ) , f − ( k ) g − ( k ) , f − ( k ) g + ( k )).The topological nature of the ground state | u ( ~k ) i can be charactered by non-zero I T KNN ,which reads I T KNN = − π Z BZ d k Ω ( ~k ) , (11)where Ω ( ~k )= − D ∂u ∂k x (cid:12)(cid:12)(cid:12) ∂u ∂k y E is the ground-state Berry curvature in BZ. The results aresummarized in Table II, which definitely shows chiral topological order with its windingbehavior depending on the choice of exchange-field parameters. From the bulk-edge corre-spondence, the nontrivial bulk topological number implies gapless edge states in the systemwith finite size. TABLE II. The TKNN numbers for effective exchange fields and corresponding EOP amplitudes ( V e , V h ) ∆ (∆ uu , ∆ ud , ∆ du , ∆ dd ) I T KNN (1 ,
1) 0 . . , , ,
0) 1( − ,
1) 0 . . , , , − − , −
1) 0 . . , , , − , −
1) 0 . . , , ,
1) 1In order to confirm the existence of the gapless edge states, we assume that the square-lattice system has two edges in y direction and is boundless in x direction. Correspondingly,we choose open boundary condition in y direction and periodic boundary condition in x direction of the lattice Hamiltonian in Eq (1) in mean-field approximation. The calculatedenergy spectrum at a typical case of ( V e , V h )=(1 ,
1) is illustrated in Fig. 4 (a). The red-solidand blue-dashed lines correspond to the different edge states with contrary chirality. It iseasy to find that the number of the gapless edge states is consistent with the bulk theorycharacterized by I T KNN . 10
IG. 4: (Color online) (a) The energy spectrum of the bilayer square-lattice system with two edgesat the y direction. k x denotes the momentum in the x direction. The magnetization parametersare set at ( V e , V h )=(1 , IV. TRANSPORT PROPERTY OF EDGE STATES
The nontrivial transport phenomena can be predicted due to the emergence of the edgestates in our system. From Fig. 4 (a), we can find that the edge sates in different chiraltopological order propagate on each boundary with opposite velocities and can be describedby the following Hamiltonian H ηedge = ± X k x ≥ λ η v F k x γ † η ( k x , y ) γ η ( k x , y ) , (12)where ± represents different edges and η =1,...,4 labels four different kinds of magnetic config-urations, namely, λ = λ =1 and λ = λ = − , v F is the Fermi velocity and k x is the momentummeasured from the Fermi surface. The quasiparticle operators for case ( V e , V h )=(1 ,
1) read γ ( k x , y ) = u ( k x , y ) e ↑ ( y ) + v ( k x , y ) h †⇑ ( y ) . (13)11he other cases have the similar forms. Due to the missing of the PHS, the quasiparticlesare not Majorana fermions.The edge states in the AQHE systems can be usually detected through the Hall conduc-tance responding to the external electromagnetic field [16][17]. However, the edge statesin our system are excitons which are charge neutral. A simple approach is to use thermaltransport measurement which is often used to judge the pair properties in high- T c supercon-ductors [18][19]. The six-terminal Hall bar showed in Fig. 4 (b) for detecting the edge statesof quantum (spin) Hall effect can be used to detect the thermal conductance. The samesetup has been used by Sato et.al [20] to detect the edge state in topological superconductor.We give the similar considerations with that in Ref. [20] as follows. The temperature mustbe sufficiently lower than the exciton gap ( T ≪ ∆ ) in order to suppress the contributionsfrom the fermionic excitations (electrons and holes) in the bulk and bosonic (phonons) ex-citations. The thermal conductance is defined by G ( T )= I ( T ) / (∆ T ) , where I ij ( T ) is athermal current between contacts i and j , and (∆ T ) ij is the temperature difference betweenthese contacts. In the low temperature limit, the T -dependence of G ( T ) have three origins:the linear law ∝ T from edge states for phase η , the exponentially low ∼ e − ∆ /T from bulkquasiparticles and the power law ∝ T from phonons. Furthermore, in analogy with thequantum spin Hall current discussed in Ref. [21], there is no temperature difference betweencontacts 2 and 3 (5 and 6) because the edge current is dissipationless. V. CONCLUSION
In conclusion, we have presented a scheme to realize the chiral topological excitonic insu-lator in the double quantum wells adjacent to two ferromagnetic films. We have predicteddifferent topologically nontrivial orders emergent along with changes in the magnetizationorientations in the ferromagnetic films. The topologically nontrivial orders can be charac-terized by the chiral topological numbers defined with TKNN numbers in bulk system orchiral edge states in edged system. Furthermore, we have given an experimental scheme todetect the excitonic gapless edge states. 12 cknowledgments
This work was supported by NSFC under Grants No. 90921003, No. 10574150 andNo. 60776063, and by the National Basic Research Program of China (973 Program) underGrants No. 2009CB929103, and by a grant of the China Academy of Engineering andPhysics. [1] F. D. M. Haldane, Phys. Rev. Lett. , 2015 (1988).[2] D. J. Thouless, M. Kohmoto, M. P. Nightingale and M. den Nijs, Phys. Rev. Lett. , 405(1982).[3] B. I. Halperin, Phys.Rev. B , 2185 (1982).[4] Y. Hatsugai, Phys. Rev. Lett. , 3697 (1993)[5] C. X. Liu, X. L. Qi, X. Dai, Z. Fang, and S. C. Zhang, Phys. Rev. Lett. , 146802 (2008).[6] N. Read and D. Green, Phys. Rev. B ,10267 (2000)[7] J. Nitta, T. Akazaki, and H. Takayanagi, and T. Enoki, Phys. Rev. Lett. , 1335 (1997).[8] J. P. Lu, J. B. Yau, S. P. Shukla, M. Shayegan, L. Wissinger, U. R¨ossler, and R. Winkler,Phys. Rev. Lett. , 1282 (1998).[9] D. Grundler, Phys. Rev. Lett. , 6074 (2000).[10] S. J. Papadakis, E. P. De Poortere, H. C. Manoharan, M. Shayegan, R. Winkler, Science ,2056 (1999)[11] Lev P. Gor’kov and E. I. Rashba, Phys. Rev. Lett. , 037004 (2001).[12] G. Engels, J. Lange, Th. Sch¨apers, and H. L¨uth, Phys. Rev. B , R1958 (1997).[13] R. Winkler, Phys. Rev. B , 4245 (2000).[14] R. Winkler, H. Noh, E. Tutuc, and M. Shayegan, Phys. Rev. B , 155303 (2002).[15] C. L. Yang, H. T. He, L. Ding, L. J. Cui, Y. P. Zeng, J. N. Wang, and W. K. Ge, Phys. Rev.Lett. , 186605 (2006).[16] P. L. McEuen, A. Szafer, C. A. Richter, B. W. Alphenaar, J. K. Jain, A. D. Stone, R. G.Wheeler, and R. N. Sacks, Phys. Rev. Lett. , 2062 (1990).[17] J. K. Wang and V. J. Goldman, Phys. Rev. Lett. , 749 (1991).[18] R. W. Hill, Cyril Proust, Louis Taillefer, P. Fournier, and R. L. Greene, Nature , 711 ,014506 (2005).[20] M. Sato and S. Fujimoto, Phys. Rev. B , 094504 (2009).[21] B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Science 314, 1757 (2006)., 094504 (2009).[21] B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Science 314, 1757 (2006).