Fractional charge and statistics in the fractional quantum spin Hall effect
aa r X i v : . [ c ond - m a t . s t r- e l ] D ec Fractional charge and statistics in the fractional quantum spin Hall effect
Yuanpei Lan and Shaolong Wan ∗ Institute for Theoretical Physics and Department of Modern PhysicsUniversity of Science and Technology of China, Hefei, 230026,
P. R. China (Dated: October 13, 2018)In this paper, we consider there exist two types of fundamental quasihole excitation in the frac-tional quantum spin Hall state and investigate their topological properties by both Chern-Simonsfield theory and Berry phase technique. By the two different ways, we obtain the identical chargeand statistical angle for each type of quasihole, as well as the identical mutual statistics betweentwo different types of quasihole excitation.
PACS numbers: 73.43.-f, 71.10.Pm, 05.30.Pr, 03.65.Vf
1. INTRODUCTION
In condensed matter systems, most states of mattercan be characterized by Landau’s symmetry breakingtheory. However, topological states of quantum mattercannot be described by this theory. The first examplesof topological quantum states discovered in nature arethe quantum Hall (QH) states [1, 2] which opened up anew chapter in condensed matter physics. In the nonin-teracting limit, the integer quantum Hall (IQH) state ischaracterized by the TKNN invariant [3] or first Chernnumber. For the fractional quantum Hall (FQH) state,interaction between electrons turns out to be crucial, andan Abelian FQH state can be characterized by an integersymmetric K matrix and an integer charge vector t , upto a SL ( n, Z ) equivalence [4]. However, both the IQHand FQH states appear in a strong magnetic field whichbreaks time reversal symmetry.In a seminal paper, Haldane [5] has shown that theIQH state can be realized in a tight-binding graphenelattice model without net magnetic field, this state, how-ever, breaks time reversal symmetry due to a local mag-netic flux density of a zero net flux through the unit cell.Recently, Kane and Mele [6, 7] proposed a novel classof topological state, i.e. the integer quantum spin Hall(IQSH) state, which can be viewed as two spin-dependentcopies of Haldane’s model and that preserve time rever-sal symmetry. Bernevig and Zhang proposed the QSHstate for semiconductors [8], where the Landau levelsarise from the strain gradient, rather than the externalmagnetic field. In the Landau level picture, the QSHstate can be understood as the state in which the spin- ↑ and spin- ↓ electrons are in two opposing effective orbitalmagnetic field ∓ B realized by spin-orbit (SO) coupling,respectively. For the QSH state, the bulk is gapped andinsulating while there are gapless edge states in a timereversal invariant system with SO coupling. Bernevig,Hughes and Zhang predict a QSH state in HgTe/CdTeheterostructure [9], which has been confirmed by experi- ∗ Corresponding author; Electronic address: [email protected] ment [10].By analogy with the relation between the IQHE andFQHE, it is natural to ask the question whether therecan exist a fractional QSH (FQSH) state. An explicitwave function for the FQSH state was first proposed byBernevig and Zhang [8], and the edge theory was dis-cussed by Levin [11]. Recently, a set of exactly solu-ble lattice electron models for the FQSH state were con-structed [12], and the generic wave function for the FQSHstate was proposed through Wannier function approach[13].As we all known that quasihole (or quasiparticle) ex-citations above the FQH ground states have fractionalcharge and fractional statistics [14, 15], such as the ν =1 /k Laughlin states have quasihole excitations withcharge e/k and statistical angle θ = π/k . So we have toask how the properties of the excitations will be in FQSHstate. In order to answer this question, we investigatethe charge and statistics of the quasihole excitations inFQSH state. All five quantities for the fractional chargeand statistics are identically given by two different ways.It is the main task in this paper.The article is organized as follows. In Sec.2, we intro-duce the theoretical model of the QSHE and the wavefunction for the FQSH state proposed by Bernevig andZhang. In Sec.3, we first analyze the quasihole excita-tions in the FQSH state and write down the wave func-tion for each type of fundamental quasihole excitation.And then the identical charge and statistics of each typeof excitation as well as the mutual statistics between dif-ferent types of excitations are obtained by the two dif-ferent method in Secs.3.1 and 3.2, respectively. Finally,Sec.4 is devoted to conclusions.
2. WAVE-FUNCTION OF FRACTIONALQUANTUM SPIN HALL STATE
Now, we will brief review the theoretical model pro-posed by Bernevig and Zhang [8]. The simplest case ofthe QSHE can be viewed as superposing two QH sub-sysyems with opposite spins. The spin- ↑ QH state haspositive charge hall conduce ( σ xy = + e /h ) while thespin- ↓ QH state has negative charge Hall conductance( σ xy = − e /h ). As such, the charge Hall conductance ofthe whole system vanishes. However, the spin Hall con-ductance remains finite and quantized in units of e/ π since the spin- ↑ and the spin- ↓ QH states have oppositechirality.To realize this QSH state we need a spin-dependenteffective orbital magnetic field, which can be created bythe spin-orbit coupling in conventional semiconductors inthe presence of a strain gradient. When the off-diagonal(shear) components of the strain symmetric tensor are ǫ xy ( ↔ E z ) = 0, ǫ xz ( ↔ E y ) = gy , ǫ yz ( ↔ E x ) = gx ,respectively, with g denotes the magnitude of the straingradient, a single electron in a symmetric quantum wellin the xy plane can be described by H = p x + p y m + 12 C ~ g ( yp x − xp y ) σ z + D ( x + y ) , (1)where the second term corresponds to the spin-orbit cou-pling ( −→ p × −→ E ) · −→ σ , and the third term is the confiningpotential. The constant C / ~ is 8 × m/s for GaAs.We introduce l = (8 mD ) − / and ω = (8 D/m ) / , whichhave length dimension and energy dimension separately,then the above Hamiltonian can be expressed as H = 12 [ p x + p y + 14 ( x + y ) + R ( yp x − xp y ) σ z ] , (2)with R = C g p m/ D . For notation convenience, we usethe same notation for dimensionless x ( y ), p x ( p y ) and H ;the correct units can be restored by introducing factors of l , l − and ω . Since the eigenvalue of σ z is a good quantumnumber, we can use the spin along the z direction tocharacterize the state and have H = (cid:18) H ↑ H ↓ (cid:19) ,H ↓ , ↑ = 12 [ p x + p y + ( x + y ) ± R ( xp y − yp x )] . (3)We focus on the special case of R = 1, i.e. D = D = mg C /
8, where the Hamiltonian becomes H = 12 m ( −→ p + e −→ A σ z ) , −→ A = mgC e ( y, − x ) = B y, − x ) , (4)where B = mgC /e , −→ B = ∇ × −→ A = − B ˆ z playsthe role of effective magnetic filed, which is essentiallyarisen from the gradient of the strain. We notice, more-over, that when R = 1, the two quantities l and ω become l = (8 mD ) − / = p /eB and ω = ω =(8 D /m ) / = eB/m , which are exactly the definitionsof magnetic length l B and cyclotron frequency ω c inQH. Thus we can imagine the electrons of the two-dimensional system experience a spin-dependent effectivemagnetic field − B ˆ zσ z . We introduce the complex coor-dinate z = x + iy = re iθ , z ∗ = x − iy = re − iθ , and define the harmonic-oscillator ladder operators as a ↑ = 1 √ z ∂ z ∗ ) , a †↑ = 1 √ z ∗ − ∂ z ) ,a ↓ = 1 √ z ∗ ∂ z ) , a †↓ = 1 √ z − ∂ z ∗ ) . (5)They satisfy [ a σ , a † σ ′ ] = δ σ,σ ′ , [ a σ , a σ ′ ] = 0, ( σ, σ ′ = {↑ , ↓} ). In the vicinity of R = 1, the Hamiltonian can bewritten as H ↑ = a †↑ a ↑ + 12 , H ↓ = a †↓ a ↓ + 12 . (6)We denote m ↑ and m ↓ as the eigenvalues of a †↑ a ↑ and a †↓ a ↓ respectively. The z component of the angular momentumoperator is given as L z = − i∂ θ = z∂ z − z ∗ ∂ z ∗ = a †↓ a ↓ − a †↑ a ↑ ,which commutes with both H ↑ and H ↓ , and has eigen-values m ↓ − m ↑ . For spin- ↑ electron, the lowest Landaulevel (LLL) corresponds to m ↑ =0 and the single particlewave function is ψ ↑ m ↓ = z m ↓ √ π m ↓ m ↓ ! exp( − | z | ) with theangular momentum m ↓ . While for spin- ↓ electrons, theLLL corresponds to m ↓ =0 and the single particle wavefunction is ψ ↓ m ↑ = ( z ∗ ) m ↑ √ π m ↑ m ↑ ! exp( − | z | ) with the angu-lar momentum − m ↑ . It is easy to see that the QSHsystem is equivalent to a bilayer system: in one layer wehave spin- ↑ electrons in the presence of a down-magneticfiled( − B ˆ z ), the spin- ↑ electrons are chiral and have pos-itive charge Hall conductance, while in other layer wehave spin- ↓ electrons in the presence of an up-magneticfield(+ B ˆ z ), the spin- ↓ electrons are anti-chiral and havenegative charge Hall conductance.In addition, the wave functions of spin- ↑ and spin- ↓ are holomorphic and anti-holomorphic functions re-spectively. If we consider the intra-layer correla-tions and ignore inter-layer correlations, the many-bodywave function would be Q i 3. FRACTIONAL CHARGE AND FRACTIONALSTATISTICS IN THE FQSH SYSTEM As we mentioned in Sec.2, the FQSH system is anal-ogous to a bilayer FQH system, but the electrons in theFQSH system experience spin-dependent effective mag-netic field. For bilayer FQH system, there are two kindsof fundamental quasihole excitation which is in eachlayer, respectively [17]. Hence it is natural to speculatethat there exist two types of fundamental quasihole ex-citations in the FQSH state. A quasihole in the spin- ↑ layer is described by the wave functionΨ ( ↑ ) ( ξ ↑ ) = Y i ( ξ ↑ − z ↑ i )Ψ m ↑ m ↓ n ( { z ↑ i } , { z ∗↓ i } ) , (8)and in the spin- ↓ layer is described byΨ ( ↓ ) ( ξ ∗↓ ) = Y i ( ξ ∗↓ − z ∗↓ i )Ψ m ↑ m ↓ n ( { z ↑ i } , { z ∗↓ i } ) , (9)where ξ ↑ and ξ ↓ denote the coordinates of quasiholes ofthe spin- ↑ and spin- ↓ layers separately. In following, wecalculate the charge and statistics of the two types ofthe quasihole excitation as well as the mutual statisticsbetween a quasihole in the spin- ↑ layer and a quasiholein the spin- ↓ layer, and give all five quantities for thefractional charge and statistics by two different methodsin Secs.3.1 and 3.2, separately. The topological structure of the FQH states can beunderstood by several approaches and the most generalis the low energy effective Chern-Simons theory [4, 18].Wen and Zee [4] pointed out that the Abelian FQHliquids can be characterized by the integer valued K -matrices and the integer valued charged vectors, up to SL ( n, Z ) equivalences, and the quasiparticle quantumnumbers, such as fractional charge and fractional statis-tics, can be calculated from them. So we will searchfor the corresponding K -matrix and the charged vector t of the FQSH state. Now we first construct the Chern-Simons theory for the the FQSH state described by wavefunction (7). When n =0, the spin- ↑ electrons and spin- ↓ electronsdecouple, the FQSH state is just two independent FQHstates with opposite magnetic fields. We introduce twoU(1) gauge fields a ↑ µ , a ↓ µ to describe the conservedelectromagnetic current J µ ↑ = π ε µνλ ∂ ν a ↑ λ and J µ ↓ = π ε µνλ ∂ ν a ↓ λ , separately. Coupling the system to an ex-ternal electromagnetic gauge potential A µ (here the no-tation A µ is not the “gauge potential” of the effectiveorbital magnetic field in FQSH system), then the totaleffective Lagrangian is L = L ↑ + L ↓ , (10)with L ↑ = − m ↑ π ǫ µνλ a ↑ µ ∂ ν a ↑ λ + e π ǫ µνλ A µ ∂ ν a ↑ λ , (11) L ↓ = + m ↓ π ǫ µνλ a ↓ µ ∂ ν a ↓ λ + e π ǫ µνλ A µ ∂ ν a ↓ λ . (12)When we integrate out a ↑ and a ↓ from Eqs.(11) and(12), one can obtain the linear response of the both lay-ers to the external electromagnetic fields. In order tocorrectly reflect the fact that the spin- ↑ FQH layer haspositive charge Hall conductance while the spin- ↓ FQHlayer has negative charge Hall conductance, we must usethe negative sign in front of a ↑ ∧ da ↑ and the positive signin front of a ↓ ∧ da ↓ .We now turn to the case for n = 0, i.e. there existthe inter-layer correlations between the spin- ↑ electronsand spin- ↓ electrons. We start with the spin- ↓ FQH layer,namely Q i 4. CONCLUSIONS In this article, we study the topological properties ofthe two types of fundamental quasihole excitation in theFQSH state by Chern-Simons filed theory and Berryphase technique. We use the two different approachesto calculate the fractional charge and statistical angle ofeach type of quasihole excitation, as well as the mutualstatistics between the two different types of excitation,respectively. All results obtained from the two methodsare identical. Acknowledgement This work is supported by NSFC Grant No.10675108. [1] K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev.Lett. , 494 (1980)[2] D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev.Lett. , 1559 (1982)[3] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M.den Nijs, Phys. Rev. Lett. , 405 (1982)[4] X. G. Wen and A. Zee, Phys. Rev. B ,2290 (1992)[5] F. D. M. Haldane, Phys. Rev. Lett. , 2015 (1988)[6] C. L. Kane and E. J. Mele, Phys. Rev. Lett. , 226801 (2005)[7] C. L. Kane and E. J. Mele, Phys. Rev. Lett. , 146802(2005)[8] B. A. Bernevig and S. C. Zhang, Phys. Rev. Lett. ,106802 (2006)[9] B. A. Bernevig, T. A. Hughes, and S. C. Zhang, Science , 1757 (2006)[10] M. K¨onig, S. Wiedmann, C. Br¨une, A. Roth, H. Buh-mann, L. W. Molenkamp, X. L. Qi, and S. C. Zhang, Science , 766 (2007).[11] M. Levin and A. Stern, Phys. Rev. Lett. , 196803(2009)[12] M. Levin, F. J. Burnell, M. K. Janusz, and Ady Stern,arXiv:1108.4954[13] X. L. Qi, Phys. Rev. Lett. , 126803 (2011)[14] F. Wilczek, Fractional Statistics and Anyon Supercon-ductivity (World Scientific, Singapore, 1990)[15] C. Nayak, S. H. Simon, M. Freedman, S. D. Sarma, Rev.Mod. Phys. , 1083 (2008)[16] B. Halperin, Helv. Phys. Acta , 75 (1983)[17] X. G. Wen and A. Zee, Phys. Rev. Lett. , 1811 (1992) [18] S. C. Zhang, H. Hansson, and S. Kivelson, Phys. Rev.Lett. , 82 (1989)[19] F. Wilczek and A. Zee, Phys. Rev. Lett. , 2250 (1983)[20] A. M. Polyakov, Mod. Phys. Lett. A3 , 325 (1988)[21] D. Arovas, J. R. Schrieffer, and F. Wilczek, Phys. Rev.Lett. , 722 (1984)[22] R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983)[23] P. J. Forrester and B. Jancovici, J. Phys. (Pairs) Lett. , L583 (1984)[24] D. Arovas, J. R. Schrieffer, F. Wilczek, and A. Zee, Nucl.Phys. B251