Condensation of fractional excitons, non-Abelian states in double-layer quantum Hall systems and Z_4 parafermions
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J u l Condensation of fractional excitons, non-Abelian statesin double-layer quantum Hall systems and Z parafermions Edward Rezayi
Department of Physics and Astronomy, California State University, Los Angeles, CA 90032
Xiao-Gang Wen
Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139
N. Read
Department of Physics, Yale University, P.O. Box 208120, New Haven, CT 06520-8120 (Dated: November 4, 2018)In this paper, we study a method to obtain non-Abelian FQH state through double-layer FQHstates and fractional exciton condensation. In particular, we find that starting with the (330)double-layer state and then increasing the interlayer tunneling strength, we may obtain a single-layer non-Abelian FQH state S (330). We show that the S (330) state is actually the Z parafermionRead-Rezayi state. We also calculate the edge excitation of the S (330) state. I. INTRODUCTION
Non-Abelian fractional quantum Hall (FQH) states area new class of FQH states whose excitations carry non-Abelian statistics.
The internal order of non-Abelianstates is so different from the symmetry breaking ordersthat totally new approaches are needed to study them.Since the very beginning, two very effective yet very dif-ferent approaches were introduced; one is based on con-formal field theory and the other is based on projec-tive construction. Both approaches allow us to calcu-late non-Abelian quasiparticles and edge states.
Theprojective construction also allows us to calculate the ef-fective bulk topological field theory for the non-AbelianFQH states. This class of quantum states is proposed tobe a medium for fault tolerant quantum computation.
However, it is non trivial to realize non-Abelian states;in particular, most observed FQH states are believedto be Abelian FQH states. But by utilizing the spe-cial form of electron-electron interaction in the secondLandau level, a non-Abelian FQH state, the Moore-ReadPfaffian state, may be realized as the ν = 5 / The ν = 12 / , although more experimental studiesare needed to be sure.In this paper, we will study another possible mecha-nism for realizing non-Abelian states, namely the pos-sibility to make them in double layer systems. Experi-mentally, the hierarchal FQH states observed in doublesystems are quite different from those in the first Lan-dau level. So there may be non-Abelian FQH states indouble-layer systems.In general, double-layer FQH states contain a classof fractional exciton excitations. A fractional exciton(f-exciton) is formed by a pair of fractionally chargedquasiparticle and quasihole in each layer. Such a frac-tional exciton may carry fractional statistics. If we startwith an Abelian double-layer state, the condensation offractional excitons may generate a non-Abelian state. The condensation of fractional excitons and the resultingnon-Abelian Moore-Read Pfaffian state has been studiedfor the 331 double-layer state. Such a phenomenon isclosely related to the transition between the weak-pairing p -wave to strong pairing p -wave BCS superconductors. In this paper, we show that the fractional exciton conden-sation in the 330 double-layer state generate the k = 4Read-Rezayi parafermion state. II. F-EXCITONS AND THEIR CONDENSATION
Let us consider a double-layer quantum Hall systemof spin polarized electrons at filling fraction 2 /n , where n is odd. We will consider only the T = 0 quantumground states. First we assume the interlayer tunnelingand interlayer interaction to be zero. In this case theelectrons in the two layers form two independent ν = 1 /n Laughlin states, which is denoted as ( nn
0) state. Such astate is a special case of more general ( mnl ) double-layerstates whose wave function is given byΦ mnl ( { z i } , { w i } ) (1) ∝ Y i 0) state) induced by interlayer repulsion, let usconsider the following excitation in the ( nn 0) state whichis formed by a charge e/n quasiparticle in one layer anda charge − e/n quasihole in the other layer. We will callsuch an excitation a fractionalized exciton (f-exciton).Clearly, the presence of a finite density of f-excitons willcause a charge imbalance between the two layers. Thus astrong interlayer repulsion will generate many f-excitonsand cause a charge imbalanced state such as the single-layer ν = 2 /n state.This consideration suggests that if we increase inter-layer repulsion in the ( nn 0) state, the energy gap for thef-excitons will be reduced. When the energy gap of thef-exciton is reduced to zero, the ( nn 0) state will becomeunstable and a quantum phase transition will happen.Such a quantum phase transition can be studied usingthe approach developed in Ref. 16. The effective the-ory for the transition is found to be a Ginzburg-Landau-Chern-Simons (GLCS) theory L = | ( ∂ + ia ) φ | − v | ( ∂ i + ia i ) φ | − m | φ | − g | φ | − n π a µ ∂ ν a λ ǫ µνλ . (2)Such a low energy effective theory describes the criticalpoint at the transition.We would like to make a few remarks:(a) Near the phase transition, both f-excitons andanti-f-excitons become gapless. This is why the effectiveGLCS theory is relativistic. The field φ describes bothf-excitons and anti-f-excitons.(b) The f-excitons obey fractional statistics with θ = 2 π/n . The U (1) gauge field a µ with the Chern-Simons term is introduced to reproduce the fractionalstatistics.(c) The energy gap of the f-excitons is m .(d) As we increase the interlayer repulsion from zero, m will decrease. For large interlayer repulsion, m will be reduced to a negative value, and cause a phasetransition from the φ = 0 phase to φ = 0 phase.(e) We note that in the φ = 0 phase the densities off-excitons and anti-f-excitons are equal. So both the φ = 0 phase and φ = 0 phases have an equal electrondensity in the two layers and are charge balanced states.The charge imbalanced state mentioned above willappear for even stronger interlayer repulsion.In the presence of inter-layer repulsion, the disper-sion of the f-excitons may have several different possible inter kE Imb stateVWC2e Laughlinnn0 state FIG. 1: Possible dispersions of low lying charge neutral exci-tations. In the nn nn e Laughlin state. At larger V inter , the Wigner crystal (WC)and a charge imbalanced state (imb state) may appear. E k inter nn0 state VImb stateWC FIG. 2: Another set of possible dispersions of low lying chargeneutral excitations. In the nn nn forms. One possible dispersion relation is illustrated inFig. 1. The effective theory (2) applies to such a situa-tion.Another possible dispersion relation for the f-excitonsis illustrated in Fig. 2. In this case, the effective theory(2) does not apply. Such case will likely lead to the WCstate. III. DOUBLE-LAYER QUANTUM HALLSTATES AND THEIR PHASE TRANSITIONS Let us assume that the situation described by Fig. 1is realized and the effective theory (2) is valid. Clearly,the φ = 0 phase is the ( nn 0) state. What is the φ =0 state? In Ref. 16, it is shown that the φ = 0 stateis the Laughlin state for charge 2 e electron pairs. Theeffective filling fraction for the 2 e pairs is ν eff = 1 / n .The φ = 0 to φ = 0 transition at m = 0 is believed tobe a continuous quantum phase transition. Such atransition is a transition between the ( nn 0) double layerstate and the charge-2 e Laughlin state, which exists inthe absence of interlayer tunneling.In the presence of a finite interlayer tunneling, the elec-tron numbers in each layer are no longer conserved sep-arately. The effective theory (2) will contain extra termsto reflect this reduced symmetry. Note that an f-excitonis created by the operator φ ˆ M , where ˆ M is an operatorthat creates 2 /n units of a µ flux. Thus φ n ˆ M n creates n f-excitons, which correspond to an electron in one layerand a hole in the other layer. So the electron tunnel- inter 220 doubleS(220) stateS(220) statet Laughlin statecharge−2elayer state V FIG. 3: The phase diagram for the (220) state, the charge-2 e Laughlin state, and the S (220) state. The S (220) state is thePfaffian state. t is the amplitude of the interlayer electrontunneling and V inter is the strength of interlayer repulsion. inter VS(330) stateS(330) state330 doublelayer state charge−2eLaughlin statet FIG. 4: The proposed phase diagram for the (330) state, thecharge-2 e Laughlin state, and S (330) state. ing operator correspond to φ n ˆ M n + h.c. . Thus, with in-terlayer electron tunneling, the effective theory near thetransition becomes L = | ( ∂ + ia ) φ | − v | ( ∂ i + ia i ) φ | − m | φ | − g | φ | − t ( φ n ˆ M n + h.c ) − n π a µ ∂ ν a λ ǫ µνλ (3)where t is the amplitude of the interlayer electron tun-neling.When n = 2, the f-excitons happen to have Fermistatistics. So the effective theory for the (220) state canbe mapped to a fermion model without the Chern-Simonsterm. The fermionic effective theory is exactly solublewhich allows us to calculate the phase diagram for the(220) state (see Fig. 3). For the critical point at t = m = 0, the gapless excitations are all neutral andare described by free massless Dirac fermions. For thecritical point at t = 0, the neutral gapless excitations aredescribed by free massless Majorana fermions.When n > 2, the effective theory for the ( nn 0) statecannot be solved. So we do not know the phase diagram,except that when t = 0 we belive that there is a continu-ous phase transition at m = 0. It is possible that such acontinuous phase transition is stable against a small in-terlayer tunneling. However, a large interlayer tunnelingmay induce a phase transition from the ( nn 0) state to a new state which will be called the single-layer ( nn 0) stateand denoted as S ( nn 0) state. This leads to the proposedphase diagram for the ( nn 0) state (see Fig. 4). IV. THE PROPERTIES OF THE S ( nn STATE What are the properties of the S ( nn 0) state? For largeinterlayer tunneling t , the electron state created by ψ † e + ψ † e has a lower energy and the state created by ψ † e − ψ † e has a higher energy. (Here ψ e and ψ e are the electronoperators in the two layers.) So we expect that for large t , the ( nn 0) state changes into a single-layer state whereelectrons are always in the even state ψ e + ψ e . Thisconsideration allows us to guess the groundstate wavefunction of the S ( nn 0) state (which is induced by largeinterlayer tunneling from the ( nn 0) state). Then fromthe guessed groundstate wave function, we can obtainthe physical properties of the S ( nn 0) state.We first note that the wave function of the ( nn 0) statecan be expressed asΦ nn ( { z i } , { w i } ) = h | Y i ψ e ( z i ) ψ e ( w i ) | nn i (4)where | i is the state with no electron and the state | nn i is the ( nn 0) state. An electron can be in a mixed even ψ e + ψ e and odd state ψ e − ψ e . After the electrons areprojected into the even state ψ e + ψ e , the ( nn 0) statechanges into the single-layer S ( nn 0) state. The wavefunction for the S ( nn 0) state is then given byΦ S ( nn ( { z i } ) = h | Y i [ ψ e ( z i ) + ψ e ( z i )] | nn i (5)For even n , we haveΦ S ( nn ( { z i } ) = S Y i,j ( z i − z j ) n ( z i +1 − z j +1 ) n e − P i | z i | (6)and for odd n Φ S ( nn ( { z i } ) = A Y i,j ( z i − z j ) n ( z i +1 − z j +1 ) n e − P i | z i | . (7)Here S is the symmetrization operator and A theanti-symmetrization operator. Note that Q i,j ( z i − z j ) n ( z i +1 − z j +1 ) n e − P i | z i | is the wave functionfor the ( nn 0) state with z i being the electron coor-dinates in the first layer and z i +1 the coordinates inthe second layer. So the symmetrization S or the anti-symmetrization A perform the projection into even states ψ e + ψ e .Once we find an expression of the ground state wavefunction in terms of correlation, such as eqn. (5), thenwe can use the parton construction developed in Ref. 4to find the bulk effective theory and the edge excitations.In other words, we can determine the topological orderfrom the expression of the ground state wave function(5).In this paper, we will concentrate only on the edgestates. What is the spectrum of the edge excitationsof the S ( nn 0) state? Let M be the total angular mo-mentum of the ground state wave function Φ S ( nn . Thenumber of low-energy edge states with a fixed total num-ber of electrons and at angular momentum M + l is givenby D l . In appendix A, we present a calculation of D l forthe S (220) and S (330) state.We find that the edge spectrum for the S (220) state(with an even number of electrons) is given by D l : 1 , , , , , , , , h ψ e ( t, ψ † e (0 , i ∼ t g e , g e = 2 . We find that the minimal charged quasiparticle ψ q inthe S (220) state carries a charge e/ 2. The quasiparticleoperator has the following correlation h ψ q ( t, ψ q (0 , i ∼ t g q , g q = 3 / . The wavefunction of the S (220) state is just the the ν = 1 Pfaffian state. The edge theory for the ν = 1 / S (220) state discussedabove.The edge spectrum for the S (330) state (with an 4 × integer number of electrons) is given by D l : 1 , , , , , , , , , , , ... (8)The electron operators have a correlation h ψ e ( t, ψ † e (0 , i ∼ t g e , g e = 3The minimally charged quasiparticle operator ψ q carriesa charge e/ h ψ q ( t, ψ † q (0 , i ∼ t g q , g q = 1 / . It turns out that the wavefunction of the S (330) stateis nothing but the k = 4 parafermion state introducedby Read and Rezayi. The edge excitations of the stateare described by a charge density mode ρ c and a k = 4parafermion conformal field theory. V. SUMMARY In this paper, we discuss another route to obtain non-Abelian FQH states through double-layer FQH states.In particular, we propose a possibility to start with the(330) double layer state, and then increase the interlayer tunneling strength. We argue that such a process maychange the (330) state to the S (330) state. Throughthe ideal wave function of the S (330) state, we find thatthe S (330) state is actually the Z parafermion statesproposed by Read and Rezayi (RR). We demonstratethe equivalence of the two states in appendix B.This research is supported by NSF grant no. DMR-0706078 (XGW), by DOE grant DE-SC0002140 (ER),and NSF grant DMR-0706195 (NR). Appendix A: The edge excitations of the Z Read-Rezayi parafermion state The edge excitations for the ( nn 0) state can be de-scribed through ρ I ( x ), I = 1 , ρ is the electron density for thefirst layer and ρ for the second layer. The Hilbert spaceand the dynamics of the edge excitations are describedby the following current algebra (in the k -space)[ ρ Ik , ρ Jk ′ ] = 1 n π kδ k + k ′ δ IJ (A1)and the Hamiltonian H = X k> V IJ ρ I, − k ρ J,k . (A2)eqn. (A1) and eqn. (A2) provide a complete description ofthe edge excitations. Note that eqn. (A1) and eqn. (A2)just describe a collection of harmonic oscillators labeledby k > I = 1 , 2, with the lowering operator a Ik ∝ ρ Ik and raising operator a † Ik ∝ ρ I, − k .The electron operators on the edge are given by ψ eI = e i nφ I ( x ) where π ∂ x φ ( x ) = ρ I ( x ). φ e is for the first layer and φ e the second layer. The electron operators have thefollowing correlation h ψ eI ( t, ψ † eI (0 , i ∼ t g e , g e = n. After we obtain the edge theory for the ( nn 0) stateand identify the electron operator, we are ready to dothe projection into the even states and to obtain the edgetheory for the S ( nn 0) state. To do so, we first identify anew electron operator ψ e and then use the new electronoperator and only the new electron operator to create theedge excitations of the single-layer S ( nn 0) state.Since the S ( nn 0) state only contains electrons in theeven state, so the new electron operator is ψ e ( x ) = ψ e ( x ) + ψ e ( x ) . The other combination ψ e ( x ) − ψ e ( x ) does not generategapless edge excitations and is dropped. If we use ψ e and ψ e to generate gapless edge excitations, we will generateall the edge excitations of the ( nn 0) state. However, toobtain the gapless edge excitations for the S ( nn 0) state,we can only use ψ e = ψ e + ψ e . So the S ( nn 0) state willhave fewer gapless edge excitations.What kind of edge excitations does ψ e generate? Toanswer such a question, we introduce the total electrondensity ρ c and the relative electron density ρ r of the twolayers: ρ c = ρ + ρ , ρ r = ρ − ρ .ρ c and ρ r satisfy the following current algebra[ ρ ck , ρ ck ′ ] = 2 n π kδ k + k ′ [ ρ rk , ρ rk ′ ] = 2 n π kδ k + k ′ In terms of φ c and φ r , defined through π ∂ x φ c = ρ c and π ∂ x φ r = ρ r , the new electron operator has the form ψ e = e i nφ c / cos( nφ r / . From the relation between the FQH wave function andCFT, the correlation of the above electron operatorreproduces the S ( nn 0) wave functionΦ S ( nn ( { z i } ) = h V ( z ∞ ) Y i ψ e ( z i ) i . (A3)First, let us consider the case for n = 3. We have shown(see appendix B) that Φ S (330) ( { z i } ) is the Z Read-Rezayi parafermion state. This means that Φ S (330) ( { z i } )can be expressed as a correlation function in the Z parafermion CFT:Φ S (330) ( { z i } ) = h V ( z ∞ ) Y i ψ ( z i ) e i3 φ c ( z i ) / i , (A4)where ψ is the simple current operator that generatesthe Z parafermion CFT. We see that ψ e ( z ) = ψ ( z ) e i3 φ c ( z ) / (A5)and we can identify cos(3 φ r / 2) with ψ . In fact,cos(3 φ r / 2) has a scaling dimension 3 / ψ .The operator product expansion of the ψ e ( z ) = ψ ( z ) e i nφ c ( z ) / will generate the operator ρ c , ψ ψ † ∼ ψ ( ψ ) etc. So the edge excitations (with a fixed totalelectron number) form a Hilbert space H which is thedirect product of two Hilbert spaces: H = H U (1) ⊗ H Z . H U (1) is generated by ρ c and H Z is generated by ψ ( z ) ψ ( z ) ψ ( z ) ψ ( z ).What is the spectrum of the edge excitations generatedby ρ c and ψ ( z ) ψ ( z ) ψ ( z ) ψ ( z )? Let M be the to-tal angular momentum of the ground state wave functionΦ S (330) . The number of the low-energy edge states at an-gular momentum M + l is given by D l . We can introducea function ch( ξ ) = ∞ X l =0 D l ξ l to describe the edge spectrum D l . The function is calledthe character of the edge excitations.If we only use ρ c to generate edge excitations, the char-acter will be ch c ( ξ ) = 1 Q ∞ l =1 (1 − ξ l ) . If we only use ψ ( z ) ψ ( z ) ψ ( z ) ψ ( z ) to generate edgeexcitations, the character will be that of Z parafermionCFT. The character (in the vacuum sector) for the Z k parafermion CFT is given bych Z k ( ξ ) = [ch c ( ξ )] × (A6) ∞ X r,s =0 ( − r + s ξ r ( r +1)+ s ( s +1)+ rs ( k +1) h − ξ ( k +1)(1+ r + s ) i . If we apply both ρ c and ψ ( z ) ψ ( z ) ψ ( z ) ψ ( z ) to gen-erate edge excitations, the character will bech( ξ ) = ch c ( ξ )ch Z ( ξ ) . (A7)The character ch( ξ ) describes the edge spectrum of the S (330) or Z Read-Rezayi parafermion state.We find that the edge spectrum for the S (330) state isgiven by D l : 1 , , , , , , , , , , , ... The electron operators have the following correlation h ψ e ( t, ψ † e (0 , i ∼ t g e , g e = n. Thus g e = 3 for the S (330) state.A quasiparticle operator ψ q must satisfy the followingcondition: in the operator product between ψ q and ψ e ψ e ( x ) ψ q (0) = X i x α i O i (0) , where the exponents α i must all be integers. In this case,we say that the quasiparticle operator ψ q is mutuallylocal with respect to the electron operator ψ e . One ofthe quasiparticles in the S ( nn 0) state is created by ψ ( , ) q = e i φ c / cos( φ r / e/n . The quasiparticle operator hasthe following correlation h ψ ( , ) q ( t, ψ ( , ) † q (0 , i ∼ t g q , g ( , ) q = 1 /n. For the Z state, we find g ( , ) q = 1 / Q ( , ) q = e/ ψ ( k,l ) q defined by ψ ( k,l ) q = e i kφ c cos[ lφ r ] , k = integer , l = integer , are also valid quasiparticle operators, which carry charge Q ( k,l ) = 2 ke/n . The exponent for such a quasiparticle isgiven by g ( k,l ) q = n ( k + l ).However, the quasiparticle ψ ( , ) q is not the one thatcarries minimal charge. The minimally charged quasi-particle has a charge Q q = e/ h q = + = . So the exponent for such a quasipar-ticle is g q = 2 h q = 1 / n = 2. We canfermionize the ρ r sector. Introducing a complex fermionfield ψ ( x ), the states generated by ρ r ( x ) can be equallygenerated by ψ † ( x ) ψ ( x ). In the fermion description e i φ r ( x ) ∼ ψ ( x ). Thus cos[2 φ r ( x )] ∼ ψ ( x ) + ψ † ( x ) ≡ λ ( x )where λ ( x ) is a Majorana fermion field which satisfies λ ( x ) λ (0) = 1 x + x [ ∂λ (0)] λ (0) + · · · Using ψ e = e i2 φ c λ , we find that ψ e ( x ) ψ † e (0) = x − + i4 πρ c (0) x − − π ρ c (0) − λ (0) ∂λ (0) + · · · Thus the edge excitations for the n = 2 case are generatedby ρ c and λ∂λ . These edge excitations are described bya density mode ρ c and a Majorana fermion λ . If we only use ρ c to generate edge excitations, the char-acter will be ch c ( ξ ) = 1 Q ∞ l =1 (1 − ξ l ) . If we only use λ∂λ to generate edge excitations, the char-acter will be ch Z ( ξ ) . If we apply both ρ c and λ∂λ togenerate edge excitations, the character will bech( ξ ) = ch c ( ξ )ch Z ( ξ ) . (A8)We find that D l : 1 , , , , , , , , , , , ... Appendix B: Relation to the Z RR state To show the equivalence the of S (330) and Z RR statewe will compare the boson version of the two states bydividing out a Jastrow factor Q i 2. Starting with S (330) we write it asΨ S (330) ( z , z , . . . , z N ) (B1)= X Q ( − ) Q Y Qi 0] (and [0 , , , (cid:18) N − N/ (cid:19) , (cid:18) N − N/ − (cid:19)(cid:18) (cid:19) , and 2 (cid:18) N − N/ − (cid:19)(cid:18) (cid:19) . It is straightforward to show that the sum of the aboveis (cid:0) NN/ (cid:1) , which is the number of terms in eqn. (B1).First consider the [5 , 0] terms. After dividing out theJastrow factor it is clear that these terms vanish if anytwo of the five particles coordinates are equal. For the[4 , 1] terms, a little algebra shows that they vanish if 4of the coordinates are set to be equal. We are left with[3 , 2] terms which contain factors such as( ω − ω ) ( ω − ω ) ( ω − ω ) ( ω − ω ) . (B2)In the rest of the wavefunction we set the ω i ’s to theircommon values ω without loss of generality. We can thencollect the terms in Q that permute the five particlesamongst themselves ( (cid:0) (cid:1) = 10 terms). Dividing out theJastrow factor, we obtain − X i 3] dis-tribution of the ω ′ s . To see that [3 , 2] and [2 , 3] are iden-tical for even N/ Q = I : (1 , , , . . . , N/ N/ , N/ , +2 , . . . , N ), Q ′ : ( N/ , N/ , . . . , N : 1 , , . . . , N/ . Clearly, these permutations produce identical terms but,depending on whether N/ N/ , 2] and [2 , N/ RR ( z , z , . . . , z N )Ψ S (330) ( z , z , . . . , z N ) = ( − N/ N/ N/ , (B4)where we have used the following form for the Z wave-function:Ψ RR ( z , z , . . . , z N )= X Q Y Qi