Anyon Condensation and Continuous Topological Phase Transitions in Non-Abelian Fractional Quantum Hall States
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Anyon Condensation and Continuous Topological Phase Transitionsin Non-Abelian Fractional Quantum Hall States
Maissam Barkeshli and Xiao-Gang Wen
Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
We find a series of possible continuous quantum phase transitions between fractional quantumHall (FQH) states at the same filling fraction in two-component quantum Hall systems. These canbe driven by tuning the interlayer tunneling and/or interlayer repulsion. One side of the transitionis the Halperin ( p, p, p −
3) Abelian two-component state while the other side is the non-Abelian Z parafermion (Read-Rezayi) state. We predict that the transition is a continuous transition in the 3DIsing class. The critical point is described by a Z gauged Ginzburg-Landau theory. These resultshave implications for experiments on two-component systems at ν = 2 / ν = 8 / One of the most challenging problems in the studyof quantum many-body systems is to understand tran-sitions between topologically ordered states [1]. Sincetopological states cannot be characterized by brokensymmetry and local order parameters, we cannot usethe conventional Ginzburg-Landau theory. When non-Abelian topological states are involved, the transitionsthat are currently understood are essentially all equiv-alent to the transition between weak and strong-pairedBCS states [2, 3]. Over the last ten years, while therehas been much work on the subject, there has not beenanother quantum phase transition in a physically real-izable system, involving a non-Abelian phase, for whichwe can answer the most basic questions of whether thetransition can be continuous and what the critical theoryis. Here we present an additional example in the contextof fractional quantum Hall (FQH) systems.The quasiparticle excitations in FQH states carry frac-tional statistics and fractional charge [1]. In particular,in a ( ppq ) bilayer FQH state [1, 4], there is a type of ex-citation, called a fractional exciton (f-exciton), which isa bound state of a quasiparticle in one layer and an op-positely charged quasihole in the other layer. It carriesfractional statistics. As we increase the repulsion be-tween the electrons in the two layers, the energy gap ofthe f-exciton will be reduced; when it is reduced to zero,the f-exciton will condense and drive a phase transition.When the anyon number has only a mod n conserva-tion, this can even lead to a non-Abelian FQH state [2],yet little is known about “anyon condensation” [5–8]. Abetter understanding of these phase transitions may aidthe quest for experimental detection of non-Abelian FQHstates, because one side of the transition – in our case the(330) state at ν = 2 / ν = 1 / ppq ) state, when the energy gap of the f-excitonat k = 0 is reduced to zero, the f-exciton will condense[2].The transition can be described by the φ = 0 → φ = 0 transition in a Ginzburg-Landau theory with a Chern-Simons (CS) term: L = | ( ∂ + ia ) φ | − v | ( ∂ i + ia i ) φ | − f | φ | − g | φ | − πθ π a µ ∂ ν a λ ǫ µνλ , where θ is the statisti-cal angle of the f-exciton. Such a transition changes theAbelian ( ppq ) FQH state to another Abelian charge-2 e FQH state.[2, 3]In the presence of interlayer electron tunneling, thenumber of f-excitons is conserved only mod p − q . Anew term δ L = t ( φ ˆ M ) p − q + h.c. must be included, whereˆ M is an operator that creates 2 π flux of the U (1) gaugefield a µ . With this new term, what is the fate of the φ = 0 → φ = 0 transition?When p − q = 2, the f-excitons happen to be fermions( ie θ = π ), so we can map the L + δ L theory to a freefermion theory and solve the problem.[2] The problemis closely related to the transition from weak to strong-pairing of a p x + ip y paired BCS superconductor [3]. Theinterlayer electron tunneling splits the single continuoustransition between the ( p, p, p −
2) and the charge-2 e FQHstates into two continuous transitions. The new phasebetween the two new transitions is the non-Abelian Pfaf-fian state [12]. This is the only class of phase transitioninvolving a non-Abelian FQH state for which anything isknown.When p − q = 2, the f-excitons are anyons. The prob-lem becomes so hard that we do not even know where tostart. But we may guess that even when p − q = 2, aninterlayer electron tunneling may still split the transitionbetween the ( ppq ) and charge-2 e FQH states. The newphase between the two new transitions may be a non-Abelian FQH state [2]. When p − q = 3, it was suggestedthat the new phase is a Z parafermion (Read-Rezayi[13])FQH state [14]. This is because anti-symmetrizing the(330) wave function between the coordinates of the twolayers yields the Z parafermion wave function, in directanalogy to the known continuous transition from (331) toPfaffian, where anti-symmetrizing the (331) wave func-tion yields the single-layer Pfaffian wave function.In this letter, we show that the Abelian ( p, p, p − state can change into the Z parafermion state through acontinuous quantum phase transition. The transition is in the 3D Ising class. The critical point is described by a Z gauged Ginzburg-Landau theory. These results are experimentally relevant in the case p = 3. The (330) state has been experimentally realizedin double layer and wide quantum wells [10]. The exis-tence of a neighboring single layer non-Abelian state inthe phase diagram, which can be realized by tuning theinterlayer tunneling/repulsion, suggests that experimentshave a chance of realizing this transition. Furthermore,recent detailed experimental studies of the energy gapsof the ν = 8 / Z parafermion state lies close to the experimentally ob-served (330) state in the quantum Hall phase diagram,suggests that the Z parafermion state ought to be con-sidered as a candidate – in addition to other proposedpossibilities (e.g. [16]) – in explaining the plateau at ν = 8 /
3. In the case of the 5 / Z parafermion state at ν = 8 / Z parafermion state, which was found to be a U (1) × U (1) ⋊ Z Chern-Simons (CS) theory (a U (1) × U (1) CS theorycoupled with a Z gauge symmetry).[18] This is closelyrelated to the effective theory for the ( p, p, p −
3) state,which is a U (1) × U (1) CS theory. So effective theories forthe Z and the ( p, p, p −
3) states only differ by a Z gaugesymmetry. Thus the transition between the ( p, p, p − Z states may just be a Z “gauge symmetry-breaking” transition induced by the condensation of a Z charged field. We find that the U (1) × U (1) ⋊ Z CS the-ory contains a certain electrically neutral, bosonic quasi-particle that carries a Z gauge charge. We argue thatthis bosonic quasiparticle becomes gapless at the transi-tion and its condensation breaks the Z gauge symmetryand yields the ( p, p, p −
3) state.To obtain the above results, without losing generality,let us choose p = 3, and consider the (330) state and thecorresponding filling fraction ν = 2 / Z parafermionstate. The same results would also apply to filling frac-tions ν = 2 n + 2 /
3, where n is an integer. We begin byexplaining the quasiparticle content of the Z states, thenwe show that there exists an electrically neutral bosonicquasiparticle in the Z state whose condensation yieldsthe (330) state and that carries a Z gauge charge inthe low energy effective theory. Finally we discuss someconsequences for physically measurable quantities. One way to understand the topologically inequivalentexcitations is through ideal wave functions, which admita great variety of powerful tools for analysis of their phys-ical properties [12, 13, 19–25]. In the ideal wave functionapproach, the ground state and quasiparticle wave func-tions of a FQH state are taken to be correlation functionsof a 2D CFT: Φ γ ( { z i } ) ∼ h V γ (0) Q Ni =1 V e ( z i ) i , where Φ γ is a wave function with a single quasiparticle of type γ located at the origin and z i = x i + iy i is the coordinateof the i th electron. V γ is a quasiparticle operator in theCFT and V e are electron operators. The electron op-erator, through its operator product expansions (OPE),forms the chiral algebra of the CFT. Quasiparticles cor-respond to representations of the chiral algebra. Twooperators V γ and V γ ′ correspond to topologically equiv-alent quasiparticles if they differ by electron operators.The Z parafermion states, which exist at ν = 2 / (2 M +1) have 5(2 M + 1) topologically distinct quasiparticles.These can be organized into three representations ofa magnetic translation algebra,[21] which each contain2(2 M + 1), 2(2 M + 1), and 2 M + 1 quasiparticles – seeTable I, where we also listed a representative operator inthe corresponding CFT description of these states. TheCFT description of these states is formulated in termsof the Z parafermion CFT [26] and a free boson CFT.The Z parafermion CFT can be formulated in terms ofan SU (2) /U (1) coset CFT [27] or, equivalently, as thetheory of a scalar boson ϕ r , compactified at a special ra-dius R = 6 so that ϕ r ∼ ϕ r + 2 πR , and that is gaugedby a Z action: ϕ r ∼ − ϕ r [28]. Such a CFT is calledthe U (1) /Z orbifold CFT. In Table I, we have includedlabellings of the operators in the CFT using both of theseformulations.The fusion rules of these quasiparticles can be obtainedfrom the fusion rules of the Z parafermion CFT: Φ a × Φ lm = Φ lm + a , Φ × Φ = Φ + Φ , and Φ × Φ = I + Φ +Φ . Φ lm exists for l + m even, 0 ≤ l ≤ n , and is subjectto the following equivalences: Φ lm ∼ Φ lm +2 n ∼ Φ n − lm − n ,where n = 4 for the Z parafermion CFT.Another useful way to understand the topological or-der of the Z FQH state is through its bulk effec-tive field theory, for which there are several differentformulations.[18, 29, 30] Here we use the U (1) × U (1) ⋊ Z CS theory.[18] This is the theory of two U (1) gauge fields, a and ˜ a , with an additional Z gauge symmetry that cor-responds to interchanging a and ˜ a at any given pointin space. The Lagrangian is L = p π ( a∂a + ˜ a∂ ˜ a ) + q π ( a∂ ˜ a + ˜ a∂a ), where a∂a is shorthand for ǫ µνλ a µ ∂ ν a λ .For p − q = 3, it was found that the ground state degen-eracy on genus g surfaces of this theory agrees with thatof the Z parafermion FQH states at ν = 2 / (2 p − Z vortices, which corre-spond to defects around which a and ˜ a are interchanged,correspond to the non-Abelian Φ m quasiparticles ( − in Table I).[18] It was also found that quasiparticle ,which corresponds to the operator Φ (see Table I), is Φ lm e iQ √ ν − ϕ c Z Orbifold Label { n l } h pf + h ga I ∼ Φ e i √ / ϕ c I ∼ φ N e i √ / ϕ c e i / √ / ϕ c e i / √ / ϕ c e i / √ / ϕ c e i / √ / ϕ c Φ j r ∼ ∂ϕ r Φ e i / √ / ϕ c j r e i / √ / ϕ c Φ e i / √ / ϕ c j r e i / √ / ϕ c Φ e i / √ / ϕ c σ e i / √ / ϕ c + Φ e i / √ / ϕ c σ e i / √ / ϕ c + Φ e i / √ / ϕ c σ e i / √ / ϕ c + Φ e i / √ / ϕ c τ e i / √ / ϕ c + Φ e i / √ / ϕ c τ e i / √ / ϕ c + Φ e i / √ / ϕ c τ e i / √ / ϕ c + Φ e i / √ / ϕ c cos( ϕ r √ ) e i / √ / ϕ c + Φ e i √ / ϕ c cos( ϕ r √ ) e i √ / ϕ c + Φ e i / √ / ϕ c cos( ϕ r √ ) e i / √ / ϕ c + TABLE I: Quasiparticles in the Z parafermion FQH stateat ν = 2 / M = 1). The different representations of themagnetic translation algebra[21] are separated by horizontallines. Q is the electric charge and h pf and h ga are the scalingdimensions of the Z parafermion field Φ lm and the U (1) ver-tex operator e iαϕ c , respectively. ϕ c is a free scalar boson thatdescribes the charge sector. { n l } is the occupation numbersequence associated with the quasiparticle pattern of zeros. charged under the Z gauge symmetry. This latter re-sult is suggested by the orbifold formulation of the Z parafermion CFT [28], where Φ corresponds to a U (1)current j r ∼ ∂ϕ r . In the Z orbifold, the scalar boson ϕ r is gauged by the Z action ϕ r ∼ − ϕ r . The Z gaugesymmetry in the bulk CS theory is the Z gauging in theorbifold CFT, which suggests that the quasiparticle Φ would carry a Z gauge charge.From Table I, we see that the Z states contain a spe-cial quasiparticle, (Φ ), which is electrically neutral,fuses with itself to the identity, and has Abelian fusionrules with all other quasiparticles. Φ has scaling dimen-sion 1 and is a bosonic operator. In the following we showthat the condensation of this neutral bosonic quasiparti-cle yields the topological order of the (330) phase. Beforecondensation, two excitations are topologically equiva-lent if they differ by an electron, which is a local excita-tion. After condensation, all allowed quasiparticles mustbe local with respect to both the electron and Φ , andtwo quasiparticles will be topologically equivalent if theydiffer either by an electron or by Φ . In the CFT lan-guage, this means that Φ has been added to the chiralalgebra and will appear in the Hamiltonian. Such a situ-ation was analyzed in a general mathematical setting fortopological phases in Ref. 5.From the OPE of Φ and the other quasiparticle op-erators in the CFT description, we find that Φ is mu- tually local with respect to the quasiparticles in the firstand third representations of the magnetic algebra, whichconsist of the quasiparticles made of Φ m and Φ m (seeTable I). However, its mutual locality exponent with theΦ m quasiparticles is half-integer, which means that Φ is non-local with respect to those quasiparticles. This isexpected, because the Φ m quasiparticles were found tocorrespond to Z vortices in the U (1) × U (1) ⋊ Z CStheory while Φ was found to carry Z charge. Thus wewould expect that Φ would be non-local with respect tothe Φ m quasiparticles, with a half-integer mutual localityexponent. As a result, quasiparticles − are no longervalid (particle-like) topological excitations after conden-sation.Since quasiparticles that differ by Φ are regarded astopologically equivalent after condensation, quasiparti-cles , , and become topologically equivalent to quasi-particles , , and (see Table I), leaving three topo-logically distinct quasiparticles from this representation.Furthermore, the three quasiparticles in the third repre-sentation split into 6 topologically distinct quasiparticles.The reason for this was discussed in Ref. 5. Consider thefusion of a Φ quasiparticle, which we will label as γ , andits conjugate: γ × ¯ γ = + + . After condensation, weidentify (Φ ) with the vacuum sector, so if γ does notsplit into at least two different quasiparticles, then therewould be two different ways for it to annihilate into thevacuum with its conjugate. A basic property of topologi-cal phases is that particles annihilate into the vacuum ina unique way, so γ must split into at least two differentquasiparticles. Since the quantum dimension of γ is 2,it must split into exactly two quasiparticles, each withquantum dimension 1: γ → γ + γ . γ and γ are nowAbelian quasiparticles because they have unit quantumdimension. Therefore, we see that the 15 quasiparticlesin the Z parafermion state become, after condensationof Φ , the 9 Abelian quasiparticles of the (330) state.As the energy gap to quasiparticle (Φ ) is reduced,the low energy effective theory will simply be the theoryof this bosonic field coupled to a Z gauge field. Thephase transition is a Higgs transition of this Z chargedboson (at least in the case where we explicitly break theglobal Z symmetry of layer exchange). Note that thereis no U (1) symmetry that conserves the density of thiskind of excitation because Φ can annihilate with itselfinto the vacuum. Such a theory of a real scalar cou-pled to a Z gauge field was studied in Ref. 31, and itwas found that the transition is continuous and in the3D Ising universality class. As we mentioned before, thesame transition, when viewed as a transition from the(330) state to the Z state, is induced by an anyon con-densation with a mod-3 conservation. This suggests thatthe anyon condensation can, surprisingly, be describedby a Z -charged boson condensation.Given this result, a natural question is why fluctuationsof Φ should physically be related to interlayer densityfluctuations, which can be tuned by the interlayer tun-neling and interlayer repulsion. One answer to this ques-tion comes from the analysis of the wave functions. The(330) wave function in real space is Φ (330) ( { z i , w i } ) = h | Q Ni =1 ψ e ( z i ) ψ e ( w i ) | i , where ψ ei is the electronannihilation operator in the i th layer. When interlayertunneling is increased, the gap between the single-particlesymmetric and anti-symmetric states is increased, untileventually all electrons occupy the symmetric set of or-bitals, created by ψ e + , where ψ e ± ∝ ψ † e ± ψ † e . Thenatural wave function to guess in the limit of infiniteinterlayer tunneling is thus the projection onto the sym-metric states: Φ( { z i } ) = h | Q Ni =1 ( ψ e ( z i )+ ψ e ( z i )) | i ,which in this case is the Z parafermion wave function[14]. Thus starting from the (330) state, the wave func-tion analysis suggests that interlayer tunneling will yieldthe Z parafermion state. Since the (330) state can beobtained from the Z parafermion state by condensingΦ , we are led to take this wave function analysis seri-ously and are led to conclude that the gap to Φ canbe controlled by interlayer tunneling. Note that tuninginterlayer tunneling will tune fluctuations of the opera-tor ψ † e + ψ e − , which is related to interlayer density fluctu-ations (since ψ † e + ψ e − + h.c. = ψ † e ψ e − ψ † e ψ e ). Sincethe dimensionless parameters that we are concerned withare t/V inter and V inter /V intra , we see that properly tun-ing the inter/intra -layer Coulomb repulsions (given by V inter/intra ) should also drive the interlayer density fluc-tuations and the corresponding transition. For yet a dif-ferent perspective, we refer to[32].Since the transition between these two FQH states isdriven by the condensation of a neutral quasiparticle, itwill be difficult to observe in experiments. Experimentshave observed a phase transition in bilayer systems at ν = 2 / Z parafermion transitiondirectly by probing the gapless neutral mode. The anal-ysis here predicts that the bulk of the sample should re-main an electrical insulator but become a thermal con-ductor at the transition. 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