From Luttinger liquid to non-Abelian quantum Hall states
We formulate a theory of non-Abelian fractional quantum Hall states by considering an anisotropic system consisting of coupled, interacting one dimensional wires. We show that Abelian bosonization provides a simple framework for characterizing the Moore Read state, as well as the more general Read Rezayi sequence. This coupled wire construction provides a solvable Hamiltonian formulated in terms of electronic degrees of freedom, and provides a direct route to characterizing the quasiparticles and edge states in terms of conformal field theory. This construction leads to a simple interpretation of the coset construction of conformal field theory, which is a powerful method for describing non Abelian states. In the present context, the coset construction arises when the original chiral modes are fractionalized into coset sectors, and the different sectors acquire energy gaps due to coupling in “different directions”. The coupled wire construction can also can be used to describe anisotropic lattice systems, and provides a starting point for models of fractional and non-Abelian Chern insulators. This paper also includes an extended introduction to the coupled wire construction for Abelian quantum Hall states, which was introduced earlier.
pacs:73.43.-f, 71.10.Pm, 05.30.Pr
The search for non-Abelian states in electronic materials is an exciting frontier in condensed matter physicsqcreview (). Motivation for this search is provided by Kitaev’s proposalkitaev () to use such states for topological quantum computation. The quantum Hall effect is a promising venue for non Abelian states. There is growing evidence that the pfaffian state introduced by Moore and Readmr (); readgreen () describes the quantum Hall plateau observed at filling greiter (); heiblum (); marcus (); willett (). The Moore Read state gives the simplest non-Abelian state, with quasiparticles that exhibit Ising non-Abelian statistics. While the observation and manipulation of Ising anyons is an important goal, Ising anyons are not sufficient for universal quantum computationfreedman (). The parafermion state introduced by Read and Rezayireadrezayi () is a candidate for the quantum Hall plateau at . The quasiparticles of the Read Rezayi state are related to Fibonacci anyonsslingerland (); trebst (), which have a more intricate structure that in principle allows universal quantum computationfreedman (); bonesteel ().
There is currently great interest in realizing quantum Hall physics in materials without an external magnetic field or Landau levels. This possibility was inspired by Haldane’s realizationhaldane () that a zero field integer quantum Hall effect can occur in graphene, provided time reversal symmetry is broken. Though such an anomalous quantum Hall effect has not yet been observed, related physics occurs in topological insulatorshasankane (); qizhang (), which have been predicted and observed in both two and three dimensional systems. Recently, there have been suggestions for generalizations of this idea to zero field fractional quantum Hall statesneupert (); sheng (); qiqhall (); regnault (), as well as fractional topological insulatorslevinstern (); maciejko (); swingle (); levin2 (). The question naturally arises whether it is possible to engineer zero field non-Abelian quantum Hall stateskapit (); bernevig2 (), which exhibit the full Ising, or even Fibonacci anyons.
A difficulty with answering this question is a lack of methods for dealing with strongly interacting systems. Moore and Read, and later Read and Rezayi, built on Laughlin’s idealaughlin () and constructed trial many body wave functions as correlators of non trivial conformal field theoriesmr (); readrezayi (). This allowed most properties of the quasiparticles and edge states to be deduced, and it had the virtue of allowing the construction of interacting electron Hamiltonians with the desired ground state. However, since this approach relied on the structure of the lowest Landau level, it is not clear how it can be applied to a lattice system. Effective topological quantum field theoriesgirvin (); zhang (); read (); lopez (); wenzee () and parton constructions wenparton () provide an elegant framework for classifying quantum Hall states and provide a description of their low energy properties. However, since the original electronic degrees of freedom are replaced by more abstract variables, these theories provide little guidance for what kind of electronic Hamiltonian can lead to a given state.
In this paper we introduce a new method for describing non Abelian quantum Hall states by considering an anisotropic system consisting of an array of coupled one dimensional wires. Study of the anisotropic limit of quantum Hall states dates back to Thouless et al.tknn (), who used this limit to evaluate the Chern invariant in the integer quantum Hall effect. The Chalker Coddington modelchalker () for the integer quantum Hall effect also has a simple anisotropic limit, which is closely related to the coupled wire model. A coupled wire construction for Abelian fractional quantum Hall states was introduced in Ref. kml, . Here we build on that work and show how the coupled wire construction can be adapted to describe the Moore Read pfaffian state, as well as the more general Read Rezayi sequence of quantum Hall states.
The coupled wire construction has a number of desirable features. First, it allows for the definition of a simple Hamiltonian, expressed in terms of electronic degrees of freedom, that can be transformed, via Abelian bosonization, into a form that for certain special parameters can be solved essentially exactly. The quasiparticle spectrum, as well as the edge state structure follow in a straightforward manner. The coupled wire construction thus provides a direct link between a microscopic electron Hamiltonian and the low energy conformal field theory description of the edge states. As such, it provides an intermediate between the wave function approach to quantum Hall states and the effective field theory approach.
The coupled wire construction also provides a simple picture for the quantum entanglement present in quantum Hall statesdong (); lihaldane (); qikatsura (). When the electron Hamiltonian is transformed via Abelian bosonization, it becomes identical to a theory of strips of quantum Hall fluid coupled via electron tunneling between their edge states. It thus provides a concrete setting for the more abstract coupled edge state models considered by Gils, et al.gils (). The coupled wire model is also similar in spirit to the AKLT model of quantum spin chainsaklt (), which provides a similarly intuitive and solvable model for understanding fractionalization in one dimension. For the non Abelian quantum Hall states, the coupled wire construction provides a concrete interpretation for the coset construction, which is a powerful (albeit abstract) mathematical tool for describing non Abelian quantum Hall statesfradkin ().
A final virtue of the coupled wire construction is that it can be applied to zero field anisotropic lattice models. The effect of the magnetic field in the coupled wire model is to modify the momentum conservation relations when electrons tunnel between wires. A similar effect could arise due to scattering from a periodic potential. The coupled wire construction may thus provide some guidance for the construction of lattice models for the fractional and non Abelian quantum Hall states. Note that our construction is somewhat different from the proposals for Chern insulators Refs. neupert, ; sheng, ; qiqhall, ; regnault, because we do not require nearly flat bands with an non zero Chern number.
The outline of the paper is as follows. We will begin in section II with an extended introduction to the coupled wire construction for Abelian quantum Hall states. Much of this material was contained either explicitly or implicitly in Ref. kml, . Here, since we are free from the constraints of a short paper, we will fill in some details that were absent in Ref. kml, . In particular, we will explain the generalization of the coupled wire construction to describe systems of bosons, we will demonstrate the Abelian fractional statistics of quasiparticles described in our approach, and we will explicitly construct second level hierarchical fractional quantum Hall states.
Section III is devoted to the Moore Read state. We will begin with a construction of this state for bosons at filling . This leads to a bosonized model that can be solved using via fermionization. We will then show that for a special set of parameters the model has a particularly simple form, which can be interpreted in the framework of the coset construction of conformal field theory. We will conclude section III by showing how to construct the more general Moore Read state at filling for even (odd) for bosons (fermions).
In section IV we will generalize our construction to describe the Read Rezayi sequence at level . Again, the formulation is simplest for bosons at filling , where the coupled wire model is closely related to the coset construction of these state. We show that the coupled wire model leads maps to a bosonized representation of the critical point of a statistical mechanics model, which for reduces to the 3 state Potts model. This bosonized representation allows us to identify the parafermion primary fields and fully characterize the edge states of the Read Rezayi states. Finally, as in section II, we conclude by generalizing our results to describe bosonic (fermionic) level Read Rezayi states at filling for even (odd).
Some of the technical details are presented in the appendices. Appendix A gives a careful treatment of Klein factors, while Appendices B and C contain some of the conformal field theory calculations discussed in section IV.
Ii Abelian Quantum Hall States
ii.1 Coupled Wire Construction for Fermions
In this section we review the coupled wire construction for fermions introduced in Ref. kml, . We begin by considering an array of identical uncoupled spinless non interacting one dimensional wires, as shown in Fig. 1a, with a single particle electronic dispersion . We assume each wire is filled to the same density, characterized by Fermi momentum . The two dimensional electron density is then , where is the separation between wires. A perpendicular magnetic field, represented in the Landau gauge , shifts the momentum of each wire. The right and left moving Fermi momenta of the ’th wire are then
where . The filling factor is then
The low energy Hamiltonian, linearized about the Fermi momenta is
where describe the fermion modes of the ’th wire in the vicinity of the Fermi points .
We next bosonize by introducing bosonic fields and that satisfy
where we use the shorthand notation . is a bosonic phase field, while describes density fluctuations. The long wavelength density fluctuations on the ’th wire are
The electron creation and annihilation operators may be written
where is a regularization dependent short distance cutoff and is a Klein factor that assures the anticommutation of the fermion operators on different wires. Eq. 4 hides the zero momentum parts of and , which must be accounted for in order to correctly treat the Klein factors. Since this issue tends to obscure the simplicity of our construction, we will not dwell on it in the text of the paper. Appendix A contains a careful discussion of the zero modes and Klein factors, which shows when they can be safely ignored.
In terms of the density and phase variables, the Hamiltonian for non interacting electrons is
Interactions between electrons as well as electron tunneling between the wires can be added. In general, there are two classes of terms : forward scattering and inter channel scattering. The forward scattering terms conserve the number of electrons in each channel and can be expressed as the interactions between densities and currents. This leads to a Hamiltonian that is quadratic in the boson variables,
Here the matrix , where describes the forward scattering interactions. describes a gapless anisotropic conductor in a sliding Luttinger liquid phaseohern (); emery (); vishwanath (); sondhi (); ranjan ().
Symmetry allowed inter channel scattering terms must be added to . They can open a gap and lead to interesting phases. The allowed terms are built from products of single electron operators, and have the form
where are integers such that ( appears times for . It is convenient to write in terms of a new set of integers,
Then takes the form
where is a non universal constant. The product of Klein factors is
The oscillatory factor describing the net momentum of is
The operators define a term in the Hamiltonian,
There are physical constraints on the allowed . Since must be integers, we require
Charge conservation requires that
Momentum conservation implies
so that the oscillatory term in (14) vanishes.
can be studied perturbatively using the standard renormalization group analysis. The lowest order RG flow equation for is
The scaling dimension depends on the forward scattering interactions in . When the operator is irrelevant and does not destabilize the gapless sliding Luttinger liquid fixed point. When , is relevant and grows at low energy, destabilizing the sliding Luttinger liquid. In principle can be parameterized given an underlying model of the electron-electron interactions. However, may be renormalized by irrelevant and/or momentum non conserving operators, so it may not resemble the bare interactions. Here we follow the approach of Ref. kml, and assume that have values such that a particular operator (or set of operators) is relevant. Our object is to characterize the resulting non trivial strong coupling phases. There are special values of that lead to particularly simple boson Hamiltonians that can be solved exactly. These solvable points provide a powerful way to characterize the resulting strong coupling phases.
As shown in Ref. kml, a number of non trivial 2D phases can be analyzed using this approach, including Abelian fractional quantum Hall states, superconductors and crystals of electrons, quasiparticles or vortices. In particular, Abelian quantum Hall states are described by a single relevant operator satisfying . From (2) and (18) this corresponds to a filling factor
In Section II.C we will review this construction for the Laughlin states and the Abelian hierarchy states. But first, we will show that the coupled wire construction can also be straightforwardly applied to systems of bosons.
ii.2 Coupled Wire Construction for Bosons
We now consider coupled wires of one dimensional bosons. The low energy excitations of a single wire can be described by “bosonizing the bosons”, to express them in terms of a slowly varying phase and a conjugate density variable satisfying (4). The density fluctuations have important contributions near wavevectors that are multiples of the average 1D density .
As with the fermions, the long wavelength density fluctuation is . The density wave at is,
The Hamiltonian for bosonic wires coupled only by long wavelength interactions has exactly the same form as . The only difference is that the non interacting Pauli compressibility term is absent. Tunneling a boson between wire and in the presence of a magnetic field is described by the operator
where is the boson creation operator. Due to interactions this process can involve scattering from the density fluctuations of the bosons. The most general coupling term thus has the form
where is given in (14). This is almost identical to the inter channel scattering terms for fermions. The only differences are the absence of Klein factors and the constraints on the allowed values of . Charge and momentum conservation still requires (17, 18), but unlike for fermions, where and obey (16), the corresponding constraint for bosons is
ii.3 Laughlin States
Here we will examine the coupled wire construction for the Laughlin states in some detail. We include the details here because the Laughlin states provide the simplest non trivial application of the coupled wire construction. We begin by introducing the relevant interaction term, and then characterize the bulk quasiparticles and edge states.
ii.3.1 Tunneling Hamiltonian
The Laughlin sequence of quantum Hall states at filling are characterized by the correlated tunneling operators involving two neighboring wires. The relevant operator is associated with a link between wires and ,
Using (21) it can readily be seen that this term is allowed for magnetic fields corresponding to . Moreover, from (16) and (26), it is clear that odd (even) corresponds to a fermionic (bosonic) state, as is expected for the Laughlin state. In Eq. 27 we have suppressed the Klein factors, which are necessary for fermions. They are treated carefully in Appendix A. This term is represented schematically in Fig. 2a. The notation for this diagram is slightly different from the one used in Ref kml, . The vertical arrows describe the tunneling of charge between the wires (represented by ), while the circular arrows describe backscattering within a wire (represented by ). Note that the number of ’s is constrained by (16) or (26).
The tunneling operators defined above have the special property that they all commute with eachother. In particular, . This means that the components from wire in those two operators commute with one another. This invites us to introduce right and left moving chiral fields on wire that distinguish the two contributions to . We thus write
The decoupling can be explicitly seen from the commutation algebra,
The interaction term is now . The charge density is .
It is also convenient to introduce new density and phase variables defined on the links between wires,
These satisfy and
The charge density associated with the link can be written
In terms of the new variables, the Hamiltonian becomes,
where without loss of generality we have assumed is real. As shown in Appendix A, there is no Klein factor, provided the zero momentum component of is correctly defined.
Provided the forward scattering interactions defining are such that is relevant, the system will flow at low energy to a gapped phase in which is localized in a well of the cosine potential. As argued in Ref. kml, , it is always possible to find such interactions. In particular, consider a simple interaction such that has the decoupled form
The scaling dimension of is . It follows that for , is relevant. It should be emphasized that can be expressed in terms of the original fermion operators, which includes a specific four fermion forward scattering interaction. For special values of this model can be solved exactly. In the limit , the variable becomes a stiff classical variable, so that the approximation of replacing by becomes exact. For larger , we rely on our understanding that is renormalized downward by , so that at stiffens at low energy. For there is another exact solution because it is possible to define new variables such that the Hamiltonian has precisely the form of (7). The problem can then be refermionized and expressed in terms of non interacting fermions which have a single particle energy gap. We will not dwell on these exact solutions any further in this paper. We will be content with our understanding that any leads to a gapped state.
The gapped phase is the Laughlin statelaughlin (). This can be seen by examining the quasiparticle excitations and the edge states.
ii.3.2 Bulk Quasiparticles
Quasiparticles occur when has a kink where it jumps by . From (32) it can be seen that such a kink is associated with a charge . This makes the charge fractionalization in the fractional quantum Hall effect appear similar to the fractionalization that occurs in the one dimensional Su Schrieffer Heeger (SSH) modelssh (). However, there is a fundamental difference. The solitons in the SSH model occur at domain walls separating physically distinct states. This prevents solitons from hopping between wires via a local operator. In contrast, the states characterized by and are physically equivalent. They are related by a gauge transformation in which, say, , which takes to . This allows quasiparticles to hop via a local operator without the nonlocal string. Though SSH solitons and Laughlin quasiparticles are distinct, they become equivalent on a cylinder with finite circumference. The Tao Thouless “thin torus” limittaothouless () can be described the extreme case in which the “cylinder” consists of a single wire with electron tunneling “around” the cylinder. In this case, our theory maps precisely to an state version of the SSH model.
The local operator that hops quasiparticles between links and is simply the backscattering of a bare electron on wire , (or equivalently for bosons the density operator). Using the transformations (28) and (30) it is straightforward to show that
From (31) it can be seen that this operator takes to , transferring a quasiparticle from to . The operator that transfers a quasiparticle along wire from to along wire link is
which can also be expressed in terms of the bare electron densities and currents.
A quasiparticle operator may be defined as
In the bulk, since is gapped, we have . Of course, since can not be locally built out of bare electron operators, it is not by itself a physical operator. However, the operator that transfers a quasiparticle from one location to another can be built from a string of local operators like (II.3.2) and (36). This allows the fractional statistics of the quasiparticles to be seen quite simply.
To move a quasiparticle from to on link and then to on , use the operator
Since is gapped, this can be written
The string of is responsible for the fractional statistics.
Consider moving a quasiparticle through a closed loop. The operator that takes a quasiparticle around the rectangle formed by , , and can be constructed by doing (38) twice, which eliminates the quasiparticle operators. This then gives a phase
where is the number of quasiparticles enclosed by the rectangle. Here we have used the fact that simply counts the number of quasiparticles on link between and .
ii.3.3 Edge States
For a finite array of wires with open boundary conditions, the edge states are apparent, since there are extra chiral modes left over on the first and last wire. From (29), it can be seen that these modes have precisely the chiral Luttinger liquid structure of edge stateswenll ().
with . The electron operator on the edge is
It is straightforward to show that this operator has the expected dimension , characteristic of the chiral Luttinger liquid.
One can view the change of variables (28) as a transformation between a sliding Luttinger liquid built out of bare electrons and a sliding Luttinger liquid built out of edge states. The correlated tunneling term for the bare electrons becomes the electron tunneling operator coupling the edge states. The array of wires then becomes an array of strips of quantum Hall fluid coupled by electron tunneling, as shown in Fig. 2. When the electron tunneling is relevant the strips merge to form a single bulk fluid, leaving behind chiral modes at the edge.
The quasiparticle operator at the edge is
As discussed above, since can not be made out of bare electron operators, it is not by itself a physical operator. However, quasiparticle tunneling from the top to the bottom edge can be built from a string of backscattering operators (II.3.2). When the gapped bulk degrees of freedom are integrated out, this string of operators becomes
ii.4 Hierarchy States
In this section we show how the coupled wire construction describes hierarchical Abelian fractional quantum Hall stateshaldanehierarchy (); halperin (). We restrict ourselves to second level states, which are characterized by a matrixwenzee (). Generalization to higher levels is straightforward.
2nd level hierarchy states arise from an interaction term that involve three coupled wires. A generic term is shown in Fig. 3, and can be described by the operator
Here and are any integers, while () is an even integer for bosons (fermions). Again, we defer discussion of the Klein factors to Appendix A. From 21, this interaction conserves momentum at a filling factor
This set of states corresponds to the standard Haldane-Halperin hierarchy states at filling ( is even (odd) for bosons (fermions) and is even) for the choice, , and .
To analyze this state, we group the wires into pairs and . Pair is connected to pair by two tunneling terms, and . As in (28) we define new variables that decouple right and left moving modes on the pairs of wires.
The new fields obey the commutation algebra
where the matrix is
The charge density is
We next define variables on links ,
These satisfy, and
In terms of these new variables, the Hamiltonian in the presence of the correlated -electron tunneling operators becomes
If flows to strong coupling, we have a gapped bulk, describing a quantum Hall fluid characterized by the matrix (49). As in Section II.C.3, this can be interpreted as quantum Hall strips with edge states coupled by the charge tunneling operators
Quasiparticles, given by kinks in or are created by
They have charge . The bare electron backscattering operator corresponds to quasiparticle tunneling,
Iii Moore Read State
We now generalize the coupled wire construction to describe the Moore Read statemr (). Our approach was motivated by the observation by Fradkin, Nayak and Shoutensfradkin () that the Moore Read state for bosons at filling has a simple interpretation in terms of two coupled copies of bosons at . Each copy is described by a Chern Simons theory, and the coupling between them introduces the symmetry breaking fradkin2 ().
We therefore first consider the problem of coupled wires of bosons at filling , where the bosons on each wire have two flavors, each at . The allowed boson tunneling and backscattering terms in our construction have a simple representation in the low energy bosonized theory. Moreover, by fermionizing the bosons, the Majorana fermions associated with the Moore Read statereadgreen () emerge naturally.
There is a special set of values for the interactions in which the problem is particularly simple. In this case, the Hilbert space associated with the two right (left) moving chiral modes on each wire decouples into two sectors. One of the sectors is coupled to the corresponding sector of the left (right) moving modes on the same wire, while the other sector is coupled to the corresponding sector of the left (right) moving modes on the neighboring wire. Both of these couplings introduce gaps, but the two sectors are gapped in “opposite directions”. This gives a kind of hybrid between the insulating phase, in which all chiral modes are paired on the same wire, and the quantum Hall states, in which all the chiral modes are coupled on neighboring wires. What gets left behind on the edge is a fraction of the original chiral modes.
This fractionalization of the original chiral modes described mathematically in terms of the coset construction in conformal field theorycoset (); cft (). The original pair of chiral modes are described by a theory with central charge . These modes decompose into three sectors: , and with , and , respectively. The independent sectors are then gapped “in different directions”. We will describe this construction in Section III.B. This, in effect, gives a concrete and somewhat more explicit implementation of the Fradkin, Nayak, Shoutens construction.
After establishing the Moore Read state for bosons, we will go on to generalize our construction to account for fermions, and the -pfaffian state at filling , where is even (odd) for bosons (fermions).
iii.1 Bosons at
We begin with a Hamiltonian describing coupled wires of two component bosons, which can be viewed as a double layer system, as in Fig. 4a. Each component has a density corresponding to filling . has the same form as (8), except now each wire has two bosons, and , for , which satisfy
The interaction terms consist of boson tunneling and backscattering operators that are consistent with momentum conservation. We consider three such terms, depicted in Fig. 4.
The first term involves coupling between channel a on one wire and channel b on the neighboring wire.
This term is similar to (27). The coefficient 2 of the terms is fixed by the filling factor. In addition, there are allowed terms that couple the two channels on a single wire. These include a Josephson like coupling between the two channels,
as well as an interaction that locks the “” densities of the two channels,
These three terms (as well as combinations of them) are the only allowed interaction terms at that include up to first neighbor coupling.
It is now useful to introduce right and left moving chiral fields,
as well as “charge” and “spin” fields,
The latter fields satisfy
For simplicity, we will first focus on the case in which , independent of and , and and are real. We will comment on the more general case later. In this case, in terms of the new variables, we have
Note that in passing between the first and second lines of (67)-(69) getting the factors of right requires care in splitting the exponential. This is explained in Appendix A.3, where the zero momentum components of are properly taken into account.
Since the interaction term is a sum of non commuting terms, analysis of this state is more complicated than it was for the Abelian quantum Hall states. However, a tremendous simplification occurs when the forward scattering interactions in are such that and are decoupled, and the Hamiltonian for has the non interacting form, in (7). In this case, the operator has precisely the form of a bosonized Dirac fermion. This allows us to fermionize, by writing
where is a Dirac fermion operator, and , are Majorana fermion operators. For the charge sector, we define
They satisfy and
The Hamiltonian may now be written,