Elementary Excitations in Fractional Quantum Hall Effect from Classical Constraints

Classical constraints on the reduced density matrix of quantum fluids in a single Landau level, termed as local exclusion conditions (LECs) [B. Yang, arXiv:1901.00047], have recently been shown to characterize the ground state of many FQH phases. In this work, we extend the LEC construction to build the elementary excitations, namely quasiholes and quasielectrons, of these FQH phases. In particular, we elucidate the quasihole counting, categorize various types of quasielectrons, and construct their microscopic wave functions. Our extensive numerical calculations indicate that the undressed quasielectron excitations of the Laughlin state obtained from LECs are topologically equivalent to those obtained from the composite fermion theory. Intriguingly, the LEC construction unveils interesting connections between different FQH phases and offers a novel perspective on exotic states such as the Gaffnian and the Fibonacci state.


I. INTRODUCTION
Low lying elementary excitations in topologically ordered systems are fascinating objects that capture the essential topological features of the corresponding ground states. In fractional quantum Hall (FQH) systems, these excitations can carry fractionalised charges and obey anyonic or non-Abelian braiding statistics [1][2][3][4] . On one hand, discerning the nature of these excitations offers important insights into the underlying mechanisms of incompressibility of the FQH states. On the other hand, manipulating these excitations in a controlled manner also holds promises for storing and processing quantum information that is topologically protected [5][6][7][8] . Thus, a detailed understanding of the nature of the elementary excitations of FQH states is important from both the fundamental as well as the practical standpoint.
The elementary excitations in FQH systems can be categorized into charged and neutral excitations. The former includes the positively charged quasiholes and the negatively charged quasielectrons. The quasihole excitation can be realised by inserting flux quanta into the incompressible quantum fluid. For FQH states with model Hamiltonians 2,4,[9][10][11] , the quasihole states usually form a zero energy manifold degenerate with the ground state. The counting of the quasihole states on the sphere encodes the topological order of the FQH phase, from which one can derive the edge modes on the disk, or ground state degeneracy on the torus. Whether a FQH state is Abelian or non-Abelian is also determined by the quasihole properties. In particular, a state is non-Abelian if just specifying the positions of the quasiholes does not uniquely determine the wave function for the multiquasihole state.
Following the terminology used in Ref. 12 , we use the term "quasielectrons", instead of "quasiparticles", for el-ementary charged excitations carrying fractionalised negative charges in a FQH system. The quasielectrons can be created by adding electrons to, or removing fluxes from the FQH fluid. A neutral excitation is composed of a quasielectron and a quasihole. The low-lying branch of the neutral excitations of FQH fluids forms a collective mode called the magnetoroton mode 13,14 . A given FQH phase can host different types of quasielectron and neutral excitations 12,15 . For the FQH state to be incompressible, all the quasielectrons as well as the neutral excitations have to be gapped in the thermodynamic limit.
While the construction of the quasihole states is relatively straightforward, the construction of model states to represent quasielectrons and neutral excitations is much more involved. This is because, in contrast to the ground state and quasiholes, model Hamiltonians for quasielectrons and neutral excitations are not known. For many FQH states, the composite fermion (CF) 16 theory or the Jack polynomial 17 and conformal field theory (CFT) 18 approach can be used to construct excitations involving quasielectrons. These different approaches generically lead to different microscopic wave functions for the same topological state of quasielectrons 19 .
Previously, one of us introduced the idea of local exclusion conditions (LECs) in the FQH effect, and demonstrated that the LECs in conjunction with the requirement of translational invariance can determine the topological properties of many FQH states 20 . An LEC is a classical constraint on the reduced density matrix of a quantum Hall fluid. Each LEC is specified by a triplet of integersn = {n, n e , n h }. An imposition ofn on a quantum Hall fluid dictates that for any circular droplet within the fluid containing n fluxes, a physical measurement can neither detect more than n e number of electrons, nor n h number of holes. Topological indices including filling factor, shift 21 and particle clus-tering 10 can emerge from quantum Hall fluids satisfying the LECs. Furthermore, LECs also determine the microscopic model wave function of the many-body ground state for these topological phases.
In this work, we show that the LECs can also determine the elementary excitations of the FQH states. For FQH phases where the CF construction or the Jack polynomial/CFT approach is applicable, the model wave functions obtained from LECs agree qualitatively, and semi-quantitatively, with the wave functions obtained using these traditional methods. The LEC approach can also be applied to many FQH phases that do not lend themselves to a description in terms of CFs or a CFT. We shall present some examples of such states below.
The construction of the model wave functions for elementary excitations using the LECs offers a new perspective on the nature of these excitations. In particular, we find that a set of LECs that defines the quasielectrons of one FQH phase, could also define the ground state and the quasiholes of a different FQH phase. This not only sheds light on the relationship between different FQH phases, but also reveals interesting links between phases that were previously believed to be unrelated. More specifically, we show strong evidence that the Gaffnian state at ν = 2/5 22 is built from a particular type of quasielectrons of the Laughlin state at 1/3. Similarly, the Laughlin state at 1/3 can be viewed as arising from the condensation of the quasiholes of the Gaffnian state. Another example is the Fibonacci state in the Read-Rezayi series 10  We deploy the spherical geometry 9 for all our calculations. The LECs, being physical constraints, can in principle be applied to any geometry. However, the spherical geometry is the most convenient one since circular droplets at the north or south poles can be easily defined in this geometry. The two good quantum numbers on the sphere are the total orbital angular momentum L and its z-component L z . For a rotationally invariant Hamiltonian, states with a given L form a (2L + 1)-degenerate multiplet with −L ≤ L z ≤ L. The state with L z = L is defined to be the highest weight (HW) state in this multiplet. We only consider fully spin polarised quantum fluids here. Furthermore, we only look at the Hilbert space of a single Landau level (LL), assuming that the relevant physical processes (e.g. arising from small LL mixing) can be well captured by more complicated dynamics within a single LL (e.g. three or higher-body interactions).
The paper is organised as follows: In Sec. II we show how the quasiholes of the FQH states can be constructed with the LEC formalism, and discuss the resulting bulkedge correspondence between the ground state entanglement spectrum and the quasihole counting. Then, in Sec. III we show how different types of quasielectrons and neutral excitations can be constructed with the LEC formalism. In Sec. IV, we show that the construction of quasielectron states using LECs reveals connections between the Gaffnian and the Laughlin state, as well as between the Fibonacci and the Moore-Read state. This leads to a new interpretation of the nature of these exotic topological states (Gaffnian and Fibonacci state) as consisting of charged excitations of simpler FQH systems (Laughlin and Moore-Read state). In Sec. V we compare and contrast the elementary excitations constructed from the LEC formalism with those obtained from the composite fermion (CF) theory. We show that the excitations obtained from these two microscopically different approaches agree qualitatively in the region of filling factors between ν = 1/3 and ν = 2/5. We conclude the paper in Sec. VI with a summary of our results and provide an outlook for the future.

II. QUASIHOLE STATE CONSTRUCTION
Let us first generalise the LEC construction of the FQH ground states 20 to the quasihole states. The number of orbitals N o and the number of electrons N e , of the ground states for incompressible quantum Hall systems satisfy the relation Here the filling factor ν = p/q (p and q are positive integers), and S e , S o are integer topological shifts for the electrons and fluxes respectively. Extensive numerical evidence shows that the constraint imposed by one or a combination of LECs (denoted asĉ) on a rotationally invariant quantum Hall fluid determines a set of topological indices [p, q, S e , S o ] satisfying the following commensurability conditions 20 : For FQH states with a CFT description or a model Hamiltonian, the quasihole counting obtained from the LECs scheme matches exactly with the CFT or model Hamiltonian approach. For all states constructed from LECs, there is also an interesting bulk-edge correspondence between the counting of the ground state entanglement spectrum and the edge modes derived from bulk quasihole counting. This holds true empirically for all the system sizes we have checked. Such bulk-edge correspondence is well-known in the CFT construction of the FQH states: the counting of levels in the entanglement spectrum of the ground state agrees with the edge state counting of the quasiholes constructed from the same CFT model 23,24 . The LEC construction indicates that such a correspondence holds even for states with no known CFT description, suggesting that the bulk-edge correspondence may be an intrinsic property of the algebraic structure of the truncated Hilbert space.
As an example, let us look at different topological phases at filling factor ν = 3/7. Consider two differ- On the one hand, the quasihole counting obtained from these two different constraints are identical. Moreover, the counting of levels in the entanglement spectra of the ground state wave functions obtained from these two constraints also agree with each other [see Fig. 2a),2b)]. On the other hand, the ground state wave functions obtained fromĉ 1 andĉ 2 have vanishingly small overlap with each other. We conjecture thatĉ 1 andĉ 2 realize different topological phases, which could possibly be distinguished by analysing their topological entanglement entropy (TEE) 25,26 in the thermodynamic limit. At the moment, due to technical challenges we do not have access to the ground state wave functions of these LECs for very large system sizes, which precludes a reliable extrapolation of their TEEs. A third topological phase at ν = 3/7 can be realised bŷ  [3,7,1,5], and thus has different shifts, and different quasihole counting [leading to a different entanglement spectrum for the ground state (see Fig. 2c))] compared to the phases obtained fromĉ 1 and c 2 .    In all three cases, there are no apparent CFT descriptions of the FQH states, but the bulk-edge correspondence holds. In particular, the quasihole counting (and thus the edge state counting) is identical forĉ 1 andĉ 2 , whileĉ 3 has a different counting. Empirically, this suggests that while different LECs determine different topological phases, the quasihole counting is only determined by the shifts and the filling factor.

III. QUASIELECTRON AND NEUTRAL EXCITATION CONSTRUCTION
We will now move on to the quasielectron and neutral excitations, and illustrate the construction for these states in the simplest case of the Laughlin state. Our methodology can be easily extended for other sets of LECs. It is instructive to first recall the construction of single quasielectron states from the Jack polynomial formalism. The starting point in the Jack formalism is the root configuration for the ground state, 100100100100 · · · , where · · · denotes repeated patterns of 100. A single quasielectron at the north pole can be added by flipping the leftmost 0 to 1 (adding one electron), and inserting two fluxes (or 0's). This results in a net addition of charge (−e)/3 (−e is the charge of the electron) consistent with the charge of the quasielectron at ν = 1/3. Different ways of inserting two fluxes leads to different types of quasielectrons as listed below: We name the quasielectron types as the (5, 2) Type or (6, 3) Type etc. for reasons that would become apparent later. The solid and empty circles below the root configuration indicate the locations of −e/3 quasiparticles and +e/3 quasiholes respectively. Here, the quasiparticles (or quasiholes) are located in regions where three consecutive orbitals in the root configuration accommodate more (or less) than a single electron. While a quasihole is an elementary excitation with charge +e/3, a quasielectron (not a quasiparticle) is the elementary excitation with charge −e/3. The quasielectron has a nontrivial internal structure as a bound state of two quasiparticles and one quasihole. Each of the quasielectron states only consists of the squeezed basis 19 from the respective root configurations shown in Eq. (4). They can be uniquely determined by the highest weight condition, with the constraint that the state relaxes back to the Laughlin ground state away from the north pole. Such quasielectron states are identical to the composite fermion construction, where different types of quasielectrons correspond to adding a single composite fermion in different CF Landau levels 19 . The basis squeezed from the (5, 2) Type root configuration manifestly satisfiesĉ α = {2, 1, 2} ∨ {5, 2, 5}, which is the set of LECs that define the Gaffnian ground state and its quasiholes 20 . Thus the (5, 2) Type quasielectron state can also be understood as a condensation of the Gaffnian quasiholes. The (5, 2) Type quasielectron is a charged excitation of the Laughlin state (defined by {2, 1, 2} 20 ) since the quasielectron at the north pole breaks the {2, 1, 2} constraint. However, the (5, 2) Type quasielectron still satisfiesĉ α . This justifies the use of our terminology for the various types of quasielectrons in Eq. (4). In fact, if we imposeĉ α on the entire Hilbert space of L z = N e /2 states, together with the highest weight condition, we obtain a large number of Gaffnian quasihole states. We can now interpret these states as containing a single quasielectron and potentially multiple neutral excitations (a neutral excitation is composed of a quasielectron and a quasihole) on top of the Laughlin state at ν = 1/3. These different states can be resolved by diagonalizing the highest weight subspace with the V 1 Haldane pseudopotential interaction (referred to as V 1 Hamiltonian from here on) 9 .
The two-quasielectron (of (5, 2) Type) states can be constructed by imposingĉ α on the Hilbert space of Each species of quasielectrons in Eq. (4) is similarly defined by its respective LECs. The (6, 3) Type is defined byĉ β = {2, 1, 2} ∨ {6, 3, 6}, while the (7, 4) Type is defined byĉ γ = {2, 1, 2} ∨ {7, 4, 7}, and so on. We note that Wĉ α No,Ne ⊆ Wĉ β No,Ne ⊆ Wĉ γ No,Ne , since the constraintĉ α is stricter thanĉ β , which in turn is stricter than c γ . All such Hilbert spaces can be explicitly computed numerically for allowed values of N e and N o . Let us define Q (5,2) No,Ne := Wĉ α No,Ne as the Hilbert space spanned by all states containing only (5, 2) Type quasielectrons. Similarly, Q (6,3) No,Ne is spanned by states that contain at least one (6, 3) Type quasielectron. Note that Q (6,3) No,Ne can contain some (5, 2) Type quasielectrons, and these (5, 2) Type quasielectrons may or may not be dressed with (5, 2) Type or (6, 3) Type neutral excitations. Numerically, Q (6,3) No,Ne can be easily determined by projecting out Wĉ α No,Ne from Wĉ β No,Ne . We can also define Q (7,4) No,Ne analogously by projecting out Wĉ β No,Ne from Wĉ γ No,Ne and so on. Therefore we have the following relations: Q (6,3) No,Ne ⊆ Wĉ β No,Ne , Q (6,3) No,Ne ⊥ Wĉ α No,Ne . Q (7,4) No,Ne ⊆ Wĉ γ No,Ne , Q (7,4) No,Ne ⊥ Wĉ β No,Ne . . . ., where ⊥ denotes the two spaces are orthogonal. In this way, the entire Hilbert space of the Laughlin phase can be systematically organized. Each quasielectron manifold Qt No,Ne (wheret denotes the LEC type) are physically distinct and orthogonal to each other. This is because different types of quasielectrons have different intrinsic angular momentum, and thus, in principle, could be experimentally distinguished. In Fig. 3  No,Ne as an example, all states in this Hilbert space contain (5, 2) Type quasielectron(s) and possibly some Laughlin quasiholes. The microscopic Hamiltonian crucially decides whether these quasielectrons can be effectively treated as "elementary particles". The V 1 Hamiltonian can be diagonalised within Q (5,2) No,Ne to resolve states containing different number of quasielectrons. In a multi-quasielectron state, interactions between quasielectrons, as well as between quasielectrons and quasiholes, can make it difficult to ascertain the precise number of quasielectrons in a particular state. This is because the variational energies are no longer just an integer multiple of the single quasielectron creation energy. The deviation, however, is small if the quasielectrons are far away from each other.
The magnetoroton mode (one (5, 2) Type quasielectron dressed by one quasihole) can be clearly seen in Fig. 3a) for a neutral system. For large momenta when the quasielectron is well-separated from the quasihole, the creation energy of a single quasielectron is roughly equal to the variational energy of the state (since a quasihole far away from other excitations costs negligible energy with the V 1 Hamiltonian). This variational energy also matches a single undressed quasielectron state in Fig. 3b), as the lowest energy state at L = N e /2. All other higher energy states contain one single quasielectron dressed by (5, 2) Type neutral excitations and therefore contain multiple quasielectrons. In Fig. 3c) and Fig. 3d), the lowest energy states contain two and three undressed quasielectrons respectively. Similar analysis can be done by diagonalising the V 1 Hamiltonian within Q (6,3) No,Ne or sectors of other types of quasielectrons, and states containing undressed quasielectrons of different types can also be determined. With q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q  V 1 Hamiltonian, the creation energies of these quasielectrons are higher than that of the (5, 2) Type. Additional branches of neutral excitations discovered in Ref. 20 can also be constructed just like the magnetoroton mode in Q (5,2) No,Ne . The key message here is that quasielectrons can be treated as weakly interacting particles for short range Hamiltonians dominated by V 1 . In addition, different types of undressed quasielectrons have different intrinsic angular momenta since the highest weight states live in different L 2 sectors 12 . Thus a rotationally invariant Hamiltonian cannot couple a single quasielectron of one type to another quasielectron of a different type.
In Fig. 3 we show the exact energy spectra of different systems (specified by N e and N o , with V 1 Hamiltonian), comparing them with the energy levels (we denote as variational spectra) obtained from diagonalising the same V 1 Hamiltonian in each quasielectron manifold Qt No,Ne separately. In each L sector, the low-lying states predominantly coming from Q (5,2) No,Ne match very well with the exact spectra, both in terms of the variational energies and wave function overlaps (> 99%). The same is true for the highest energy states, which consist of different types of quasielectrons depending on the system size and the orbital angular momentum. The mismatch between the exact spectra and variational spectra is due to the fact that the Hamiltonian couples different Qt No,Ne , at least for finite systems with V 1 interaction. Even though a single undressed quasielectron of one type cannot scatter into another type of undressed quasielectron because they have different quantum numbers, it can scatter into other types of dressed quasielectrons. Matrix elements coupling multiple quasielectrons of different types are non-zero if they carry the same quantum number. Thus with realistic interactions, only a dilute gas of quasielectrons can be resolved in terms of the different quasielectron types. If the interaction is dominated by V 1 the low lying quasielectrons are of the (5, 2) Type (if they are present). On the other hand, in large L sectors, quasielectrons of other types (e.g. the (6, 3) Type) are the low lying ones (since the (5, 2) Type is absent). These other types of quasielectrons can be experimentally detected as excitations with large angular momentum transfer from the probing particles (e.g. neutrons or photons) to the FQH fluid 28,29 .

IV. CONNECTIONS BETWEEN DIFFERENT FQH PHASES
We have shown that a set of LECs can define both the ground state and quasihole states of one FQH phase, and at the same time define the quasielectron and neutral excitations of a different FQH phase. This leads to a new perspective on the nature of many topological phases, as well as the connections between them. For example, c α = {2, 1, 2} ∨ {5, 2, 5} defines the Gaffnian state and its quasiholes 20 , as well as the (5, 2) Type quasielectron and neutral excitations (the constituent quasielectron of the neutral excitation is of the (5, 2) Type) of the Laughlin state at ν = 1/3. Thus this family of quasielectrons and neutral excitations can also be understood as the Gaffnian quasihole states, when the quasiholes are uniformly spaced far away from the local defect at whicĥ n = {2, 1, 2} is violated. Similarly, the Laughlin state can also be understood as a condensation of the Gaffnian quasiholes.
The Gaffnian ground state is also closely linked to the (5, 2) Type quasielectrons of the Laughlin states, which is very much reminiscent of the hierarchical or the CF construction. We shall report in detail elsewhere on the connections between the Gaffnian and CF states at ν = 2/5. Similarly, the (6, 3) Type quasielectrons at ν = 1/3 establishes a connection between the FQH phase at ν = 1/3 and ν = 3/7. This is because the (6, 3) Type quasielectrons and the ν = 3/7 ground state (together with its quasiholes) are all defined byĉ β = {2, 1, 2} ∨ {6, 3, 6}. We find that the ν = 3/7 ground state determined by theĉ β LEC has a high overlap with the Jain 3/7 state as well as the LLL Coulomb ground state [see Table II]. The fact that the Gaffnian state can be reinterpreted as the Laughlin (5, 2) Type quasielectron state is illustrated in Fig. 4a). Starting with the Laughlin ground state at ν = 1/3 with N e electrons and N d o = 3N e − 2 orbitals, we can add n qe number of (5, 2) Type quasielectrons by adding n qe electrons and 2n qe orbitals. Diagonalising the respective Q (5,2) No,Ne Hilbert space with the V 1 Hamiltonian, states containing n qe undressed quasielectrons can be resolved. In particular, when n qe = N e + 2, there is only one highest weight state in Q (5,2) No,Ne , which is translationally invariant. It turns out that this state is precisely the model Gaffnian state.
For even and odd number of electrons, the neutral excitations form the well-known magnetoroton mode and the neutral fermion mode respectively. These quasielectron and neutral excitations can be completely defined byĉ m = {3, 2, 3} ∨ {5, 3, 5}. It turns outĉ m also identifies the Fibonacci state at ν = 3/5, which is the next state after the MR state in the Read-Rezayi sequence 10 . Starting with the MR ground state at ν = 1/2 with N e electrons and N d o = 2N e − 2 orbitals, every time we add one electron and one flux (orbital), we are adding two quasielectrons, each with charge −e/4. We continue this process by adding two quasielectrons at a time. The undressed (5, 3) Type quasielectrons can be resolved by first obtaining the highest weight quasielectron Hilbert space Q (5,3) No,Ne withĉ m , then diagonalising within this Hilbert space with the model three-body Hamiltonian for the MR state. After adding N e electrons and N e orbitals, the only highest weight state in Q (5,3) No,Ne is the translationally invariant Fibonacci state, or the state containing 2N e quasielectrons of the (5, 3) Type [see Fig. 4b)].

V. COMPARISON OF THE COMPOSITE FERMION AND LEC CONSTRUCTIONS
In this section we compare the wave functions obtained from the composite fermion (CF) and LEC constructions in detail. We show that for quasiholes and undressed quasielectrons, these two theories, which are microscopically different, lead to model wave functions that agree with each other qualitatively and semi-quantitatively. For the sake of completeness, we will first provide a primer on the CF theory.
A vast majority of the LLL FQHE phenomena is captured in terms of emergent topological particles called composite fermions, which are bound states of electrons and an even number (2p) of quantized vortices 16 . CF theory postulates that a system of interacting electrons at filling factor ν = ν * /(2pν * ± 1) can be mapped onto a system of weakly interacting composite fermions at a filling factor ν * . In particular, integer filling of CF-LLs (termed ΛLs), i.e ν * = n, leads to FQHE of electrons at ν = n/(2pn ± 1). The mapping to integer quantum Hall effect (IQHE) leads to the following CF/Jain wave functions for interacting electrons in the LLL 16 : Here α labels the different eigenstates, Φ ν * is the Slater determinant wave function of electrons at ν * (with where overline denotes complex conjugation) and P LLL implements projection to the LLL. Throughout this work we carry out projection to the LLL using the Jain-Kamilla method 32,33 , details of which can be found in the literature 31,34-36 . The quasihole and quasielectron excitations of the FQH systems are obtained as composite fermion hole (CFH) and composite fermion particle (CFP) respectively. A CFH is a missing composite fermion in an otherwise full ΛL, while a CFP is a composite fermion in an otherwise empty ΛL. Wave functions of the CFP and CFH can be constructed along the lines of Eq. (7) using the analogy to the particle and hole excitations in the corresponding IQH state. The CF theory has been shown to be in excellent qualitative as well as quantitative agreement with exact diagonalization studies of the LLL Coulomb problem for both the ground states as well as the excitations 31,32,37,38 .
We shall focus our attention on the simplest FQH state at ν = 1/3, which in the CF theory maps to a ν * = 1 state. The ground state wave function of Eq. (7) for this case reduces to the Laughlin state 2 . Furthermore, the wave function of the CFH at ν = 1/3 is identical to that of the Laughlin quasihole 2 which is identical to the quasihole obtained from the LEC construction. Thus, to compare the CF and LEC constructions we shall mainly focus on the CFP. The wave function of the CFP at ν = 1/3 is not identical to that of the Laughlin quasielectron 2 . However, the CFP and the Laughlin quasielectron have high overlaps with each other for small systems 31 and are believed to describe the same excitation. The orbital angular momentum L of a single CFP at ν = 1/3 is L CFP 1/3 = N/2 31,39 . States consisting of multiple CFPs at ν = 1/3 are constructed using the analogy to multiple particle states at ν * = 1. The orbital angular momenta of states consisting of multiple CFPs can be ascertained by adding the angular momenta of the constituent CFPs.
The wave functions of Eq. (7) are most readily evaluated in first quantization using the Metropolis Monte Carlo method 40 . In Fig. 5 we show the LLL Coulomb spectra of multiple quasielectron states at ν = 1/3 obtained from the LEC and CF constructions. We first point out that the angular momentum quantum number for a state with a single quasielectron obtained from the LEC and CF constructions agree with each other. Moreover, their LLL Coulomb energies are close to each other, indicating high overlaps of the corresponding wave func-  tions. Furthermore, the overlaps and energies of multiple quasielectron states constructed from the LEC and CF theories are also in fairly good agreement with each other. It is important to note that the Coulomb interaction, unlike the short-range model V 1 interaction, is long-ranged. Therefore, a reasonable agreement in the LLL Coulomb energies of the CF and LEC states suggests that the two seemingly disparate constructions are consistent with each other. Finally, we mention that both the LEC and CF constructions give a good representation of the LLL Coulomb spectra as evidenced from their comparison with the spectra obtained from exact diagonalization [see Fig. 5].
The qualitative and semi-quantitative agreement between the LEC and the CF construction for the Laughlin quasielectrons has important implications. In particular, the Gaffnian and the CF states at ν = 2/5 turn out to be closely related. This is because the former is made of undressed (5, 2) Type quasielectrons, while the latter is made of undressed CF quasielectrons. Our analysis in this section shows that for undressed quasielectrons, the LEC and CF constructions produce physically equivalent states. When quasielectrons are dressed with neutral excitations, however, the LEC and the CF constructions have qualitatively different predictions in terms of the counting of the states. Typically, the manifold of dressed CF quasielectrons is much larger than the manifold of dressed LEC quasielectrons of the same type. A detailed study of these connections will be presented elsewhere.

VI. CONCLUSIONS
In this work, we extended the recently developed LEC construction 20 to study the elementary excitations, namely quasiholes, quasielectrons and neutral excitations, of many FQH phases. The quasihole model wave functions can be generated using the same set of LECs that determine the corresponding ground state. This can be achieved by imposing the highest weight condition on the truncated Hilbert space containing more orbitals than the ground state (with the same number of electrons as in the ground state). Empirically, we find the edge modes of the FQH phase derived from the quasihole counting matches exactly with the counting of the entanglement spectrum obtained from the ground state. Such agreement is known to hold for the FQH states constructed from a conformal field theory (CFT). Our studies indicate that such a bulk-edge correspondence holds more generally, even for states that do not lend themselves to a CFT description.
In contrast to the quasiholes, the set of LECs defining the quasielectrons is different from, but closely related to, the set that identifies the ground state. This is reasonable, since unlike quasiholes, quasielectrons cost a finite energy in the thermodynamic limit which arises from violating the commensurability conditions of the FQH ground state. It is well-known that a FQH phase can have different types of quasielectrons. Here we showed that the quasielectron manifold, which is composed of different types of quasielectrons, can be uniquely determined by the LEC construction. The neutral excitations can be studied analogously as bound states of quasielectrons and quasiholes. The LEC scheme thus allows us to systematically build the entire energy spectrum of a FQH phase with different types of elementary excitations.
We also identified a number of cases where a set of LECs defining the quasielectrons of one FQH phase also defined the ground state and quasiholes of a different FQH phase. In particular, we explicitly showed that the Gaffnian ground state at ν = 2/5 is made of a particular type of Laughlin quasielectrons. For undressed quasielectrons at ν = 1/3 we provided numerical evidence to show that the LEC and the standard CF constructions, which are microscopically different, lead to topologically equivalent states. Given that the Abelian Jain ν = 2/5 state is made of CF quasielectrons, we conjecture that just from the wave function itself, the Gaffnian ground state and the Jain ν = 2/5 state are physically indistinguishable. All topological indices that can be extracted from the two wave functions could be identical. Furthermore, the Fibonacci ground state at ν = 3/5 is made of a particular kind of Moore-Read quasielectrons. Thus, the LEC construction opens an avenue to find novel connections between various FQH states.
The LEC construction shows that, besides the ground state, the excitations of FQH fluids are also determined by the algebraic structure of the truncated Hilbert space. This is because the quasihole and quasielectron manifold (and thus the neutral excitation manifold) can be defined without referring to microscopic Hamiltonians. Indeed, the topological properties of a particular FQH phase characterise both the ground state as well as the excitations. For the ground state, the topological indices include the filling factor, the topological shifts, the topological entanglement entropy and particle clustering. For quasiholes and quasielectrons, the topological indices include their fractionalised charge, the quasihole counting, and the topological spins. We have shown here that the LEC construction can uniquely determine all these topological indices, and thus the universal properties of many FQH phases. It would be interesting to further explore how robust these universal topological properties are in the presence of realistic interactions. The study of the interplay between the Hilbert space algebra and a realistic interaction could have crucial experimental ramifications, especially for non-Abelian FQH phases. Figure 5. Comparison of the lowest Landau level Coulomb spectra obtained from exact diagonalization (red dashes), composite fermion theory (black dots) and local exclusion conditions (blue crosses) for systems consisting of one (a), two (b), three (c), four (d) and five (e) quasielectrons at ν = 1/3. The calculations were carried out in the spherical geometry with Ne electrons in No orbitals. The composite fermion states are shown schematically in the inset.