A non-Abelian fractional quantum Hall state at $3/7$ filled Landau level

We consider a non-Abelian candidate state at filling factor $\nu=3/7$ state belonging to the parton family. We find that, in the second Landau level of GaAs (i.e. at filling factor $\nu=2+3/7$), this state is energetically superior to the standard Jain composite-fermion state and also provides a very good representation of the ground state found in exact diagonalization studies of finite systems. This leads us to predict that \emph{if} a fractional quantum Hall effect is observed at $\nu=3/7$ in the second Landau level, it is likely to be described by this new non-Abelian state. We enumerate experimentally measurable properties that can verify the topological structure of this state.

In recent years, Balram and collaborators have demonstrated that some of Jain's parton states 26 are reasonable candidates for the second LL FQH states, such as those at ν = 5/2 27 , ν = 7/3 25 , ν = 12/5 28 , and ν = 2 + 6/13 29,30 .(We stress here that the n = 1 LL of monolayer graphene behaves differently than the SLL of ordinary semiconductors such as GaAs.The physics of the n = 1 LL of monolayer graphene is similar to that of the LLL of GaAs, described very well in terms of the CF theory 31 .)Certain other states from the parton construction have been proposed for higher graphene LLs 32,33 and wide quantum wells 34 .Many of these states support non-Abelian quasiparticles, as shown by Wen 35,36 .
In this article, we study the competition between two members of the parton family, labeled by 33 111 and 311 (meaning explained below), both of which are candidate states at ν = 3/7.The 311 state represents the familiar integer quantum Hall (IQH) state of composite fermions 9 , has Abelian quasiparticles, and is known to describe the 3/7 FQHE in the LLL.In contrast, the 33 111 state is believed to support non-Abelian quasipar-ticles that have sufficiently rich braid statistics to allow, in principle, universal topological quantum computation (similar to what is believed for the 12/5 FQHE).Our principal result is to show that in the SLL of GaAs, i.e. at ν = 2 + 3/7, the 33 111 state is variationally superior to the 311 state, and is also an excellent representation of the ground state found in exact diagonalization studies for finite systems.
It is worth remarking on the current experimental status of the nature of the state at ν = 2 + 3/7.An earlier experimental work 37 reported a shoulder in R xx near ν = 2 + 3/7 in GaAs quantum wells, suggesting an incipient FQHE, but experiments in better quality samples and at lower temperatures [38][39][40] show an R H = h/(2e 2 ) IQH plateau ("re-entrant IQH state") at ν = 2 + 3/7, indicating the formation of presumably a bubble crystal that is pinned by the disorder.Evidence for 10/7 FQHE has recently been seen in bilayer graphene 41 , although it is unclear at the moment if the LL orbital hosting this state is more similar to the lowest or the second LL of semiconductor quantum wells (the LLs of bilayer graphene are more complex than the LLs of semiconductor quantum wells).
Our numerical studies do not rule out the possibility that the actual ground state at ν = 2 + 3/7 is an FQHE liquid.This possibility would be consistent with experiments if the FQHE state is masked by disorder, given that disorder favors a crystal over a liquid; in that scenario, an FQHE at ν = 2 + 3/7 would reveal itself in still better quality samples.The other possibility, of course, is that the true ground state at ν = 2 + 3/7 is a bubble crystal.However, our numerical results suggest that the incompressible 33 111 liquid state is very competitive and might be stabilized by changing the effective interaction between electrons, which can be accomplished by changing the width of the quantum well and/or the density, or LL mixing, or by screening the interaction with the help of a nearby metallic layer.If an FQHE is observed at ν = 2 + 3/7, the considerations of this article should be relevant.

II. THE PARTON CONSTRUCTION
The parton construction 26 provides a theoretical framework for constructing new candidate FQH states from known QH states.In the parton construction, one considers breaking the electrons into m species of fictitious particles called partons, with the parton species labeled by λ.Because the density of each parton species must be the same as the electron density [given by ρ = eνB/(hc), where B is the external magnetic field], we must have eν = e λ ν λ , indicating that the charge of the λ-parton is given by e λ = ν/ν λ in units of the electron charge (−e).Because the parton charges must add to the electron charge, we have m λ=1 e λ = −e.It follows that electron filling ν is related to the parton fillings A candidate incompressible state is produced when each parton species occupies an incompressible state, in particular an IQH state with filling ν λ = n λ .We label the incompressible parton states by their integer fillings n 1 n 2 ...n m .This leads to trial wave functions of the form where Φ n is the Slater determinant wave function for n filled Landau levels of noninteracting particles, z k = x k − iy k are the complex coordinates of the kth electron, and P LLL is the lowest Landau level projector.Note that all the partons have the same coordinate as their parent electrons.This "gluing" procedure removes the artificial degrees of freedom introduced by breaking the electrons into partons.We also introduce the notation n to denote negative fillings (i.e., negative magnetic field), which correspond to factors of Φ n = Φ * n in the trial wave functions.Parton states can be Abelian or non-Abelian.The Abelian Jain CF wave functions 9 are a subset of the parton theory; they are states of the form n11 • • • .States with repeated factors of Φ n λ , where |n λ | ≥ 2, are non-Abelian 35,42,43 .
A. A non-Abelian state at ν = 3/7 At filling factor ν = 3/7, the parton theory provides a non-Abelian candidate state 33 111 described by the wave function In the second equality above, we have used Ψ CF 3/5 = P LLL Φ3Φ 2 1 to define the LLL projection of Ψ 33 111 ν=3/7 in a specific fashion; past work has indicated that different ways of accomplishing LLL projection yield very similar wave functions, and in particular, do not alter the topological nature of the state 44 .A nice feature of the 33 111 wave function as expressed in Eq. ( 2) is that it can be evaluated for large system sizes which allow a reliable extrapolation of its thermodynamic energy.This relies on the fact that the n11 states can be evaluated for hundreds of electrons using the Jain-Kamilla (JK) method of projection 3,[45][46][47][48] .For the n11 states, which involve reverse flux attachment (i.e., negative filling factors), it is time-consuming to evaluate the states for systems sizes larger than 50 electrons since the wave function calculation requires high-precision arithmetic which slows down the computation considerably; nonetheless, as we see below, reliable thermodynamic limits can be obtained with systems accessible to numerical evaluation.We note that the wave functions we consider are always constructed in the LLL since they can readily be evaluated in this form.The second LL physics will be simulated in the LLL by using an effective interaction whose Haldane pseudopotentials in the LLL are very nearly the same as the second LL pseudopotentials of the Coulomb interaction.
The 33 111 state has a Wen-Zee 49 shift S = −3 in the spherical geometry, defined below.Due to repeated factors of 3, the 33 111 state is a non-Abelian state hosting quasiparticles whose non-Abelian fusion rules are described by an SU (2) −3 Chern-Simons (CS) theory 28,35,50 .The braiding properties of these so-called Fibonacci anyons are rich enough to potentially carry out universal fault-tolerant topological quantum computation 51,52 .

III. NUMERICAL RESULTS
We perform variational Monte Carlo (VMC) and exact diagonalization (ED) calculations in the spherical geometry 53 wherein N electrons are confined to the surface of a sphere.The radial magnetic field is generated by a magnetic monopole placed at the center of the sphere.The strength of the magnetic monopole is denoted by the integer 2Q, which produces a magnetic flux of 2Qφ 0 , where φ 0 = hc/e is the magnetic flux quantum.The radius of the sphere is given by √ Q where = c/(eB) is the magnetic length.On the sphere, the single-particle orbitals in the Landau level indexed by n = 0, 1, • • • are eigenstates of angular momentum operator with eigenvalue l = |Q| + n, and can be labeled by the z-component of their orbital momentum, l z , which ranges from −(|Q| + n) to |Q| + n.For incompressible states, the strength of the magnetic monopole is the sum of the effective magnetic monopoles required to create each parton species's IQH effect (IQHE), i.e. 2Q = λ 2Q * λ = λ (N/n λ − n λ ).In constructing states on the sphere, we note the shift 49 is given by S = ν −1 N − 2Q.The incompressible states are uniform on the sphere i.e., have total orbital angular momentum L = 0.
Throughout this work, we assume that the magnetic field is sufficiently large to fully spin polarize the electrons.We also neglect the effects of LL mixing and disorder.We will assume zero width for most of our work, but will also consider how finite quantum well width affects our results.In the LLL, we use the bare Coulomb interaction to calculate the energy of the zero-width system.For the n = 1 LL of GaAs and graphene, we use the effective interactions employed in Refs. 54and 31 , respectively, to simulate the physics of the n = 1 LL in the LLL.

A. Ground state
In Fig. 1 we compare the VMC energies of the 33 111 and the 311 wave functions at ν = 3/7 in the LLL and the n = 1 LLs of GaAs and monolayer graphene.These energies include the electron-electron, electron-background, and background-background contributions [the last two collectively contribute −N 2 /(2 √ Q ) to the total energy]; we further multiply these energies by 2Qν/N , which corrects for the finite size deviation of the density from its thermodynamic value and thus minimizes finite-size effects 55 .All energies are given in units of e 2 /( ) where is the dielectric constant of the host material.As anticipated, in the LLL the 311 state has lower energy than the 33 111 state.In contrast, in the SLL of GaAs, we find that the 33 111 state has lower energy.In the n = 1 LL of monolayer graphene, we find the 311 state has lower energy, consistent with the fact that FQHE states in the n = 1 LL of monolayer graphene conform to the CF paradigm 31,56,57 .Under our working assumption of neglecting the effects of finite width and LL mixing, the results for the n = 0 LL of graphene are identical to those in the LLL of GaAs.
Next, we present results obtained from ED at the appropriate flux of 2Q = 7N/3 + 3. The exact ground state in the SLL for the three smallest systems of N = 9, 12, 15 electrons are all uniform, i.e., have L = 0.The next system size of N = 18 electrons, which has a Hilbert space dimension of above 25 billion, is not accessible to ED.The exact SLL Coulomb ground state has an overlap of 0.92 with the 33 111 state for N = 9. (For this purpose, we have obtained an exact Fock-space representation of the 33 111 state using the method described in Ref. 59 .)This overlap is small compared to those we encounter for n11 states in the LLL, but quite high for typical comparisons in the SLL; for example, the 7/3 Coulomb ground state for N = 9 electrons has an overlap of 0.48 with the Laughlin 111 state.The calculation of the Fock-space representation of the 33 111 state with N ≥ 12 is beyond our computational reach, which precludes a determination of its overlap with the exact ground state for these systems.
We next compare the pair-correlation function of the exact SLL Coulomb ground state with that of the 33 111 state for N = 15 electrons (see Fig. 2).The paircorrelation functions of both states show oscillations that decay at long distances, which is a typical characteristic of an incompressible state 60,61 .Moreover, both states show a "shoulder"-like feature in the pair-correlation function at short distances, which is a signature of clustering in non-Abelian states 13,62 .Furthermore, these two pair-correlation functions are in remarkably good agreement with each other.To test whether the results depend sensitively on the choice of the pseudopotentials, we have also evaluated the exact ground state in the SLL with the truncated planar disk pseudopotentials for N = 15 electrons.The overlap between the ground states obtained using the disk and spherical pseudopotentials is 0.9792 which shows that the two ground states are quite close to each other.
For the N = 15 system, the exact energy for the effective interaction we use to simulate the physics of the n = 1 LL in the LLL is -0.3826.In comparison, the 33 111 state has an energy of -0.3809(1) for the same interaction (The number in the parenthesis is the statistical uncertainty in the Monte Carlo estimate of the energy of the 33 111 state.).This level of agreement is on-par with that of other trial states in the second Landau level 30 .For completeness, we have also evaluated the exact LLL Coulomb ground state for the same system.The ground state in the LLL is not uniform (has L = 4) which indicates that the nature of the ground state in the two LLs is quite different from each other for this system.
These studies show that the 33 111 state is variationally better than the 311 state in the SLL, and also provides a very good approximation for the exact ground state for systems accessible to ED.These facts establish the plausibility of the 33 111 state for FQHE at ν = 2 + 3/7 in the SLL or GaAs.

B. Excitation gaps
We can extract the charge and neutral gaps for the three smallest systems that are accessible to exact diagonalization.The charge gap here is defined as ∆ c = [E(2Q = 7N/3 + 4) + E(2Q = 7N/3 + 2) − 2E(2Q = 7N/3 + 3)]/3, where E(2Q) is the ground state energy at flux 2Q and the factor of 3 in the denominator accounts for the fact that the addition or removal of a single flux quantum produces three fundamental quasiholes or quasiparticles in the 33 111 state.The neutral gap ∆ n is defined as the difference between the two lowest energies at the flux of 2Q = 7N/3 + 3. The density-corrected charge and neutral gaps, which include the background contribution, for the individual systems are reported in the table inset in Fig. 3.The charge and neutral gaps do not fit well to a linear function of 1/N and thus we do not have a reliable estimate of them in the thermodynamic limit.Also, the neutral gap is larger than the charge gap for these systems which indicates strong finite-size effects in the SLL.(We expect the charge gap to be larger than or equal to the neutral gap in the thermodynamic limit.) We have also attempted to estimate the charge gap via a VMC.To do so we create a quasihole-quasiparticle pair by promoting one particle in one factor of Φ3 to the fourth Λ level (ΛL).(ΛLs are the Landau-like levels occupied by (3) where the superscript "exciton" refers to the creation of a quasiparticle and quasihole in this factor.Unfortunately, as we show below, the quasihole and quasiparticle are very large and so for system sizes that we can access, the quasihole and quasiparticle overlap significantly.The gap does not follow a linear fit in 1/N and we only roughly estimate it to be of order 0.01 e 2 /( ) as shown in Fig. 3.We have not attempted to evaluate the charge gap using the standard procedure of inserting or removing a flux quantum since doing so creates 3 quasiholes or 3 quasiparticles in the 33 111 state, which would overlap strongly and thus the interactions between them cannot be neglected.As a result, we cannot place the particles on the sphere so that they are well separated for any reasonably sized system accessible by the JK projection.3) and ( 2).The gap shows strong finite-size fluctuations, precluding a clear extrapolation to the thermodynamic limit, but is finite for all N considered.The lack of a clear trend with 1/N is likely the result of the large overlap, and thus a significant interaction, of the quasiparticle and quasihole for small system sizes (see text and Fig. 6).We estimate from this calculation that the charge gap should be of the order 0.01e 2 /( ).For comparison, the inset shows the charge and neutral gap ∆n (see text for definition) for small systems evaluated from exact diagonalization.

C. Stability of 33 111 state
We next test the stability of the 33 111 state in the SLL to small perturbations in the interaction.We consider two starting unperturbed interactions: the spherical pseudopotentials and a set of truncated disk pseudopotentials (remember that the interaction is fully defined by the pseudopotentials).We then add small deviations to the V 1 and V 3 pseudopotentials for each case.We diagonalize these systems to find the charge and neutral gaps, as well as the overlap with the 33 111 state.In Figs. 4 and 5, we present color plots for each quantity in the δV 1 − δV 3 plane for the spherical and truncated disk pseudopotentials respectively.We find that for a wide range of perturbations, the overlap between the 33 111 state and the ED ground state remains high, and the state also supports finite neutral and charge gaps.Since for N = 9 particles the 33 111 state occurs at the same flux as the 1/3 Laughlin state, we have also calculated the overlap of the ground states with the Laughlin state.As the upper right panels of Figs. 4 and 5 show, the Laughlin state has a lower overlap than the 33 111 state at the SLL Coulomb point, but the overlap grows as the pseudopotentials become more like the LLL, i.e. as V 3 decreases relative to V 1

D. Fractional charge of the quasiparticles
The parton theory predicts that the minimal charge quasiparticle of the 33 111 state is obtained by creating a quasiparticle in the 3 factor.This excitation carries a charge of (−e)/7.It is interesting to note that the nonabelian nature of the 33 111 state does not cause a further fractionalization of its quasiparticle charge (Another example of a non-Abelian state where the charge does not fractionalize further is the 222 1111 state at ν = 2/5 28 .).Accessibility to large system sizes allows us to microscopically evaluate the charge of the quasiparticle in the 33 111 state.In Fig. 6 we show the density profile ρ(r) of the 33 111 state at ν = 3/7 with an exciton where the constituent quasiparticle is located at the north pole and the quasihole at the south pole of the sphere.We consider a system of N = 90 particles in this calculation.This state is modeled by the wave function given in Eq. ( 3).Close to the equator the density of the state goes to the density ρ 0 of the uniform 33 111 state.Note that the density distributions of the quasihole and quasiparticle are not identical since they reside in different ΛLs.Furthermore, the quasiparticle and quasihole of 33 111 have a wider extent compared to their 311 counterparts.To demonstrate that the smallest charge quasiparticle (quasihole) of the parton ansatz 33 111 has a charge equal to one-seventh of the electron charge, we calculate the integrated cumulative charge Q(r) = (−e) r 0 d 2 r 1 [ρ( r 1 ) − ρ 0 ] from the north (south) pole to the equator.Doing so for the system of N = 90 electrons shown in Fig. 6, we obtain a charge of magnitude 0.145e which is close to the ex-pected value in the thermodynamic limit of e/7 = 0.143e; the small discrepancy arises from the fact that even for a system with N = 90 particles the quasiparticle and quasihole have a finite overlap.

E. Effect of finite well-width
We have also considered the effect of the finite wellwidth w of the quantum well in the LLL and SLL of GaAs.The LLL finite width interaction is obtained via a self-consistent LDA method for a given well width and electron density 63 .We explore parameters ranging from w =18 to 70 nm and electron densities, ρ, ranging from 0.1×10 11 cm −2 to 3×10 11 cm −2 .Representative thermodynamic extrapolations for several different densities at the fixed quantum well width of 70 nm are shown in the left panel of Fig. 7.The Jain 311 state has lower energy in the LLL for all parameters we have studied.
In the SLL, we use an effective interaction parameterized by the well-width in units of magnetic length for an infinite square well confinement potential, described in detail by Shi et al. 54 .We have calculated energies for well-widths, w, up to 0.9 , shown in the right column of Fig. 7.We find that the 33 111 state has lower energy in the SLL for the entire range of widths considered.

IV. DISCUSSION AND EXPERIMENTAL RAMIFICATIONS
Our exact diagonalization studies in the spherical geometry for systems with up to N = 15 particles are consistent with a non-Abelian FQHE at ν = 2+3/7.We note that competition between an FQHE liquid and a charge density wave state such as a Wigner or a bubble crystal can be quite subtle 64 .The spherical geometry favors a liquid for finite systems, and a crystal may be stabilized in the thermodynamic limit.However, our calculations make a strong case that the 33 111 is very competitive here, and if an FQHE is seen at ν = 2 + 3/7, it likely represents a new non-Abelian state.
One may ask how the 33 111 state may be distinguished from the 311 state.We end the article with a comparison of the various topological properties of the two states.
The minimally charged quasiparticle carries a charge of magnitude e/7 for both the 33 111 and the 311 states.However, the quasiparticles of the 33 111 state obey non-Abelian braid statistics 35 , in contrast to the Abelian quasiparticles of the 311 state.
Due to the presence of the 3 factors, the 33 111 state possesses upstream neutral modes that can be detected in In summary, we have considered the feasibility of the non-Abelian 33 111 state for FQHE at ν = 2 + 3/7, i.e. when the second Landau level is 3/7 occupied.We have shown that this state has lower energy than the Abelian 311 state, and also has a high overlap with the exact b) ground state for small systems.We have also proposed experimental measurements that can reveal the underlying non-Abelian topological order of the 33 111 state and distinguish it from the Abelian 311 state.

FIG. 1 . 7 = 3 ] * [Φ 3 ]
FIG. 1. (color online) Thermodynamic extrapolations of the per-particle Coulomb energies for the 311 and (blue triangles) and the 33 111 states (red circles) at ν = 3/7.The panels a.), b.) and c.) show energies in the n = 0 LL, n = 1 LL of GaAs, and in the n = 1 LL of monolayer graphene, respectively.The extrapolated energies, obtained from a linear fit in 1/N , are quoted in Coulomb units of e 2 /( ) on the plot and the number in the parentheses indicates the uncertainty in the linear fit.These energies include contributions of the electron-background and background-background interaction, and are density-corrected 55 .The Coulomb energies for the Jain 311 state in the n = 0 and n = 1 LLs of monolayer graphene have been reproduced from Refs. 58and 31 .

FIG. 3 .
FIG. 3. (color online) Charge gap ∆c calculated from the difference in energies of the trial wave functions presented in Eqs.(3) and (2).The gap shows strong finite-size fluctuations, precluding a clear extrapolation to the thermodynamic limit, but is finite for all N considered.The lack of a clear trend with 1/N is likely the result of the large overlap, and thus a significant interaction, of the quasiparticle and quasihole for small system sizes (see text and Fig.6).We estimate from this calculation that the charge gap should be of the order 0.01e 2 /( ).For comparison, the inset shows the charge and neutral gap ∆n (see text for definition) for small systems evaluated from exact diagonalization.

FIG. 4 . 3 |
FIG. 4. (color online) Overlap and gap maps obtained in the spherical geometry using exact diagonalization of N = 9 electrons with the 33 111 state at ν = 3/7 by perturbing V1 and V3 around the second Landau level (SLL) spherical pseudopotentials.The system of N = 9 electrons at a flux of 2Q = 24 aliases with the Laughlin state; therefore, for comparison, in the top right panel we show the overlap map for the ν = 1/3 Laughlin state for the same system.The center dot denotes the exact SLL Coulomb point (V1 = 0.4642 and V3 = 0.3635) for which the values in the four panels are | Ψ 33 111 3/7 |Ψ SLL 3/7 | = 0.92, | Ψ Laughlin 1/3

FIG. 6 .
FIG. 6. (color online) a) Density profile ρ(r) of a state with a far-separated quasihole and quasiparticle at ν = 3/7 modeled by the parton wave function given in Eq. (3) for N = 90 electrons on the sphere.The quasiparticle is located at the north pole and the quasihole is located at the south pole.The quantity shown is [ρ(r) − ρ0]/ρ0, where ρ0 is the density of the uniform 33 111 state at ν = 3/7.b) The quasihole [quasiparticle] cumulative charge Q(r) = (−e) r 0 d 2 r1[ρ( r1) − ρ0] [for quasiparticle we show −Q(r)] as a function of the distance r measured along the arc from the south [north] pole to the equator in units of the magnetic length .The quasihole [quasiparticle] cumulative charge approaches −1/7 [1/7], in units of the electron charge (−e), near the equator.