Introduction

The fractional quantum Hall state (FQHS) at the Landau level filling factor ν = 1/31,2 is a prototypical example of a strongly correlated topological phase that hosts quasiparticles with fractional charge3 and fractional statistics4. Improvements of the quality of the host material and advances in cryogenic technology led to the observation of an increasing number of FQHSs. A large number of FQHSs form at Landau level filling factors ν that belong to two major Jain sequences: ν = n/(2n + 1) and ν = n/(4n + 1), where n is an integer. Substituting n with 1, 2, 3,…, one obtains the prominent members of the first Jain sequence ν = 1/3, 2/5, 3/7,… and that of the second Jain sequence ν = 1/5, 2/9, 3/13,…. These sequences were discovered in GaAs/AlGaAs5,6,7,8,9,10,11, and were reported in other high quality host materials, such as graphene12,13,14,15,16,17, ZnO/MgZnO18, and AlAs19.

Jain’s composite fermion theory explains these two major sequences of FQHSs by mapping the strongly interacting system of electrons into a system of nearly free particles, the composite fermions (CFs)20,21. The success of the CF theory is rooted in the ability to describe the complex correlations of the system with analytical many-body wavefunctions. The construction of these trial wavefunctions relies on the introduction of Jastrow factors which may be interpreted as electrons capturing magnetic flux quanta. FQHSs are then understood as integer quantum Hall states of CFs moving in a magnetic field modified by the Berry phases associated with the attached fluxes and having a spectrum of quantized and degenerate energy levels called Λ-levels. Specifically, the sequence of FQHSs at ν = n/(2n + 1) owes its existence to two-flux CFs (2CFs), whereas that at ν = n/(4n + 1) to four-flux CFs (4CFs). CFs, the emergent particles of the fractional quantum Hall regime, are thus a resource for topological protection that arise from the correlated motion of flux tubes, or vortices, and the electrons themselves.

According to the CF theory, FQHSs at filling factors of the form ν = n/(6n + 1) are a consequence of the most intricate higher order correlations that stem from six-flux CF (6CF)20. The FQHSs at ν = 1/7, 2/13, 3/19,… are members of this sequence. However, as discussed in detail later in our manuscript, experiments at such filling factors associated with 6CFs so far did not find signatures of topological protection in the GaAs/AlGaAs system. To our knowledge, topological protection associated with 6CFs was not reported in any other high quality hosts such as graphene, ZnO/MgZnO, or AlAs either. Here we present results of a search for FQHSs associated with 6CFs in a region of filling factors related by the ν ↔ 1-ν symmetry22 to the filling factors of the form ν = n/(6n + 1) in a GaAs/AlGaAs sample that belongs to the newest generation of samples of the highest quality23. The electron density is 1.01 × 1011 cm−2, the low temperature mobility 35 × 106 cm2 V−1 s−1, and the width of the quantum well is 49 nm.

Topological protection is signaled by the presence of a topological invariant. In the commonly used transport measurements, the presence of Hall quantization Rxy = h/fe2, where f is a simple fraction, is the indicator for topological protection. In addition, the opening of an energy gap is a necessary condition for topological protection. In transport, this latter property translates to a decreasing longitudinal magnetoresistance Rxx with a decreasing temperature T of the activated form: Rxx~exp{-Δ/2kBT}. Here h is the Planck constant, e the elementary  charge, kB the Boltzmann constant, and Δ the magnitude of the energy gap. A quantized Hall resistance and an activated Rxx, together with a vanishing Rxx, are indeed the canonical transport signatures of FQHSs.

We now review experimental data available in the 1/7 ≤ ν < 1/5 range of filling factors where FQHSs associated with 6CFs are predicted to form. Early transport in the GaAs/AlGaAs system found weak minima in Rxx at ν = 2/117 and ν = 1/78. More recent transport confirmed local minima in Rxx at ν = 1/7, 2/1310,24 and at 2/1110 and found local minima in the high frequency conductance at ν = 1/7 and 2/1125. But these local minima did not persist to the lowest temperatures attained in those experiments. The transport signatures observed at ν = 1/7, 2/13, and 2/11 are clearly different from those of canonical FQHSs; they were interpreted as evidence of FQHSs at finite temperatures which were engulfed by a competing Wigner solid ground state at the lowest temperatures10,11,12,13,14,15,16,17,18,19,20,21,22,23,24. This interpretation, however, left lingering questions. First, during the long history of the field, developing FQHSs exhibited an inflection in the Rxy versus B traces. Evidence for such a Hall signature was seen in an early work at ν = 1/78 but was not reproduced later. In fact recent transport did not provide any Rxy data10,24. Second, past improvements in the mobility resulted in developing FQHSs transitioning into fully developed ones, which exhibited Hall quantization and an energy gap. Well-known examples of this behavior occurred at ν = 1/31, 1/56,7,8, 5/25, and 4/1126. In contrast, a significantly larger mobility of the sample in ref. 24, by more than a factor 25 larger than that in early work8, did not result in opening of an energy gap at either ν = 1/7 or 2/13. Third, signatures at the prominent members of this sequence were observed in transport only and not in other experiments. Indeed, microwave absorption27,28, surface acoustic wave propagation29, and screening efficiency measurements30 did access the filling factors ν = 1/7, 2/13, and 2/11, but no signatures of FQHSs were reported. To conclude, measurements at ν = 1/7, 2/13, and 2/11 reveal a complex behavior. The lack of observation of a quantized Rxy or at least of an inflection in the Rxy versus B curves in recent high quality samples remains a weak point of a fractional quantum Hall interpretation at filling factors, such as ν = 1/7, 2/13, or 2/11. The ground state at these filling factors in macroscopic samples is an insulator, either the Wigner solid or a crystal of CFs. Even though at these filling factors there is limited evidence for incipient quantum correlations at finite temperatures, these correlations so far are insufficient to establish topological protection associated with 6CFs.

Numerical experiments in the extreme quantum limit reported a similarly complex behavior. Early simulations considered the competition of Laughlin states and the Wigner solid2,31,32,33,34,35,36,37,38. The ground state at ν = 1/7 was overwhelmingly the Wigner solid32,33,34,35,38. However, the Wigner solid and the FQHS are so close in energy that tuning a parameter of the system, such as the short-range part of the electron-electron interaction36 or the inter-CF interaction37, induces a phase transition between the two ground states. More recent numerical work considered the competition between the FQHSs with crystals of CFs, correlated electron solids that can assume lower energies than the Wigner solid39,40,41,42. References 39,40,41 found the CF crystal more stable at ν = 1/7. A variational Monte Carlo simulation and a density matrix renormalization group calculation could not clearly discriminate between a fractional quantum Hall and a CF crystal ground state and results of an exact diagonalization study were interpreted as evidence for incompressible fractional quantum Hall ground states42. However, this latter technique is known to disfavor crystalline phases. These calculations were performed under certain approximations and therefore it remains to be determined which way the delicate energy balance between these competing phases will be tilted under conditions mimicking those of realistic samples. Melting of an insulating ground state into a finite temperature FQHS as the temperature is raised remains a possibility34,41.

Results

Figure 1 shows the longitudinal resistance Rxx and the Hall resistance Rxy as function of the magnetic field B obtained at T = 7.6 mK. For these measurements, our sample was mounted into a He-3 immersion cell43; the temperature was measured with a carbon thermometer44 calibrated against a quartz tuning fork He-3 viscometer43. The region  above B = 2.095 T marks the lowest orbital Landau level, whereas the region at lower fields is the second Landau level. The high quality of the sample is highlighted by the observation in the second Landau level of all eight reentrant insulators45. Furthermore, we also observe the even-denominator FQHSs at ν = 5/25, a clear signature of the ν = 2 + 6/13 FQHS46, and a prominent ν = 7/11 FQHS26. Near B = 2.332 T there is a reentrant integer quantum Hall state associated with the Wigner solid47.

Fig. 1: Longitudinal magnetoresistance Rxx and Hall resistance Rxy as a function of the magnetic field B.
figure 1

Data were collected at the temperature of T = 7.6 mK. Several prominent integer and fractional quantum Hall states are marked by their filling factor. In addition, the locations of Fermi seas of CFs are indicated by vertical lines at ν = 3/2 and at ν = 3/4. The region above B = 2.095 T corresponds to the lowest orbital Landau level, whereas the region at lower fields is the second Landau level.

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The dominating feature in Fig. 1 is a very wide ν = 1 integer quantum Hall plateau. At higher magnetic fields, near B = 5.240 T, we observe the well-known ν = 4/5 FQHS5. Other FQHSs are observed at ν = 7/9 and 10/13, and there is a pronounced local Rxx minimum at ν = 13/17. These four filling factors are of the form ν = 1-n/(4n + 1), where n = 1, 2, 3, and 4. This sequence of filling factors is related to the major Jain sequence ν = n/(4n + 1) introduced earlier via the ν ↔ 1-ν particle-hole conjugation21,22. Therefore FQHSs at these filling factors originate from the formation of 4CFs for which the vacuum is the ν = 1 FQHS. As a consequence, these 4CFs can be thought of as formed via the flux attachment procedure applied to holes, rather than electrons21,22. The Fermi sea of the ν = 1-n/(4n + 1) sequence at ν = 3/4 is also marked in Fig. 1. Figure 2 provides a magnified view of the region of interest between B = 4.8 T to 5.7 T.

Fig. 2: A magnified view of magnetotransport traces Rxx and Rxy in the range of 4.8 T < B < 5.7 T.
figure 2

Numbers mark filling factors of prominent FQHSs at ν = 4/5, 7/9, and 10/13 that belong to the 1-n/(4n + 1) sequence, with ν = 1, 2, and 3. In addition, there is a pronounced local Rxx minimum at ν = 13/17. The most interesting feature of the data is the FQHS at ν = 9/11 which does not belong to the 1-n/(4n + 1) sequence. The vertical arrow indicates the location of a Fermi sea of CFs at ν = 3/4.

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Between ν = 1 and ν = 4/5 there is an unfamiliar magnetoransport feature at B = 5.120 T. As seen in Fig. 2, an examination of this feature reveals a Hall resistance plateau. This magnetic field corresponds to the filling factor ν = 9/11 and the value of the Hall plateau is quantized at Rxy = 11 h/9e2, to within 0.14%. At the same time, Rxx nearly vanishes at B = 5.120 T. These transport features are canonical signatures of a FQHS at ν = 9/11. Hall quantization is evidence for an edge state dominated transport and thus establishes 9/11 as a topological invariant at this filling factor.

A study of the temperature dependence of our data, shown in Fig. 3, reveals that the FQHS at ν = 9/11 is one of the most feeble FQHSs we measure. Nonetheless, Rxx at ν = 9/11 decreases with a decreasing temperature. This behavior is shown in Fig. 4. Data for these two figures was obtained after cycling the sample to room temperature; in this cool-down the sample was thermalized through the measurement wires, rather than with He-3. The linear part of the Arrhenius plot shown in Fig. 4 present at the lowest temperatures, i.e. at 1/T higher than 0.0137, establishes that Rxx in this region has a thermally activated form Rxx~exp{-Δ9/11/2kBT}. The lowest temperature data of Fig. 4 therefore provide evidence for the opening of a gap in the energy spectrum at ν = 9/11. By performing a linear fit, we extract the magnitude of the energy gap Δ9/11 = 32 mK. The presence of an energy gap at ν = 9/11 indicates incompressibility and ensures an edge-dominated transport. We thus conclude that the observation of a Hall plateau quantized to Rxy = 11 h/9e2, of an Rxx local minimum with a nearly vanishing value, and of the opening of an energy gap provide strong evidence for the formation of a topologically protected FQHS at ν = 9/11.

Fig. 3: Temperature dependence of the longitudinal magnetoresistance Rxx between ν = 0.77 and 0.845.
figure 3

Red regions mark areas of low resistance and are therefore indicative of fractional quantum Hall ground states that possess an energy gap in their excitation spectra. The FQHS at ν = 9/11 is the weakest, albeit fully developed, of the FQHSs observed in this range of filling factors.

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Fig. 4: Arrhenius plots of the Rxx versus the temperature T at three filling factors of interest.
figure 4

Energy gaps Δ are obtained from the linear region of these plots at the lowest temperatures from linear fits. Dashed lines show these linear fits to data.

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Discussion

We now focus on the origins of the ν = 9/11 FQHS. As discussed, other nearby FQHSs develop at filling factors of the form ν = 1-n/(4n + 1). The filling factor ν = 9/11 is not part of either the ν = 1- n/(4n + 1) or the ν = 1-n/(2n + 1) sequences, therefore correlations embodied by 4CFs or 2CFs cannot account for a FQHS at this filling factor. We conclude that higher order correlations must be at play. In the following we describe two constructions within the CF theory that can explain the formation of this FQHS, both relying on the formation of 6CFs. Using the prescriptions of the CF theory, trial wavefunctions can be written down for both constructions21.

For the first construction, we notice that  9/11 = 1-2/11, the filling factor ν = 9/11 is thus related via the ν ↔ 1-ν particle-hole symmetry to ν = 2/1122. As discussed earlier, at ν = 2/11 a fully gapped FQHSs has not yet been demonstrated. Nonetheless, the CF theory prescribes a many-body wavefunction for a FQHS at this filling factor. The ν = 2/11 quantum number belongs to the well-known n/(6n + 1) Jain sequence, with n = −2. A negative integer n indicates that the corresponding FQHS develops at a effective magnetic field that points against the direction of the externally applied magnetic field21. Within this construction, the ν = 9/11 FQHS is described as the ν*=−2 integer quantum Hall effect of 6CFs which are built using the flux attachment procedure starting out from holes, rather than electrons. We only invoked 6CFs which fill two Λ-levels. Such a FQHS would necessarily be fully spin polarized21. We note that there is a very similar construction for the ν = 5/7 which, however, is based on 4CFs. Indeed, the ν = 2/7 FQHS can be understood as the ν* = −2 integer quantum Hall effect of 4CFs and the ν = 5/7 = 1-2/7 FQHS is related by the ν ↔ 1-ν particle-hole symmetry to the ν = 2/7 FQHS.

A second construction is obtained using CFs of mixed flavor. Following steps similar to the ones employed for the FQHS at ν = 4/1126,48 and at ν = 4/549,50, we first apply the flux attachment procedure to generate 2CFs. The Λ2-level filling factor of these 2CFs at ν = 9/11 will be ν2* = −(1 + 2/7). Here the index 2 indicates a description based on 2CFs. One may consider the lower fully filled Λ2-level inert; a fraction 2/7 of the second Λ2-level is filled. If the 2CFs in the upper Λ2-level are free, a gap is not expected at the filling factor 2/7 of this energy level. However, if the 2CFs strongly interact, they may capture additional fluxes in order to form CFs of higher order and to condense into the Λ-levels of these newly formed CFs. The generation of an energy gap through this process is referred to as the fractional quantum Hall effect of composite fermions26,48. For the second step of the construction, we apply the flux attachment procedure once more to the 2CFs in the topmost partially filled Λ2-level, by attaching four more fluxes to each 2CFs. The end result of this process is 6CFs which will completely fill two Λ6-levels of 6CFs. Thus, in contrast to the first construction, this second construction for the ν = 9/11 FQHS the formation of both 2CFs and 6CFs needs to be invoked. The associated trial wavefunctions can account for both a fully spin-polarized and a partially spin-polarized FQHS. However, it is believed that the former is identical to that of the first construction21. To summarize, we presented two constructions within the confines of the CF theory to describe the FQHS at ν = 9/11. The topmost Λ-level for both of these constructions is due to the formation of 6CFs, i.e. the valence CFs for both descriptions of the ν = 9/11 FQHS are 6CFs.

We now discuss the behavior at other filling factors of the form ν = 1-n/(6n + 1). The value n = −1 yields ν = 4/5. At this filling factor there is a strong FQHS, with an energy gap of Δ4/5 =  348 mK. Similarly to the case of ν = 9/11, at ν = 4/5 one could write down a wavefunction based on 6CFs filling one Λ6 level. However, other wavefunctions of lower order CFs can also be constructed49,50. According to the CF theory, dissimilar interpretations are possible and, in such situations, the more natural construction based on CFs of lower order is typically adopted21. The value n = +1 of the ν = 1-n/(6n + 1) sequence yields ν = 1-1/7 = 6/7. This value of the filling factor is reached at B = 4.887 T. On the T = 7.6 mK curves of Figs. 1 and 2, at this value of the B-field Rxx = 0 and Rxy = h/e2. Such a transport behavior indicates a ground state with an insulating bulk that is distinct from a FQHS, such as a Wigner solid or a random insulator. Because of the presence of edge states, the former is sometimes referred to as the integer quantum Hall Wigner solid51,52. Interestingly, as the temperature is raised, this ground state with an insulating bulk at ν = 6/7 is destroyed. However, in the 7.6 – 400 mK range, in our sample we do not observe a local minimum in Rxx at ν = 6/7. Therefore we do not find any evidence for a FQHS at ν = 6/7 even at finite temperatures. We conclude that in our sample only the ν = 9/11 FQHS may be associated with 6CFs.

In the regime of formation of 6CFs, a Fermi sea is expected in the limit of n → ∞. The parent Fermi sea for the ν = 1-n/(6n−1) sequence is thus expected to form at ν = 5/6, a filling factor which in our sample is reached at B = 5.027 T. Transport at a Fermi sea of CFs exhibits a nearly B-field independent and featureless magnetoresistance Rxx. This behavior may be observed in Fig. 1 for the Fermi seas of CFs forming at ν = 3/4 and 3/2. As seen in Fig. 2, magnetoresistance at ν = 5/6 is very different. We thus suggest that in our sample a Fermi sea associated with 6CFs does not form at ν = 5/6. We think that the unusual magnetoresistance at ν = 5/6 is caused by a competing second ground state, possibly a Wigner solid or a random insulator. While our data is not consistent with a fully developed ground state with an insulating bulk, the integer quantization discussed earlier at the nearby ν = 6/7 associated with a bulk insulator suggests that the same type of bulk insulator also competes at ν = 5/6 with the Fermi sea of 6CFs. A similar competition of the Fermi sea with a bulk insulator is known to occur for 4CFs at ν = 1/41,6,7,18.

In the following we examine other filling factors from the literature at which CFs of mixed flavor, including 6CFs, may play a role. A FQHS at ν = 6/17 was proposed to originate from mixed flavor CFs, with one filled Λ2-level and another filled Λ6-level of 6CF26,48. Similarly, a FQHS at ν = 5/17 and ν = 4/13 was proposed to be generated by a filled Λ4 level and one/two filled Λ6 levels, respectively26,48. Hall quantization and the opening of an energy gap, thus topological protection, was not yet observed at any of these filling factors53,54.

We note that the nature of the ground state at ν = 9/11 may change when changing sample parameters. For example, it is known that by increasing the width of the quantum well, a Wigner solid will develop at ν = 9/11 when the electron density is larger than a critical value of about 1.4 × 1011 cm−2 51.

In conclusion, we report the observation of an incompressible fractional quantum Hall ground state at ν = 9/11, a Landau level filling factor of the form ν = 1-n/(6n + 1), with n = −2. Hall quantization and the opening of an energy gap at this filling factor indicate a topologically protected ground state with one of the most intricate electronic correlations. Our observations highlight the formation of a new type of emergent fermionic particle, the six-flux CF, a particle that cannot be perturbatively connected to any other CF particles with fewer fluxes.

Methods

Magnetotransport measurements were performed in a van der Pauw geometry, with an excitation current of 3 nA and employing a standard lock-in technique at 13 Hz. Sometimes the sample state in the GaAs system is prepared by a brief illumination using a red light emitting diode. However, in our experiments we did not employ such an illumination technique, our sample was cooled in dark. The Rxx plateaus have a small B-field dependent offset. For the energy calculations shown in Fig. 4, these offsets were subtracted.