LL-filling factors for FQHE determined by commensurability condition (paired indicates condensate of electron pairs), for the first particle subband in each of the three first LLs () for the monolayer graphene.
Fractional quantum Hall effect (FQHE) discovered experimentally in 1982 is still mysterious, not fully understood phenomenon. It fundaments are linked with a nontrivial topological effects in 2D space going beyond the standard description of FQHE with local quantum mechanics. The study of integer and fractional QHE in graphene might be helpful in resolution of this fundamental problem in many body quantum physics. FQHE has been observed both in monolayer and bilayer graphene with an exceptional accuracy due to advances in experimental techniques and purity of graphene samples. Recent experimental observations of FQHE in the bilayer graphene reveal different FQHE behavior than in the monolayer samples or in conventional semiconductor 2D materials. This unexpected phenomena related to Hall physics in the bilayer systems allows to better understand more than 30 years old puzzle of FQHE. In the chapter we will summarize the recent and controversial experimental observations of FQHE in bilayer graphene and describe the topology foundations which may explain the oddness of correlated multiparticle states in the bilayer system. These topological arguments shed also a new light on understanding of heuristic CF concept for FQHE and deeper the topological sense of the famous Laughlin function describing this strongly correlated state.
- monolayer graphene
- bilayer graphene
- hierarchy of filling fractions
- braid group for bilayer graphene
- correlated state
Fractional quantum Hall effect (FQHE) discovered experimentally in 1982  (Nobel Prize 1989 for Tsui and Störmer) is still a mysterious phenomenon robust against its complete understanding. The famous Laughlin function  proposed for theoretical description of this effect (Nobel Prize 1989 for Laughlin, together with Tsui and Störmer) was the illuminating concept towards explanation of the FQHE. This function was the next start-point for further attempts to understand and describe in detail the experimentally observed FQHE behavior with still growing number of new observations. Despite the breakthrough character of the Laughlin concept, the complete explanation of the FQHE has not been achieved as of yet. The basic problem is linked here with nontrivial topology-type effects in two-dimensional (2D)-charged systems exposed to strong perpendicular magnetic field going beyond the standard description of FQHE upon the local quantum mechanics but generating specific long-range correlations in the whole 2D system.
The Laughlin function has not been derived but it was rather proposed as variational trial multiparticle wavefunction and verified next by numerical exact diagonalization of the interaction of electrons. The archetype for the Laughlin function was the Slater determinant built of zeroth Landau level (LL) single-particle functions. The polynomial part of this Slater determinant has the form of the Vandermonde determinant polynomial and a heuristic change to the Jastrow polynomial results in the Laughlin function. This simple mathematical maneuver has, however, a pronounced physical consequence. The Laughlin function describes the strongly interacting particles, contrary to the Slater function and, moreover, pretty well describes multiparticle correlations of FQHE specific for 2D space topology. The ostensible simplicity of the Laughlin function triggered next efforts to describe FQHE in terms of quasiparticles common in solids, that is, in terms of effective single-particle objects dressed with interaction and defined as the poles of the single-particle-retarded Green function . Nevertheless, in the case of the FQHE, the Coulomb interaction-induced mass operator is discontinuous in 2D at magnetic field presence, and this property precludes any definition of the quasiparticle as dressed electrons with the interaction.
Despite this inability to rigorously define quasiparticles in the FQHE system, a heuristic concept of single-particle picture for this system has been formulated in terms of the so-called composite fermions (CFs) . CF is proposed as complex object consisting of the electron with attached local field flux (even the number of an auxiliary magnetic field flux quanta pinned to the electron). The origin of the auxiliary magnetic field as well as the manner of how the local flux is fixed to a moving electron is not explained, but the CF model works surprisingly well . CFs defined as electrons with fictitious magnetic field flux quanta fixed to them gain the Laughlin function phase shift by the Aharonov-Bohm-type effect when composite particles interchange their position. Note, however, that the Aharonov-Bohm effect is of non-local character. The main advantage of CF effective single-particle picture resolves itself to the mapping of FQHE onto IQHE in the resultant magnetic field, that is, the true external magnetic field reduced by the averaged field of fluxes pinned to electrons. This mapping allows for the definition of the Jain-like hierarchy of FQHE in partial agreement with experimental data. In view of the above-mentioned discontinuity of the Coulomb interaction mass operator, CFs cannot be named as quasiparticles (moreover, an auxiliary magnetic field fluxes dressing electrons cannot be “produced” by the non-magnetic electric Coulomb repulsion of electrons). One can conclude that CFs describe phenomenologically a different and more fundamental property of the system in close relation to the specific topology of 2D charged system in magnetic field and connected with this topology specific long-range correlation induced by electron repulsion.
In order to decipher this enigmatic character of CFs and the topological correlations in FQHE, the algebraic-topology braid group-based approach has been developed and applied to FQHE . Within this approach, the trajectories of interchanging positions of particles on the 2D plane in the system of
Nevertheless, in 2D for charged-repulsing electrons in the presence of strong perpendicular magnetic field, the topological situation changes again. For magnetic field strong enough, the only possible on the plane cyclotron classical orbits may be too short to match neighboring particles uniformly distributed on the plane (with classical positions fixed by the repulsion of electrons). Too short braids must be thus removed from the braid group. The remaining braids form the subgroup of the former group called the cyclotronic braid subgroup. The cyclotronic subgroup collects only those braids which can match neighboring particles.
To clarify the structure of the cyclotronic braid subgroups, let us note that in the presence of strong magnetic field the 2D braids must be built from pieces of cyclotron orbits and this orbit must precisely fit to interparticle spacing. Otherwise, the definition of braids is precluded. Hence, the commensurability of planar braids with particle spacing is an unavoidable condition to define the braid group and establish the quantum statistics (by 1DURs of the braid group). The quantum statistics is a necessary prerequisite to multiparticle-correlated state. Therefore, the commensurability condition determines the magnetic field strength (or equivalently, Landau level-filling rate) at which correlated state can be formed. The archetype of the commensurability in 2D-charged
For stronger magnetic fields,
2. FQHE in graphene
Graphene is ideally two-dimensional but with some specific modification of LL structure due to pseudo-relativistic band structure of this gap-less semiconductor. Thus, examining FQHE in graphene is especially challenging. The specific for graphene electronic dynamics is referred to the Dirac point in all corners of the hexagonal first Brillouin zone. This specific pseudo-relativistic local band structure does not influence, however, the topology constraints imposed on trajectories and the FQHE has been observed in graphene. Easy methods for the manufacturing of monolayer and bilayer graphene ought to be experimentally noticeable. And indeed, several recent observations of FQHE in bilayer graphene [10–14] demonstrate strong distinctions in comparison to FQHE hierarchy in monolayer graphene [15–18].
The observed FQHE phenomena in bilayer graphene supply an opportunity to verify the cyclotron braid group commensurability approach because the topology of double sheet in bilayer graphene causes different commensurability conditions in comparison to the monolayer case. This will result in specific hierarchy of filling rates for FQHE in bilayer graphene not observed in monolayer graphene and in conventional semiconductor 2D systems.
In the present chapter, we summarize the recent and controversial experimental observations of FQHE in bilayer graphene and describe the topology foundations, which may explain the oddness of correlated multiparticle states for the bilayer system. The significance of such a discussion consists in experimental confirmation of new topological-based theory of FQHE going beyond the phenomenological model of CFs.
Advances in experiment allow now for measurements of longitudinal and transversal resistivity in Hall configurations in various graphene sample arrangements. The great progress in experiment has been achieved by the utilization of measurement technique in the so-called suspended graphene, where small scraping of the graphene sheet has been hanged in free air without any contact with substratum and thus without disturbing phonons in crystals and therefore with extremely high mobility triggering FQHE organization in relatively higher temperatures. It is possible to suspend both monolayer and bilayer sheets. Another experimental achievement is the utilization of homogeneous with graphene crystal substrate of boron nitride (BN) crystal similar as graphene hexagonal surface lattice geometry. The application of BN substrate to support graphene layer also allows for the observation of FQHE in higher temperatures because of reducing stress and perturbations caused by lattice mismatching. The new data for fractional quantum Hall effect both in suspended graphene scrapings [10–14, 17, 18] and in graphene samples on crystalline substrate of boron nitride [15, 16] have been recently collected.
The specific structure of Landau levels (LLs) in graphene leads to distinct than in conventional semiconductor 2DEG scheme for integer quantum Hall effect in graphene referred to as “relativistic” of its version . The graphene planar sheet has a regular crystalline structure with two equivalent sublattices (two carbon atoms in the Bravais cell). This feature together with vanishing of the forbidden semiconductor gap on corners (two of them—non-equivalent) of hexagonal Brillouin zone, with cone-like shape of local valence and conduction bands meeting at these points (the so-called the Dirac points), results in the fourfold spin-valley degeneracy of LLs. Moreover, for Dirac points in graphene, the specific Berry phase-like shift for chiral 2D carriers additionally modifies the LL spectrum, which finally results in series for fillings at which IQHE plateaus occur in graphene. These plateaus in the monolayer graphene occur at the centers of the consecutive LLs (not at completely filled LLs) . In bilayer graphene the extra degeneracy of
In the present paper, we propose the explanation of this exotic, beyond CF, state in bilayer graphene by the topological commensurability approach [6, 21 22]. In this braid group-based approach, we explain the structure of fractional fillings of LL subbands beyond the CF model and explain the reason for the failure of CF model in bilayer graphene. The topological braid group approach formerly developed in [23, 24] gives the hierarchy of FQHE in agreement with the available experimental data for monolayer and bilayer graphene in contrary to CF model. It is worth emphasizing the topological explanation of the even denominator-filling ratios for FQHE in bilayer graphene, including .
2.1. Commensurability condition
Let us briefly recall the concept of the commensurability of cyclotron trajectories with interparticle spacing in 2D-charged system at strong magnetic field within the braid group approach to multiparticle systems at magnetic field presence. It must be emphasized that for FQHE formation, the interaction of electrons is essential similarly as of any other correlated state. The strong Coulomb repulsion of electrons determines the steady uniform distribution of 2D electrons—in the form of triangle Wigner crystal lattice as the classical lowest energy state. Such a distribution of electrons rigidly fixed (in classical sense) by interaction is the starting point for quantization in terms of Feynman path integral including summation on braid group elements assigned with appropriate unitary weight supplied by 1DUR of the braid group . The latter is necessary in the case of multiparticle system as to any path in the multiparticle configuration space an arbitrary closed loop from the braid group can be attached and the resulting paths with such loops fall into geometrically disjoint sectors (topologically non-equivalent) of the path space. This not-continuous structure of the path space, enumerated by the braid group, forces the additional sum over the braid group elements in the general path integration. Each component to this sum must enter the sum with some unitary factor (due to causality condition) and the collection of these factors for all braid group elements establishes its 1DUR. This explains the central role of 1DURs of the braid group for the definition of quantum statistics of multiparticle system. Each different 1DUR defines different sorts of quantum particles corresponding to the same classical ones [6, 9]. On the other hand, the collective behavior of a quantum multiparticle state must be assigned by the statistics phase shift acquired by the multiparticle wave function if one considers position exchanges of particle pairs [6, 9]. The exchange of particles is understood as the exchange of the position arguments of the multiparticle wave function, and these exchanges are defined by the braid group elements. Hence, the determination of the braid group for the multiparticle system is an unavoidable prerequisite for any correlated state.
The braid group is the first homotopy group of the multiparticle configuration space and describes all possible particle exchanges including quantum indistinguishability of identical particles. This group is denoted as , where
As mentioned above, the Coulomb repulsion of electrons on the plane is a central prerequisite for the braid group definition at magnetic field presence, because the interparticle separation rigidly fixed by the Coulomb repulsion (in the classical equilibrium state) must interfere with the planar cyclotron orbits, discriminating in this way possible correlation types, by the commensurability condition.
For the same classical particles, the quantumly different particles (as, e.g., bosons and fermions in 3D or anyons in 2D) can be defined as assigned by distinct 1DURs of the related braid group. At the magnetic field presence for 2D-charged particles (as in quantum Hall configuration), there are, however, also other important consequences of 2D topology beyond solely 1DURs of the braid group. For the magnetic field strong enough that the classical cyclotron orbits are shorter in comparison to the interparticle separation on the plane, the braid group must be modified itself. The classical trajectories of charged particles obligatory defined by cyclotron orbits at the presence of a magnetic field allow for braid exchanges only when the size of the cyclotron orbit fits accurately to the separation between particles, as is illustrated in Figure 1 (the separation between particles on the plane is fixed by the Coulomb interaction). It is easy to notice that the geometrical planar commensurability between the cyclotron orbit and the interparticle spacing is an unavoidable requirement needed for the definition (implementation) of the generators
Though the generators
In Figure 2 (left), the scheme of the cyclotron orbit at magnetic field
Because the braid group generator must be defined by half of the cyclotron orbit (cf. Figure 3), thus the braid with one additional loop corresponds to the cyclotron orbits with three loops—such a generator has the form . The group generated by (new elementary braid exchanges) is obviously the subgroup of the original braid group because its generators are built from original group generators
This generalization to higher LLs resolves itself to the observation that for the filling fractions corresponding to the commensurability condition written as with , the cyclotron orbits also fit to equidistantly separated particles, though not to just any but rather to every
2.2. Hierarchy for FQHE in graphene
Due to the specific band structure in graphene with Dirac points on the corners of the hexagonal Brillouin zone , the LL spectrum is not equidistant as for ordinary 2DEG but is proportional to instead of
For the bilayer graphene, the situation slightly differs [28, 29]. Due to an interlayer hopping, the Hamiltonian for bilayer graphene attains back the quadratic form with respect to the momentum. Hence, the LL spectrum in bilayer graphene resembles that one for the ordinary 2DEG but with four subbands for each LL level except for the LLL, which has eightfold degeneration [28, 29]. As usual in graphene, the division of the LLL subbands equally among particles and holes causes the bottom for uniformly charged carriers (electrons or holes) to be placed in the center of the eightfold-degenerated LLL (let us remind that in the monolayer graphene, the LLL was only fourfold degenerated; the additional degeneracy in the bilayer graphene is caused by the inclusion of states with
2.3. FQHE hierarchy in monolayer graphene
For the magnetic field strong enough that and for Fermi level shifted (by a lateral voltage) to the conduction band, we deal with fractionally filled first conduction subband of the particle LLL (in this numeration of subbands 2 marks the valley pseudospin orientation and the arrow ↑ marks the orientation of the ordinary spin). For
An important property follows from the fact that cyclotron orbits in graphene must be accommodated to the bare kinetic energy with , similarly in the conventional semiconductor 2DEG (as in non-interacting 2D gas), despite the different “relativistic” version of Landau levels energy in graphene. This “relativistic” oddness is caused, however, by peculiar crystal field features in graphene which do not change the topology of braids. Thus, the bare kinetic part of Landau energy is not modified by the crystal field and the dimension of braid cyclotron orbits repeat the corresponding orbit size from the non-interacting gas. Therefore, for graphene the cyclotron orbit structure is governed by ordinary Landau level restrictions despite a specific band structure with Dirac points, being the result of the crystal field in graphene and not of kinetic energy origin. The difference between the conventional 2DEG system and graphene will be thus related with the different number of LL subbands in graphene in comparison to the conventional 2DEG. The uniform shift in filling factors will be caused also by the Barry phase shift, nonzero for chiral valley pseudospin in graphene .
Therefore, the cyclotron orbit size in the subband
One can generalize this fundamental fractional series by an assumption that in the case when the portion of the residual flux per last loop must be different than the portions per former loops, the last loop of the multilooped cyclotron orbit may be commensurate with every
One can observe also that other variants of commensurability may be concerned with multilooped orbits. Namely, each loop of the multilooped structure may be in principle accommodated to particle separation in a different and mutually independent manner matching nearest and next neighbors in various schemes. One of such a possibility may correspond with the situation when in
In order to include the Berry phase anomaly in graphene, the overall shift of
With lowering of the magnetic field, one can achieve the completely filled subband
In an analogous way, one can consider fillings of the following LLs. The nearest one corresponds to
Finally, let us note that at the special case of the commensurability, , one can meet at . This commensurability concerns rather the paired particles and not the single ones. The pairing does not change the cyclotron radius but reduces twice the carrier number , which gives the above commensurability for pairs at . Hence, at this filling rate, one can expect a manifestation of the IQHE but for paired electrons (the considered correlation corresponds to
An interesting new possibility for commensurability happens next for
The similar scheme of commensurability may be applied to the following subbands. For the subband , the cyclotron orbit size is again equal to , whereas the interparticle spacing is governed by plaque , for . The basic commensurability condition has thus the form: which gives the main series for FQHE (multiloop), . This series can be developed to the full hierarchy via more complicated commensurability opportunities, as described above. In this way, the condition with gives fractions with single-looped correlations of FQHE (single-loop)-type for and fully developed IQHE for
2.4. FQHE hierarchy in bilayer graphene
In bilayer graphene, the topology of braid trajectories considerably changes in comparison to the monolayer system. The bilayer graphene is not strictly two-dimensional and the bilayer graphene opens new possibility for the topology of trajectories. Two sheets of the graphene plane lie in close distance and electrons can hop between these two sheets. We deal here with electrons residing on a two-sheet structure instead of the single sheet as it was the case for the monolayer graphene.
All the above-described requirements to fulfill commensurability condition in order to define the related braid group are binding also for the bilayer graphene, with a single exception. This exception is linked with the fact that
The contribution of this sole loop is independent from other loops and must be omitted. After avoidance of this single loop, the next loops must duplicate the former ones without any rise of the surface per total cyclotron orbit no matter in which way loops are distributed among both sheets (in both sheets, the remaining loops will duplicate loops already present there). Thus, only
It must be emphasized that for multilooped orbits in bilayer graphene the total number of loops still is
For holes in this subband (holes corresponding to empty states in the almost filled subband of particle type), we can write, . The generalization to the full hierarchy of FQHE in this subband attains thus the form, , where
The considerable novelty occurs, however, in the next two subbands of the LLL, and . Because of the larger size of cyclotron orbits for
The generalization of this main series for holes in the subband and to the full FQHE hierarchy in this subband looks as follows: for subband holes, and for the full FQHE hierarchy in this subband, , (the Hall metal hierarchy may be obtained in the limit
In the subband of the LLL, the new commensurability opportunity occurs (a similar one as that which occurred in the first LL of the monolayer graphene): for . It gives filling ratios . These rates are related with single-looped cyclotron trajectories, thus with single-loop correlations similar as for IQHE (though the first two for not integer-filling rates). This is typical for LLs with
Filling of the last subband in the LLL in bilayer graphene undergoes the similar constraints as for all subbands with
The evolution of the fractional-filling hierarchy of subsequent LLs is pictorially presented in Figures 4 and 5 for the monolayer and bilayer graphene, correspondingly. It is also summarized in Tables 1 and 2.
2.5. Specific to bilayer graphene FQHE hierarchy change caused by the type of the LLL degeneracy lifting
For bilayer graphene the degeneration of
|LL subb.||FQHE (single-loop), paired, IQHE||FQHE (multiloop)()||Hall metal|
|LL subb.||FQHE (single-loop), paired, IQHE||FQHE (multiloop) ()||Hall metal|
|LL subb.||FQHE (single-loop), paired, IQHE||FQHE (multiloop) ()||Hall metal|
One can consider also the situation in the LLL of bilayer graphene, when the degeneracy of
3. Comparison with experiment
The filling factor in graphene (monolayer and bilayer) can be changed both by the external magnetic field strength variation and by the particle concentration variation via shifting of the Fermi level near the Dirac point by the application of a lateral voltage of relatively small magnitude (typically 10–60 V). Due to spin-valley degeneracy and Berry phase contribution related with the chiral valley pseudospin, the IQHE is observed in monolayer graphene for fillings for particles from the conduction band and for the mirror-negative fillings for holes from the valence band. However, FQHE was not detected in graphene samples deposited on a substrate of
FQHE in suspended graphene is observed at relatively high temperatures around 10 K , and even higher (up to 20 K) , which seems to be explained by the relative strengthening of the electric interaction due to the absence, for suspended samples, of a dielectric substrate (with the dielectric constant in case of
The nowadays experimental data of FQHE in graphene [31–35] are consistent with the predictions of the braid group commensurability approach. In the case of graphene, the specific band structure with pseudo-relativistic conical Dirac bands leads to equal division of the LLL between the valence band of holes and the conduction band of electrons, which itself via an anomalous “relativistic” IQHE [20, 36, 37]. Applying lateral gate voltage (within the range ca. 5–60 V [10, 15, 31]) allows for the control of the density of carriers at a constant magnetic field, which is exceptionally valuable from an experimental point of view. For relatively small densities of carriers, the cyclotron orbits will be too short for braid exchanges of particles (leading to FQHE (multilooped)) at a relatively low magnetic field—the weaker for smaller concentrations is). The experimental observations exactly support this prediction [31, 32]. For low concentration, while closing on the Dirac point, one may expect, on the other hand, too strong which would correspond to an emergence of the insulating state near the Dirac point in a sufficiently large magnetic field. This prediction also agrees with the experiment . In the case of the hexagonal structure of graphene, electron (or hole) Wigner crystallization  may exhibit interference between the triangular crystal sublattices, and including such resonance between two sublattices may cause blurring of the sharp transition to the insulator state, which is also consistent with observations.
The rapid progress in the recent experiment allowed for the precise observation of FQHE in graphene on the crystal substrate of boron nitride (
The mobility of carriers in graphene is lower than in traditional 2DEG, but taking into account that the carrier concentration in graphene can be lower in comparison to semiconductor heterostructure [40, 41], the corresponding mean free path in both cases well exceeds the sample dimension (of µm order, as the mobility is proportional to the concentration and to the mean free path of carriers).
It is worth noting that energy gaps protecting incompressible FQHE states in graphene are larger than in traditional semiconductor materials, reaching the high magnitude of the order of 10 K, which is linked with Dirac massless character of carriers. In conventional semiconductor heterostructure, the corresponding gaps are much lower and the observed FQHE is much more fragile in comparison to graphene.
The recent development in experiments with monolayer graphene on BN substrate [15, 16] and with suspended small sheets [17, 18] allowed for the observation of more and more Hall features referred to FQHE in subsequent subbands of two first LLs. While the sequence of fillings in the lowest subband of the LLL fits well to CF predictions (including CFs with two- and four-flux quanta attached), an explanation of the FQHE-filling structure in next subbands strongly deviates from this simple picture. CF theory fails also in all subbands of the first LL [16–18]. Various scenarios of breaking of the approximate SU(4) spin-valley symmetry in graphene do not solve this problematic situation despite many theoretical attempts, which evidences insufficiency of the CF model in this case.
For much more effective understanding of FQHE in graphene, there occurs the braid group base commensurability approach. The hierarchy for FQHE predicted in this way is consistent with all experimental data known as of yet—the corresponding filling fractions can be reproduced by the topology braid group method (cf. Table 1). From this comparison, it is also clearly visible why the CFs are efficient only in the LLL and fails in many other situations.
Partial efficiency of CF model in the LLL is linked with the fact that exclusively in the LLL cyclotron orbits are always shorter than the interparticle spacing and additional loops are necessary. These loops can be simulated by fictitious field flux quanta attached to CFs. In the case, however, when the more complicated commensurability condition supports particular FQHE states in the LLL or in higher LLs the CF model fails. The braid group approach reproduces all features described correctly by CF model and moreover explains details inaccessible for CF approach. The usefulness of the CF model is in particular strongly limited in higher LLs because in this level the simple multilooped commensurability is needed only close to the subband edges, whereas the central regions of all subbands in higher LL are of single-looped type commensurability with next-nearest neighbors beyond the CF concept. For example, in the first LL on monolayer graphene the doublets of fillings are as follows: (), (), (), (), corresponding to single-looped commensurability condition, not related to CFs. These doublets are visible in experiment [15–18]. The number of centrally located filling rates for FQHE (single loop) grows next with the LL number as 2
The commensurability braid group approach successfully reproduces all the positions of observed features in two lowest LLs of the monolayer graphene. The elongate plateaus at edges of subbands with IQHE in center seem to embrace also minimums related to FQHE(multiloop) rates which are located closely to edges – such location makes them out of the experimental resolution. In higher LL, the new features were observed in between the abovementioned doublets but with not developed vanishing longitudinal resistivity as at other FQHE states. It suggests that not all particles participate in correlated state, but might correspond to multilooped correlation commensurate to every second or every third particles. And indeed, these new features in the first LL of monolayer graphene recently reported  at are reproduced one to one by the commensurability series with as shown in Figure 6. One can notice that the FQHE-filling rates are repeated in this hierarchy. It is likely that the single-looped correlated states at the same filling rates are more stable than the multilooped ones which seem to be consistent with the experimental data presented in Figure 6 and in the upper panel of Figure 7.
The next evidence supporting the correctness and usefulness of the commensurability condition is the consistence of the related predictions with the experimental observations in bilayer graphene. The remarkable feature observed in bilayer graphene is the occurrence of FQHE in the LLL in bilayer graphene at fractions with even denominators  which is completely beyond the CF model explanation ability. The commensurability braid group approach for bilayer graphene reproduces, however, perfectly all the experimentally observed FQHE hierarchy, including the controversial state at —cf. Figure 8, the illustration in Figure 5, and in Table 2.
Note finally that the FQHE hierarchy in bilayer systems with the characteristic even denominators manifested itself previously also in bilayer 2DEG of conventional Hall setups—the existence of FQHE state at state has been discovered there [42, 43], which also is consistent with the commensurability braid group predictions.
The interesting observation in bilayer graphene is reported by the comparison of FQHE measurement in bilayer graphene in suspended samples and in samples on the BN substrate [10–14]. Surprisingly, the FQHE state at observed in bilayer structure on BN substrate disappears in the suspended sample of bilayer graphene. We propose to explain this behavior by our commensurability braid group approach, noting that the occurrence of the FQHE state depends of the order of the LLL subband degeneracy lifting, as shown in Table 3. We suppose thus that the external conditions related to the presence of the BN substrate can reverse the ordering of the LLL subbands resulting in the occurrence of the FQHE state at , which is, however, no case for the suspended sample with opposite lowest subband ordering.
The commensurability of cyclotron orbits building cyclotron braids with interparticle spacing in homogeneous 2D-charged systems in magnetic field is formulated in order to verify the possibility of arrangement of correlated multiparticle Hall states and decipher the “magic” hierarchy of filling ratios for FQHE. By application of the commensurability braid group approach to study the statistics of interacting many particle systems, the FQHE hierarchy of filling rates in graphene has been determined. The evolution of this hierarchy with growing number of the LL has been deciphered in consistence with experimental observations. The new opportunities for commensurability in higher LLs were established leading to different than ordinary FQHE (multiloop)-correlated states and the related fractional fillings of LLs have been identified beginning from the first LL. This new states have been named as FQHE (single loop) because the related correlations are described by single-looped braids. Both the monolayer and bilayer graphene have been analyzed and the essential difference of related hierarchy structures has been identified and described in detail. The even denominator main line of the fractional-filling hierarchy in the bilayer graphene is found in satisfactory consistence with the experimental observations. The presented hierarchy for the monolayer and bilayer graphene found the confirmation in available experimental data for graphene on BN substrate as well as for suspended samples including bilayer graphene for fillings up to sixth spin-valley subband. The impressive success of the commensurability braid group approach is the explanation of the disappearance of the FQHE at in bilayer graphene in suspended sample, whereas this state has been formerly observed in bilayer graphene on the BN substrate. We link this phenomenon with the inversion of the ordering of the LLL lowest subbands caused by an influence of the crystalline BN substrate.
It is worth noting that the hierarchy observed in bilayer graphene is out of reach of the conventional CF theory.
This work is supported by the NCN projects P.2011/02/A/ST3/00116 and P.2016/21/D/ST3/00958.