Hierarchy of fillings for the FQHE in monolayer graphene

In this paper, the commensurability conditions, which originated from the unique topology of two-dimensional systems, are applied to determine the quantum Hall effect hierarchy in the case of a monolayer graphene. The fundamental difference in a definition of a typical semiconductor and a monolayer graphene filling factor is pointed out. The calculations are undertaken for all spin-valley branches of two lowest Landau levels, since only they are currently experimentally accessible. The obtained filling factors are compared with the experimental data and a very good agreement is achieved. The work also introduces a concept of the single-loop fractional quantum Hall effect.

Scientific RepoRts | 5:14287 | DOi: 10.1038/srep14287 to the Berry phase), ν is counted with respect to the first empty sublevel placed in the CB (a third spin-valley branch of the LLL). This change in the terminology arises from the fact, that it is natural to define ν in terms of an electronic density measured from the Dirac (neutrality) point, so from the bottom of the conduction band 25 . Thus, a zero filling ratio in graphene ν = 0 actually corresponds to two completely filled spin-valley sublevels -↑ K 0 and ↓ K 0 . Finally, the relation between graphene and typical semiconductor filling factor can be expressed in a form, for graphene samples (Fig. 1).
Let us remind, that the effective low energy Hamiltonian for a monolayer graphene is equal to 23 , where ζ = ± 1 (plus for K Dirac point and minus for K′ Dirac point), σ is a vector of Pauli matrices (connected with sublattice pseudospin) and p is a quasimomentum. Note, that if one express the quasimomentum with its length and direction p = pn, then the Hamiltonian can be rewritten in the form, where, n = (ζcos(Jφ), ζsin(Jφ)) (for a monolayer graphene J = 1, but it is convenient to introduce this parameter at this point, because after applying J = 2, the obtained ζ , H q eff is proper for a bilayer graphene samples 27 ), ε(p) is the absolute value of the energy eigenvalue. The operator σ ⋅ n projects the pseudospin onto the direction of the momentum p. Note that the eigenstates of the Hamiltonian are also eigenstates of this projection operator with eigenvalues + 1 for electrons and − 1 for holes. In conclusion, the Dirac particles are chiral and the pseudospin is always parallel to the momentum for free electrons from conduction band and always antiparallel for free holes from the valence band 23,28 . Simultaneously, the form of the eigenfunction (spinor), where λ = ± 1 (plus for the CB, minus for the VB) and φ = tg(p x /p y ) minus π 2 stands for the angle between the quasimomentum and the abscissa. During the adiabatic evolution of such state, the quasimomentum (and so the vector n) is rotating through φ in the reciprocal space. When the quasiparticle encircles a Figure 1. The comparison of a 1/3 fractional Hall state for a monolayer graphene (a) and a typical 2DEG (b). The pink colour symbolizes filled (with electrons) parts of subsequent sublevels of the LLL. In a classical two-dimensional semiconductor the filling factor ν semi is measured in relation to the empty lowest Landau level. However, in graphene, since a half of the lowest Landau level is placed in the VB (due to the Berry phase), the filling factor ν is not counted with respect to the empty LLL, but with respect to the first empty sublevel of the LLL placed in the CB (third spin-valley branch of the LLL called ′↑ K 0 ). Thus, such filling in graphene ( ) ν = 1 3 , unlike in a traditional semiconductor, is accompanied by two completely filled sublevels of the LLL located in the VB 22,25 . The relation between graphene and typical semiconductor filling factor can be expressed in a form ν = ν semi − 2.
Scientific RepoRts | 5:14287 | DOi: 10.1038/srep14287 closed contour in the momentum space (φ = 2π), the eigenfunction acquire a nonzero phase equal to π, called the Berry phase, both n = 0 and n = 1 lie exactly at the neutrality point. Thus, the Berry phase gives rise to a unique form of the lowest Landau level in a monolayer graphene, which is placed exactly at the Dirac point. For this reason, the LLL has states (organized in four sublevels) distributed equally between the CB and the VB. Finally, filling factors for free holes from the valence band are just a mirror reflection -with a minus sign -of filling factors for free electrons from the conduction band, therefore it is convenient to determine ν in relation to the bottom of the CB.
In order to apply the topology based commensurability conditions 2,3 to graphene, let us remind that this conditions originated from the necessity of determination of generators of the braid group describing exchanges of particles on a plane. Upon the magnetic field not all trajectories are available, what leads to a redefinition of the full braid group 1 . Only if cyclotron orbits fit to the interparicle separation, the mutual exchange of neighboring particles on a 2D (two-dimensional) manifold with uniform particle distribution is possible (Fig. 2). Otherwise the cyclotron orbits are too short to match closest particles or too large to maintain a particle separation fixed by the uniform density (rigidly kept constant by Coulomb interactions). The cyclotron orbits have the same size for all particles due to a flat band condition reducing the kinetic energy competition and resulting in the same averaged velocity in a presence of a perpendicular magnetic field (though quantumly, velocity is not well defined since its coordinates do not commute). The cyclotron orbits restrict the topology of all trajectories uniformly in 2D, thus restrict the braid group structure despite of particularities of an interaction and other fields, e.g. of the crystal field. Therefore, for graphene, the cyclotron orbit structure is governed by ordinary Landau Levels restrictions despite a specific band structure with Dirac points. The latter particularities of graphene quantum dynamics are included in the ordinary part of the Feynman path integral, whereas additional summation over topologically nonequivalent trajectory classes concerns the braid group structure the same as for a 2DEG upon the magnetic field. The difference between the conventional 2DEG systems and graphene will be here related with distinct degeration of LLs in graphene compared to a typical semiconductor 2DEG.
The kinetic energy of electrons located on a graphene sample upon a strong magnetic field perpendicular to its surface can be thus established from the Landau level quantization (the crystal field does not affect the characterisation of the kinetic energy), Figure 2. This figure presents electrons placed on a 2D manifold in the presence of a strong magnetic field. The exchanges of neighbouring particles are possible only when the cyclotron orbit fits perfectly to the interparticle distance (a). When the cyclotron orbit is too short the exchanges are not allowed, since the condition of maintaining the minimum distance R min between fermions (protected by Coulomb repulsion forces) is not obeyed (b). Similar situation appears when the cyclotron orbit is larger than the spacing between electrons. However, in this case, the condition is broken not for nearest but (for example) for nextnearest neighbours (c). where ω = c eB mc is the cyclotron frequency and n = 0, 1, 2… enumerates the Landau levels. The cyclotron orbit size is proportional to E kin and clearly depends on n numerating LLs, thus it has a distinct value for electrons from different LLs. The coincidence of a cyclotron orbit and an orbit that embrace a single quantum of an external magnetic field flux in the case of completely filled LLL, allows to use this orbit area as the definition of the cyclotron orbit size.
In a graphene all LLs splits into four sublevels with the same degeneration and distinction of electrons due to the ordinary spin of electrons (↓ , ↑ ) and the valley pseudospin (K, K′ ). This sublevel degeneration equals to, where, S represent the sample surface and φ = hc e 0 is a quantum of the magnetic field flux. Preceding subsections contain a derivation of the fillings ν which give rise to integer and fractional quantum Hall states, based on an assumption that such collective state can be realized only when particle exchanges are possible (as required for determination of the statistics for a correlated multiparticle state), i.e., when the cyclotron radius matches perfectly to the integer multiple of a half of minimal distance between particles (protected by Coulomb repulsion forces).
Completely filled subband 0 K′↑ of the lowest Landau level. In the case of a completely filled sublevel of the LLL, the number of particles is equal to the value of the degeneration, From the above equation it is easy to conclude that also the number of quanta of an external magnetic field flux equals N = N 0 and so the minimum area per particle embraces one φ 0 . For this reason, we can establish the surface of the cyclotron trajectory, We assumed that for ν e = 1 the loopless exchanges (described with full braid group generators) are possible.
Partially filled 0 K′↑ sublevel of the lowest Landau level. After the applied magnetic field is raised (B > B 0 , where B 0 corresponds to the completely filled lowest sublevel of the LLL), the number of electrons filling the ′↑ K 0 Landau sublevel becomes smaller compared to the degeneracy N < N 0 . Therefore, the cyclotron radius is not long enough to reach the neighboring carrier and the statistic cannot be determined. To create a collective multiparticle state (like a fractional quantum Hall state) the cyclotron radius must be enhanced by using braids of multi-looped type (as has been proved 1 , in 2D spaces multi-looped braids can match neighboring particles and substitute single-looped generators of the braid group), where p is an odd number (a half of a closed, p-looped cyclotron trajectory needs to form a proper, open exchange trajectory, thus p cannot be even). Simple transformations allow for determining the N to N 0 ratio -the filling factor for particles, and analogically for holes in this sublevel (let us emphasize that they are not free holes from the VB in graphene, which due to a particle -hole interband symmetry in Dirac points may be assigned with the same fractions as free electrons from the conduction band with sign reflection only), The generalized hierarchy, similarly as in a 2DEG 1 , can be established if one assumes that only p − 1 loops of a cyclotron trajectory embrace exactly one external magnetic field flux quantum each, whereas the last one embrace only a fraction of φ 0 , but equal to the number of quanta per particle calculated formally for other (even fractional) fillings, Thus n does not need to be an integer number (like in the Jain's hierarchy 4,5 ) -it might be equal to other fraction ν (this leads to a fractal-like construction). Examples of the above filling hierarchy for electrons are presented in the Table 1. Only in one exceptional situation the cyclotron radius is long enough to reach neighbouring particles, The resulting integer filling factor ν = 2 corresponds to both - -completely filled (so experiencing integer quantum Hall effect -IQHE) subbands of the LLL.
If, however, the cyclotron orbits are shorter in comparison to the particle separation, the multi-loop trajectories are needed to restore the exchanges. In the p-looped case (p -odd), Table 1. Filling factors, for the first sublevel of the LLL, for electrons manifesting the FQHE determined with the commensurability condition (multi-loop trajectories). The filling factors marked with a blue colour are experimentally accessible 12,[14][15][16][17][18][19]22,24 . We have noticed that only fillings, which are separated from the nearest integer ν by more than ~0. Let us note, that the fractional quantum Hall states for the second sublevel are generated by less than half of electrons (the rest N 0 participate in a creation of the IQHE in the lower subband), hence the dips in a longitudinal and plateaus in a transverse resistivity may not be so pronounced.
Analogically, assuming that only a fraction of a flux quantum falls on a last loop from the p-looped trajectory, we can generalize the hierarchy (Table 2), Filling of the ↑ K 1 subband of the first Landau level. While the ↑ K 1 subband of the first Landau level is filling, the total number of free particles is raising from a double degeneracy 2N 0 to a triple degeneracy 3N 0 (for ν e = 3). During this process, N − 2N 0 electrons are being located on the 1LL and remaining 2N 0 are experiencing the IQHE in the zeroth Landau level. It may be misleading that we only include two sublevels of the LLL in our considerations, but remember that N denotes the number of free electrons in the conduction band and for that kind of particles only two sublevels are available. Additionally, the filling factors for free carriers from the VB (holes) are a simple mirror reflection (with a minus sign) of filling factors for free carriers from the CB (electrons).
The surface of a cyclotron orbit from the first Landau level is considerably changed and equals to, This situation (sudden growth of the cyclotron orbit size) allows for new possibilities -new allowed commensurability conditions, which are listed below 2,3 : 1. The cyclotron orbit may fit perfectly to the minimal interparticle separation in this subband, . Note, that in this case multi-looped trajectories are not needed to form a Hall state in a fractionally filled Landau level, so obtained filling factors correspond to the single-loop FQHE. This new Hall feature is possible only for n > 0. Although the quantization of the transverse resistivity is fractional and simmilar to that for ordinary FQHE ( ) ν h e 2 , the Laughlin correlations are represented with the p = 1 power in Jastrow polynomial, which displays single-loop braid exchanges similar to that in IQHE. The system is, hence, described with a full braid group. This specific FQHE state is associated with single-loop cyclotron braids, in contrast to the multi-loop braided FQHE, typical for n = 0. It seems, that the stability of this novel effect might be comparable to the stability of the IQHE, rather than of the ordinary FQHE, what was confirmed in ref. 24 and described in more details in a "Comparison with experiment" paragraph. 2. Also in this band the cyclotron radius might be too short to match the distance between particles.
Then the multi-loop trajectories have to be introduced resulting in the FQHE for electrons, We can also estimate the dual filling factor for holes The generalized hierarchy can be established in the same manner as for subbands of the zeroth Landau level (we could also derive in that manner the Hall metal hierarchy, n → ∞), Exemplary fractions are presented in the Table 3. It is worth to emphasize, that in the Jain's model the FQHE of electrons is described as the IQHE of CFs. Additionally, within this construction n stands for the number of completely filled Landau levels in a diminished effective magnetic field experienced by CFs, so it needs to be an integer number. However, in the discussed cyclotron model, n 1 is a portion of a flux quantum that falls on the last loop of a multi-looped cyclotron trajectory (an implementation of a multi-looped braid). Therefore, n 1 can be set equal to any number of magnetic field flux quanta per particle in the system that ensures, or rather allows, the creation of an collective Hall-like state. Finally, n does not need to be an integer number. We assume, that it can take the value of any fractional filling factor derived from the commensurability conditions. It is also worth to notice, that φ  We have also noticed that only fillings which are separated from the nearest integer ν by more than 0.3, are (at the moment) experimentally observable (in the table this filling factors are marked with a   Table 3. Filling factors, for the lowest sublevel of the first LL, for electrons experiencing the FQHE, determined with the commensurability condition (multi-loop trajectories). The horizontal line separates a standard set of fractions (constructed witch n = m/3, m-integer) from a set of fractal filling factors (that may also be derived from higher order commensurability conditions). Colored filling factors decribe fillings, which are separated from the nearest integer ν by more than 0.3 and are (at the moment) experimentally observable 12,15,22,24 . Others, are lost in a deep and extended minimum of a longitudinal resistivity connected with the integer ν.
Scientific RepoRts | 5:14287 | DOi: 10.1038/srep14287 blue colour). Others, are lost in a deep and extended minimum of a longitudinal resistivity connected with the integer ν.
3. It is also possible that the cyclotron surface matches to twice the minimal interparticle separation, which allows every second carrier to exchange (it allows also for establishing of the statistics), . Also this hierarchy can be generalized (assuming that on a last loop from a p-looped trajectory falls only a , ν e = 5 corresponds to the IQHE in all five ) Landau subbands. , ν e = 6 corresponds to the IQHE in all five Landau subbands.
Summarizing of filling ratios obtained with the use of the commensurability conditions. All of the filling factors obtained in preceding subsections -from various commensurability conditions -are captured in the Table 4. Note, that in a very similar manner as presented in the previous section, one can estimate the filling factors for higher Landau levels, only with the use of the commensurability conditions, 1. If the cyclotron orbit fits perfectly to the interparticle separation, the condition takes the form of, Fillings ν describing the IQHE and the single-loop FQHE are marked with three colours: a dark blue colour -the cyclotron orbit fits to the interparticle separation, a light blue colour -when it fits to the separation of every second particle and a green colour -if it fits to the separation of every third particle. Additionally, filling factors describing the ordinary FQHE are also marked with three different colours: a red colour -when the fraction is derived from the basic hierarchy, a magenta colour -when the fraction is derived from the generalized hierarchy with n = m/3 (m-integer) and a black colour -also for fractions derived from the generalized hierarchy but with n ≠ m/3 (these fillings may also be derived from higher order commensurability conditions). Note, that the generalized hierarchy can also be derived. Also, one can define which ν are filling ratios of the ordinary FQHE and which of the single-loop FQHE. Note, that an integer quantum Hall effect can be defined as a collective state that is formed without the implementation of multi-looped trajectories (multi-looped braids) to provide the particle exchanges. In the same time, the ordinary fractional quantum Hall effect is defined as a collective state with multi-looped trajectories implemented. That is why it is rather improper and may be misleading to call the collective states described with fractional filling factors, but created with cyclotron orbits with no additional loops, as the ordinary FQHE states or even the IQHE states. In this paper we named this unique ν as single-looped FQHE filling factors.
It is also worth to emphasize, that the role of the Coulomb repulsion force is crucial even within the cyclotron subgroup model, since it ensures that the minimal interparticle separation (particles cannot approach closer to each other) is rigidly kept. Otherwise, it would be kept only in average and trajectories outside the cyclotron subgroup could be occasionally realized. Besides the interactions, also the two-dimensionality of a manifold is an important prerequisite (the rich structure of a full braid group). Finally, the commensurability conditions of the cyclotron orbit and the minimal interparticle distance reflect the necessity of exchanges of neighbouring particles for creation of collective states like the fractional (ordinary or single-looped) and integer quantum Hall states.
Comparison with the experiment. Since its first isolation by Geim and Novoselov (2004), graphene was subjected to the intensive, experimental research. Due to very strong interactions between massless Dirac particles (as a result of small dielectric constant ε in comparison to a standard 2DEG) signs of a collective behaviour, resulting in the integer and the fractional quantum Hall effect formation, were highly expected. Also the possibility of modifying with a lateral gate voltage (V bg <10 V to avoid collapse of a sample induced by electrostatic attraction 14 ) of the carrier density in a fixed magnetic field strength, made experiments on monolayer graphene exceptional. Although the IQHE appeared to be extremly robust, allowing for its observation even in room temperatures 29 , the FQHE remained hidden by the cause of a high disorder level. A significant improvement in transport properties was achieved after a preparation of graphene both suspended and placed on a boron-nitride substrate. However, scientists experienced a great disappointment -measurements carried out on these samples in a standard Hall-bar geometry failed to develop quantum Hall features 30 (a recently achieved 31 implementation of non-invasive contacts, which are not strictly connected to the probed area of a sample, finally allowed for an observation of Hall-like states in four-terminal measurements). Later it was argued 16 , that the possible problem in multi-terminal devices may lie in small dimensions of samples (especially a small length to width ratio W/L). The voltage probing lead placed within a large potential drop region -a hot spot -and a proximity of the current probing lead may result in shorting out of the Hall voltage. Additionally, adsorbates may not be completely removed from a device body -but only redistributed over a monolayer graphene flake -upon the current annealing (passing a large current through a sample) 31 . This problem also arises from the interfering nature of electrodes in the Hall-bar geometry, which act like heat sinks -electrons with a very high kinetic energy can easily leak out through voltage probing terminals -and give rise to an inhomogeneous temperature profile 31 . Consequently, abandoning a typical measurment geometry and adopting a two-terminal observation metod allowed for the first observation of a ν = 1 3 FQHE state in graphene 14,19 . One should, however, know, that a detailed description of a quantum Hall state in the two-terminal geometry can be quite tricky. For exemple, the value of obtained conductance G actually depends on both longitudinal and transverse conductivities 16 . Although, in a vicinity of σ xx dips, the total conductivity G is expected to coincide with σ xy , its value may still be higher than those of traditional Hall-bar measurements 14 . Also the estimation of a FQHE excitation gap -and other quantitative characterizations -cannot be performed in a straightforward manner 18 . The latter requires an appropriate theoretic approach, like the Gaussian model with peak positions and widths, together with a sample W/L relation (effective rather than real, since its value may differ from the geometric aspect ratio of the device), as parameters 16 . It was also demonstrated, that in such devices pinning of an electron density below the contacts, which results in a inhomogenous density in the graphene sample and a p-n-p junction formation (or p-n, n-n' and others), gives rise to a fractional conductance G connected with IQHE and not FQHE as expected 16,32 .
Moreover, recently other type of a measurement setup appeared to be very successful in developing of collective features -especially in the LLL. Since graphene flakes may be much cleaner on the nanometer scale, probing a smaller area can allow for observation of the fragile FQHE states, which are absent in typical transport experiments. Because of a nonzero disorder, the typical procedure gives only an average view of the sample characteristics and may cover up some interaction induced effects 17 . The microscopic information is not blurry and can be obtained with the use of local electronic compressibility measurements, performed with single-electron transistors 12,15 .
It is also worth to emphasize, that finally some plateaus in σ xy corresponding to the integer and the fractional quantum Hall effect were observed in multi-terminal devices 18,22,24 . The progress in a Hall-bar measurements was achieved after replacing the invasive contacts, directly connected to a sample body probed by transport measurements, with the non-interfering electrodes, connected with a flake by the etched constrictions 31 . As a matter of fact, the experiment carried out in this geometry by Amet et al. 24 was the first one to reveal the plethora of fractional quantum Hall states in the first Landau level (although a few states, mostly associated with the single-loop FQHE, were already evidenced earlier 22 ).
Additionally, in high magnetic fields, in graphene samples, a transition to the insulating state was observed for low concentrations of carriers 14,19 . The latter manifest itself as a dramatic increase in the resistance. It was also pointed out by Du et al. 19 , that the fractional quantum Hall states (like ν = − 1/3) may compete with this insulating phase, leading to the disappearance of Hall plateaus in the presence of high disorder. However, the FQHE may reappear after annealing of the sample. Thus, a great value of mobility seems to play a triggering role in a collective Hall-like state creation, what stays in perfect agreement with the cyclotron subgroup model, since a sufficiently longer mean free path is needed to traverse multi-looped trajectories.
The recent measurement of the transconductance in the PET configuration of small scrapings of graphene (0.8 × 3 μm) in a varying gate voltage V bg + δV bg (t) (f = 433 Hz, V bg ~ 1 V, δV bg ~ 10 mV), has revealed a fine structure of local correlated states 17 . This structure is visible in typical transport experiments as noisy-like oscillations but in the not-noisy regime. The detailed inspection of this fluctuations was done by visualization of the transconductance in the (N, B)-plane, which revealed a highly ordered pattern ( Fig. 2 in 17) attributed to series of local correlated states closely accompanying fractions for the IQHE (and the FQHE as well) in monolayer graphene with broken SU(4) symmetry. The linear character of this new feature was discovered, including several bunches collinear to directions of main IQHE/FQHE ratios in (N, B) with n enumarating Landau bands, b depending on n and level degeneracy, first ± arising from an electron -hole symmetry and second one -from the 'eight-figure-shape' of the trajectory. These fractions are gradually shifted towards subband edges with growing n and compressed to smaller periods for higher n, which also agrees with the details of the new observation 17  are not visible in graphs contained in this paper, but they can be easily found in Fig. 1b in ref. 15 and Fig. 4c in ref. 15, respectively).
In the next subbands - (the highest subband of the 1LL is still unreachable in experiments) -the situation slightly changes and there are three, not one, filling factors corresponding to the single-looped braid group generators (two single-loop FQHE states and one IQHE state),   responsible for this phenomenon. Thus, particles are not forced to traverse path consisted of many loops (to enhance a cyclotron radius and restore particle exchanges) and the system is described with a full braid group, not a subgroup. Although, these single-loop states are combined with a noninteger value of a transverse conductivity σ xy , its nature is expected to have more in common with the integer quantum Hall effect, rather than the ordinary FQHE. Our conclusions seem to agree with a measurment presented in Fig. 1c in ref. 24. Remarkably, the dips in a longitudinal resistance R xx described with fractional filling factors ν = , , , are developing, side by side with integer ν, in Landau fan diagram for magnetic fields as low as 5T. This happens even before the famous ν = 1 3 state emerges (as well as its counterpart 4 3 from the second spin-valley branch of the LLL). This proves that states described by multiples of 1 3 (solely in the 1LL) are almost as stable as integer quantum Hall states, as it was predicted by the topology-based model.
The FQHE filling factors from the first Landau level (determined from the main and the generalised hierarchy) are gathering close to the subband rims, which are marked with the integer filling ratios. The higher the subband index is, ν converge closer to the rims (this is also the feature of the FQHE fillings in typical 2D semiconductors 2 ). The fractional filling factors placed in the vicinity of the IQHE ratio may disappear in its extended dip of the longitudinal resistivity. This mixing leads to the further flattening of the minimum already large in diameter. These predictions are consistent with the experimental observations, since a great amount of the theoretically derived Hall states stay beyond the resolution ability of measurement techniques. We have noticed, that only filling factors, which are separated from the nearest integer ν by more than 0.3 are measured. The latter is confirmed by the fact, that all filling factors -derived in the previous sections -that fulfill this condition (for example all ν marked with a blue colour in the Table 3

Discussion
The commensurability conditions are entirely based on a unique topology of two-dimensional manifolds with the magnetic field presence -and its impact on an appropriate braid group describing the multi-particle system -taken into account. In a great simplification, the commensurability conditions determine whether the cyclotron orbit fits to the interparticle distance or not. The fulfillment of fitting terms ensures that the analyzed filling factor describes the collective quantum Hall-like state. In this article we have determined the FQHE and the IQHE hierarchy for the two lowest Landau levels, which seem to stay in a perfect agreement with the data provided by experiments. It is worth to notice, that states with fractional fillings may also correspond to the single-looped fractional quantum Hall effect, when the particles do not need the multi-looped trajectories (arising from new multi-looped operators of a reduced full braid group called a cyclotron braid subgroup) for exchanges. The paper also presents a summary equations for ν calculation in the n = 0 and the n = 1 Landau level, but they can be easily generalized for higher LLs. We have also pointed out the basic differences between the typical 2D semiconductor and the monolayer graphene, with a special attention paid on a distinct definition of a filling factor in the latter case. It is also emphasised that, although the calculations were carried out for free electrons from the CB, the obtained hierarchy is actually a mirror reflection taken with a minus sign of the one for free holes from the VB, which is usually more pronounced in experimental results. Additionally, it is worth to remember that the specific character of the graphene energy spectrum and a nonzero Berry phase cause that the LLL is placed exactly in the Dirac point and only two out of its four sublevels are placed in the conduction band and are accessible for free electrons. In the future work we would like to derive and present the detailed description of the appropriate braid subgroup for all filling factors, even out of the lowest Landau level. Note, that the appropriate generators (of cyclotron subgroups) were already defined for the main line