## Abstract

The commensurability condition is applied to determine the hierarchy of fractional filling of Landau levels for fractional quantum Hall effect (FQHE) in monolayer and bilayer graphene. Good agreement with experimental data is achieved. The presence of even-denominator filling fractions in the hierarchy of the FQHE in bilayer graphene is explained, including the state at

## 1. Introduction

Recent progress in Hall experiments on graphene has revealed many new features in resistivity in Hall configurations exhibiting the fractional quantum Hall effect (FQHE), both in suspended graphene scrapings [1–3] and in graphene samples on a crystalline substrate of boron nitride [4,5]. The different structure of the Landau levels (LLs) in graphene—compared with the conventional semiconductor 2DEG—is the source of a distinct scheme of the integer quantum Hall effect (IQHE) in graphene referred to as the relativistic IQHE version. The Berry-phase-induced shift for chiral carriers in graphene, together with the fourfold spin-valley degeneracy of LLs, result in the series *n*=0 and *n*=1 LLs causes accumulation of the Barry phase which shifts the IQHE plateau positions to *ν*=4,8,…. In bilayer graphene, LLs are fourfold spin-valley degenerate except for the eightfold-degenerate lowest Landau level (LLL) [6]. Simultaneously, numerous new features have been experimentally observed at the fractional filling of LLs related to the FQHE. The new filling fractions are observed in the first six subbands of LLs with *n*=0 and *n*=1 in monolayer graphene [2–5], which do not replicate the hierarchy of the FQHE in conventional 2DEG systems. Particularly interesting is the observation of unusual even-denominator filling fractions for FQHE detected in bilayer graphene, including the most pronounced feature at

In this paper, we analyse the hierarchy of fractional filling linked to strongly correlated multiparticle states in graphene using the topological commensurability approach developed earlier for ordinary 2DEG Hall systems [7–9]. Through this technique, we explain the structure of fractional filling of LL subbands and demonstrate its evolution with the increasing number of LL. This approach provides the FQHE filling hierarchy, which is shown to be in good agreement with currently available experimental data for monolayer [10] and bilayer graphene.

The usefulness of the commensurability condition is based on the essential role of the electron interaction that fixes interparticle separation on the plane allowing the discrimination of filling fractions by comparison of the cyclotron orbit size with interparticle spacing. Only when classical cyclotron orbits match perfectly with particle spacing, the mutual classical exchange of neighbouring particles becomes possible in the presence of a perpendicular magnetic field and the appropriate braid group can be defined which allows for the quantum statistics determination [7]. The geometry of particle cyclotron orbits enables the topological classification of possible or impossible trajectories for multiparticle system at magnetic field presence and the construction of the related braid group [7,8]. In two dimensions, all cyclotron orbits are planar, and in the case of uniformly distributed particles with the same kinetic energy exposed to the perpendicular magnetic field, they either topologically admit exchanges of neighbouring particles (expressed by elementary braids—generators of the braid group) or not, depending on the commensurability of cyclotron orbits with interparticle separation. In the LLL, the size of the cyclotron orbit is defined as *A*=*S*/*N*_{0}, where *S* is the surface area of the sample and *N*_{0} is the LL degeneracy. If the planar size of the cyclotron orbit *A* equals the plane fraction per single particle (i.e. *A*=*S*/*N*, where *N* is the number of particles), a mutual exchange of particles is possible and the ordinary braid group can be defined. Otherwise, when *S*/*N*>*A* the exchange of particles is impossible along ordinary single-loop cyclotron trajectories, as they are not sufficiently long to merge with neighbours on the plane. Furthermore, when *S*/*N*<*A*, the interchange of two-dimensional particles is also impossible along cyclotron trajectories because such an exchange does not conserve uniform particle distribution with constant interparticle spacing unless *A* matches up with every second particles, every third particles, and so on (as typical for completely filled higher LLs). These three distinct commensurability situations are illustrated in figure 1L. The topological condition for interparticle exchange in uniformly distributed planar systems of charged interacting particles plays a fundamental role in eliminating trajectories from the full braid group that are not admissible at the presence of the magnetic field that do not allow for particle merging. The discrepancy between cyclotron orbit size and interparticle spacing precludes definition of the full braid group generators *σ*_{i} corresponding to the interchange of *i*th and *i*+1th particles (upon a certain selected enumeration of particles that is arbitrary in general because of particle indistinguishability). However, a related braid group must be defined to establish the statistics of quantum particles and a correlated multiparticle state. The statistics is governed by the selected one-dimensional unitary representation (1DUR) of this group [11]. According to the general rules of quantization [12], if the particle classical positions defined by the arguments of the multiparticle wave function *Ψ*(*z*_{1},…,*z*_{N}) (where *z*_{i} is the coordinate of *i*th particle on the plane, i.e. the classical position of the *i*th particle) change along a selected loop from the braid group, then this wave function acquires the phase shift *e*^{iα} given by the 1DUR of this particular braid. Note that considered exchanges of particles expressed in terms of the braid group concern in fact exchanges of arguments of the multiparticle wave function. In two dimensions, these exchanges do not resolve themselves to the permutation alone as they would in three dimensions.

In the case of the commensurability, as shown in figure 1L(a), the full braid group can be defined and the statistics typical of a two-dimensional system become available. According to the form of 1DURs for the full braid group for the plane [7] *e*^{iα} with *α*∈[0,2*π*), they correspond to fermions when *α*=*π*, to bosons when *α*=0 and to anyons for other *α*. However, the generators *σ*_{i} of the full braid group cannot be defined if *S*/*N*>*A*, as shown in figure 1L(b). If one removes from the full braid group all impossible single-looped-related trajectories (corresponding to the generators *σ*_{i} that cannot be defined because the cyclotron orbits are not sufficiently large to match the particles), one observes that the remaining braids with additional loops, i.e. multi-looped, fit to particles in two dimensions that are separated by a greater distance than the single-looped cyclotron orbit can reach. These braids are related to multi-looped cyclotron orbits with *p* loops (i.e. *pA*=*S*/*N*, where *p* is an odd integer) [8]. This property follows from the fact that multi-looped cyclotron orbits on the plane must be larger than single-loop orbits for the same magnetic field. The external magnetic field flux passing through the multi-loop two-dimensional cyclotron orbit is the same as that passing through the single-loop orbit; in the former case, only a fraction of the total flux falls per loop. Consequently, the size of each loop grows corresponding to that fraction of the flux, as illustrated in figure 1R. Figure 1R (left) shows the cyclotron orbit for magnetic field *B* as accommodated to the quantum of the magnetic field flux (i.e. *BA*=*hc*/*e*). The cyclotron orbit *A* coincides with *S*/*N* (where *S* is the sample area and *N* is the number of particles) in the case of the completely filled LLL. For a field that is larger by a factor of *p* (i.e. *pB*), the cyclotron orbit accommodated again to the flux quantum is smaller than the interparticle separation *S*/*N*, as illustrated in the centre graph in figure 1R for *p*=3. If *p*-looped orbits are considered, then in two-dimensional space the external flux *pBA* must be shared between *p* loops within the same surface *A* (i.e. *BA* for each loop). Thus, each loop accommodated to the flux quantum *hc*/*e* has an orbit with the surface *A*, resulting in a total flux of *pBA* per particle, as illustrated in figure 1R (right). The size of *A* in the right panel is identical to *A* in the left panel, which means that the *p*-looped orbits fit the interparticle separation defined by *A* though the single-looped orbits do not fit. It must be emphasized here, that in order to exchange particles according the cyclotron subgroup generator *p*=3) all *p* loops have to be completed despite one would suspect that the one loop from this *p*-loop orbit would be enough—this is, however, a pictorial illusion, because the single-loop generator *σ*_{i} does not belong to the cyclotron group (i.e. there is no such exchanges). This property, attributed exclusively to the exact two-dimensional topology, provides an explanation for the FQHE and related exotic Laughlin correlations [8] (the one-dimensional unitary representations of *pπ* when *σ*_{i} only *π*, what coincides with the phase shift of the Jastrow polynomial factor of Laughlin function with the exponent *p*).

## 2. The fractional quantum Hall effect in monolayer graphene

When the magnetic field is sufficiently strong that *ν*∈(0,1), one encounters completely filled valence band hole states in the LLL and fractionally filled subsequent particle states in the conduction subband corresponding to *n*=0,2↑ (where 2 indicates the valley pseudospin orientation—for the first electron subband of the LLL, as four subbands of the LLL are shared between the valence and conduction bands in graphene, and ↑ indicates the ordinary spin orientation). The degeneracy of each subband is *N*_{0}=*BS*/(*hc*/*e*) and *N*<*N*_{0} for fractional filling *ν*∈(0,1).

The cyclotron orbit size in the LLL is defined in the graphene case as *A*=*S*/*N*_{0}, which is the same size as in the case of conventional semiconductor 2DEG because the degeneracy of each LL subband, *N*_{0}=*BS*/(*hc*/*e*), is the same in both cases, and the archetype of the correlated incompressible multiparticle state is given by the IQHE ordering when *A*=*S*/*N*, i.e. for completely filled subband, *N*=*N*_{0}. As this property is common to all Hall systems, including also graphene, one can say that the cyclotron orbits are accommodated to the ‘bare kinetic energy’ *n*=0, where *ω*_{c}=*eB*/*mc*, despite that in the case of graphene LL energy is changed by the band effects [6]. The cyclotron orbits are of the same size for all particles in the LL as a result of the flat band condition, quenching the kinetic energy competition and resulting in the same mean velocity and cyclotron orbit size for all particles (however, in a quantum sense, velocity is not well defined because its coordinates do not commute). The cyclotron orbits restrict the topology of all trajectories uniformly in two dimensions and thus restrict the braid group structure despite particularities of the dynamics in the crystal field, because these particularities do not change trajectory topology. Therefore, in the case of graphene, the cyclotron orbit structure is governed by ‘ordinary’ LL energies as in the case for 2DEG. LL restrictions act regardless of the specific band structure. Dirac points that arise from the crystal field strongly modify LLs in graphene but not in terms of the ‘bare kinetic energy’, which determines the classical cyclotron orbit size. The difference between conventional 2DEG systems and graphene resolves in the regard of commensurability of cyclotron orbits and particle separation to a distinct structure of LL subbands present in graphene (compared with the conventional semiconductor 2DEG) and in the Berry-phase-induced shift in the LL fillings.

Thus, the cyclotron orbit size in the *n*=0,2↑ subband is (*hc*/*e*)/*B*=*S*/*N*_{0}, where *S* is the sample surface and *N*_{0} is the subband degeneracy. The multi-looped braid structure is necessary because this orbit size is smaller than the interparticle separation *S*/*N* (because *N*<*N*_{0}). From the commensurability condition, *q*(*S*/*N*_{0})=*S*/*N*, one finds that *ν*=*N*/*N*_{0}=1/*q* (where *q* is an odd integer to maintain the braid structure [8]). For holes in this subband, one can expect the symmetric filling ratios *ν*=1−1/*q*. As in the case of the ordinary 2DEG, one can generalize this simple series by assuming that the last loop of the multi-looped cyclotron orbit is commensurate with the interparticle separation for another filling ratio expressed by *l*, whereas the former loops take away an integer number of flux quanta. In this manner, one obtains the filling hierarchy of the FQHE in this subband of the LLL: *ν*=*l*/*l*(*q*−1)±1, *ν*=1−*l*/*l*(*q*−1)±1, where *l*=1,2,… and minus in the denominators indicate the possibility of an eight-figure-shape orientation of the last loop with respect to the antecedent one. The Hall metal states can be characterized by the limit *ν*=1/(*q*−1), *ν*=1−1/(*q*−1). To account for the Berry phase anomaly in graphene, one could shift *ν* by −2; however, we use the net filling fraction here instead.

For the completely filled *n*=0,2↑ subband (i.e. for *ν*=1), one arrives at the IQHE. For lower magnetic field strength (or a larger number of electrons), and when the first three LLL subbands are filled but the last LLL subband is not fully filled, the cyclotron orbit size *S*/*N*_{0} is still lower than the interparticle separation *S*/(*N*−*N*_{0}) (because *N*−*N*_{0}<*N*_{0}). As a result, the multi-looped structure is repeated from the previous subband, resulting in the same hierarchy except for a shift ahead by 1. For the case of a completely filled all subbands of the LLL, one obtains the IQHE according to the main line (i.e.

We now consider the filling of the next LL with *n*=1. This LL also has four subbands, but in this level, the bare kinetic energy is *hc*/*eB*=3*S*/*N*_{0}. For *N*∈(2*N*_{0},3*N*_{0}], we address gradual filling of the *n*=1,1↑ subband. Cyclotron orbits of size 3*S*/*N*_{0} are compared here to the interparticle separation scale *S*/(*N*−2*N*_{0}). When only a small number of electrons exist in this subband, one encounters the multi-loop structure (corresponding to the inequality 3*S*/*N*_{0}<*S*/(*N*−2*N*_{0})). When *q*(3*S*/*N*_{0})=*S*/(*N*−2*N*_{0}) (where *q* is an odd integer), which provides the main series for the FQHE(multi-loop) in this subband in the form *ν*=2+1/3*q*. Similar to the earlier case, the complete hierarchy reads *ν*=2+*l*/3*l*(*q*−1)±1, (*l*=*i*/3, *i*=1,2,…), *ν*=4−*l*/3*l*(*q*−1)±1 (for subband holes), with the Hall metal hierarchy appearing in the limit of *S*/*N*_{0}=*xS*/(*N*−2*N*_{0}) and *x*=1,2,3, we obtain *n*>0), where cyclotron orbits may be larger than interparticle separation distances and may fit to every second or every third particle. These correlated states are referred to as the FQHE(single-loop). The quantization of the transverse resistance *R*_{xy} related to these fractional filling ratios of higher LLs, *ν*=*hc*/*eB*, is the same as for the FQHE, *h*/*e*^{2}*ν*, but the correlations of Laughlin type involve the exponent *p*=1 in the Jastrow polynomial, given by 1DUR of single-looped braid exchanges as in the IQHE. The number of these new fractional filling ratios grows as 2*n* with the LL number *n*.

The commensurability condition allows also for indication of filling rates at which the paired states can be arranged. Pairing of particles does not change the cyclotron orbit size, i.e. the cyclotron orbit for the pair is the same as for the single particle, which follows from the invariance of the cyclotron energy versus pairing, *S*/*N*_{0}=1.5*S*/(*N*−2*N*_{0}), one obtains *N*−2*N*_{0} by the number of pairs, (*N*−2*N*_{0})/2, then one gets the commensurability condition for pairs 3*S*/*N*_{0}=3*S*/(*N*−2*N*_{0}), resulting in

The following subbands are filled with electrons under a similar scheme. For the *n*=1,1↓ subband, the cyclotron size is 3*S*/*N*_{0} and the interparticle distances are measured with the plaque *S*/(*N*−3*N*_{0}), where *N*∈(3*N*_{0},4*N*_{0}]. The commensurability condition *q*3*S*/*N*_{0}=*S*/(*N*−3*N*_{0}) results in the main series for the FQHE(multi-loop) (i.e. *ν*=3+1/3*q*), from which the full hierarchy can be developed in a similar manner as described above. The condition 3*S*/*N*_{0}=*xS*/(*N*−3*N*_{0}) with *x*=1,2,3 results in fractions with single-loop correlations of the FQHE(single-loop) type for *ν*=4, respectively, whereas a paired state can be realized at

## 3. The fractional quantum Hall effect in bilayer graphene

The special topology of the bilayer 2DEG structure creates an opportunity to verify the braid-group-based concept of the commensurability of cyclotron orbit size with interparticle spacing in a planar system of interacting particles. Bilayer graphene is not strictly two dimensional, which changes the topological situation considerably. Two sheets of the graphene plane lie in close vicinity with a hopping constant that mediates changes in electron position between the planes. Here, we consider the double amount of electrons that reside on a two-sheet structure instead of a single sheet (which was the case for monolayer graphene).

All described above requirements to fulfil the commensurability condition when defining the related braid groups for the correlated multiparticle states are in charge also in the case of bilayer graphene, with a single difference compared to the monolayer case. Namely, the double-loop cyclotron orbits may have the same size in bilayer graphene as the single-looped orbit. This follows from the fact that the second loop may be located in the graphene sheet opposite the first one and that the external magnetic field flux that passes through such a double-loop orbit is twice as large as through a single-loop orbit. Each loop has in this case a separate individual surface—in contrast to the double-loop orbit located on the purely two-dimensional plane (which forced each loop to take away only a fraction of the total flux because both loops share the same surface in two dimensions). Considering that a multi-looped orbit in bilayer graphene may be partially located in each two-dimensional sheet, the contribution of the one loop must be avoided, whereas the remaining loops must share the same flux as that passing through a single-looped orbit, independent of how the loops are apportioned between the two sheets. Thus, one can write out the commensurability condition for the case of overly short single-looped cyclotron orbits (for example, in the *n*=0,2↑ subband of the LLL—the first particle-type subband of the LLL) in the following form: (*p*−1)(*hc*/*eB*)=(*p*−1)(*S*/*N*_{0})=*S*/*N* for *hc*/*eB*=*S*/*N*_{0}<*S*/*N*, resulting in the main line hierarchy *N* is the total number of particles in both graphene sheets, *N*_{0} is the degeneracy counted for both sheets together, *S* is the surface area of the sample (i.e. the surface area of a single sheet) and *p* is an odd integer.

The factor *p*−1 in the formula above arises from the fact that when the effective cyclotron orbits are enlarged, the only orbits participating are those from the ideal two-dimensional sheet of bilayer graphene (no matter where the doubling loops are located; note that the largest size effective orbit is attained in this way) with the exception of a single orbit located in the sheet opposite the first one. This sole loop contributes to the total flux with an additional flux quantum because of its own surface, and this loop must be omitted. The next orbits must duplicate the former ones without departing from the surface; and which sheet they are located in is irrelevant because they will duplicate loops already present in either. Thus, only *p*−1 loops take part in the enhancement of the effective *p*-looped cyclotron orbit.

For such multi-looped orbits, the total number of loops is still *p*—thus, the generators of the corresponding cyclotron subgroup are of the form *p* exponent for the Jastrow polynomial. However, because of the distinct commensurability of orbits with interparticle separation in bilayer graphene, the related filling fractions are *ν*=1/(*p*−1) in the first particle-type subband of the eightfold degenerate LLL (i.e. the *n*=0,2,↑ subband). This even-denominator main series of the FQHE hierarchy in bilayer graphene coincides reasonably well with experimental observations [1], including the case of

For holes in the subband (these holes are not holes from the valence band but rather correspond to unfilled states in the nearly filled particle-type subband), one can write *ν*=1−1/(*p*−1), whereas generalization to the full hierarchy of the FQHE in this subband takes the form *ν*=*l*/*l*(*p*−2)±1, *ν*=1−*l*/*l*(*p*−2)±1, where *l* corresponds to a filling factor for another correlated Hall state, including the case of completely filled LLs exhibiting the IQHE. In the next subband of the LLL, corresponding to *n*=0,2↓ (assuming that this subband succeeds the former one), the hierarchy is identical but is shifted by 1 because the commensurability condition has the same form for all subbands with the same *n* due to the same cyclotron orbit size.

Different effects occur in the *n*=1,2↑, *n*=1,2↓ subbands of the LLL because when *n*=1 (2 indicates electron subbands accessible in the LLL with eight subbands, shared between the valence and conductivity bands in bilayer graphene) the cyclotron orbit size is 3*hc*/*eB*=3*S*/*N*_{0}. The FQHE main series in the first of these subbands of the LLL, *n*=1,2↑, has the form *ν*=3−1/3(*p*−1); for the full FQHE hierarchy in this subband, *ν*=2+*l*/3*l*(*p*−2)±1, *ν*=3−*l*/3*l*(*p*−2)±1 (with Hall metal hierarchy in the limit of

Nevertheless, a new commensurability opportunity occurs in the *n*=1,2↑ subband of the LLL: 3/*N*_{0}=*x*/(*N*−2*N*_{0}) for *x*=1,2,3, which yields filling ratios *ν*=3). This new Hall feature, typical for LLs with *n*≥1, we term the FQHE(single-loop). Moreover, for *x*=1.5, one can consider the twice reduction in particle number (*N*−2*N*_{0})/2 to arise from the pairing, which provides perfect commensurability of cyclotron orbits of pairs with the separation of particle pairs occurring at

The last *n*=1,2↓ subband in the LLL in bilayer graphene is filled with electrons in a similar manner as in the antecendent subband because the cyclotron orbits have the same size in both LLL particle subbands with *n*=1. Thus, the hierarchy of fractional filling for the last subband in the LLL is shifted by 1 from the antecedent subband without any other modification. However, the situation changes in the next LL (the first one above the LLL). In the first such LL (with *n*=2), the cyclotron orbits suited to the commensurability condition are determined by the bare kinetic energy for *n*=2, and the corresponding cyclotron orbit size is 5*hc*/*eB*=5*S*/*N*_{0}. An analysis similar to that of the previous LL provides the main series and the full hierarchy of the FQHE(multi-loop) in the *n*=2,1↑ subband: *ν*=4+1/5(*p*−1), *ν*=4+*l*/5,*l*(*p*−2)±1, respectively. (Inclusion of subband holes is accomplished by the replacement of 4+ by 5− in both expressions.) As before, the limit *n*=2,1,↑ subband, the satellite states occur at

The evolution of the fractional filling hierarchy of subsequent LLs is summarized in tables 1 and 2 for monolayer and bilayer graphene, respectively.

For bilayer graphene, the degeneracy of the *n*=0 and *n*=1 states results in eightfold degeneracy of the LLL, i.e. in a doubling of the fourfold spin-valley degeneracy. The degeneracy is not exact, and both Zeeman splitting and valley splitting increase with rising magnetic field amplitude. Stress, deformation and structure imperfections also cause an increase in valley splitting. Inclusion of the interaction plays a similar role. Coulomb interaction causes the *n*=0,1 states to mix, thus lifting their degeneracy. Particularly interesting is the degeneracy lifting that admits an inverted filling order of LLL subbands with distinct *n*. The order inversion of *n*=0,1 to *n*=1,0 affects the filling ratio hierarchy. Assuming that the LLL subbands with *n*=1 are filled earlier than the *n*=0 subband, we obtain the following hierarchy for the first *n*=1,2↑ subband: multi-looped orbits for *ν*=*l*/3*l*(*p*−1)±1, *ν*=1−*l*/3*l*(*p*−1)±1; single-looped orbits for *n*=0,2,↑, we obtain the hierarchy of filling in the following form: multi-looped orbits for *ν*=1+*l*/*l*(*p*−1)±1, *ν*=2−*l*/*l*(*p*−1)±1 and no single-looped orbits.

One can also consider the situation in the LLL of bilayer graphene, in which the degeneracy of the *n*=0,1 states is lifted in such a way that both levels cross at a certain filling factor *ν**<1 (see [13], where mixing between *n*=0,1 states is numerically analysed for small models on a torus or sphere). The corresponding hierarchy of fractional filling can be obtained by the insertion of crossing subbands. Depending on the value of *ν**, various patterns can be achieved by a combination of hierarchy schemes, listed in table 2.

## 4. Comparison with experiments

Despite the use of strong magnetic fields (up to 45 T), the FQHE has not been detected in graphene samples deposited atop SiO_{2} substrates. Only since the emergence of a specialized technique to manufacture the so-called suspended ultrasmall graphene scrapings, which feature extreme purity and a high carrier mobility exceeding 200 000 cm^{2} V^{−1} s^{−1}, has it been possible to observe the FQHE in graphene at net filling *ν*=4).

Furthermore, experiments on monolayer graphene on BN substrates [4,5] and in the form of suspended small sheets [2,3] have enabled observation of numerous Hall features, including fractional filling of successive subbands in the first two LLs. Although the sequence of LLL fillings follows composite fermion (CF) predictions (including CFs with two and four flux quanta attached), the filling structure of the following subbands strongly deviates from this picture. The pattern of filling rates repeated in the subbands of the first LL is apparently incompatible with the CF concept [2,3,5]. This discrepancy has been attributed to the various scenarios of approximate SU(4) spin-valley symmetry breaking in graphene [2,3,5]. Because the magnitude *E*_{Z} of Zeeman splitting is small compared with the Coulomb energy in graphene (i.e. *E*_{Z}/*E*_{C}∼0.01*ε*) and the ratio of lattice scale to magnetic length is *a*/*l*_{B}∼0.06 (where *ε*∼3.2) [4], the subbands that differ in spin and valley-pseudospin orientation are closely located and can be regarded as approximately degenerate. The SU(4) spin-valley symmetry can be subsequently broken by various factors, and one can develop arguments for an unusual filling ratio hierarchy in related symmetry-breaking and phase-like transitions.

Nevertheless, all FQHE filling fractions observed experimentally can be reproduced by the hierarchy described above (as listed in table 1). This framework also clarifies why the CFs are efficient in the LLL but not in higher LLs. This finding is linked with the fact that—exclusively in the LLL—cyclotron orbits are always shorter than interparticle separation and additional loops are always necessary. These loops can be modelled by fictitious field flux quanta attached to CFs. Although the analogy to additional loops is not exact, it is sufficient to provide a similar main line of filling hierarchy in the LLL to that given by the commensurability condition. However, the usefulness of the CF model breaks down in higher LLs because starting from the first LL the multi-looped commensurability criterion is needed only close to the subband edges, whereas the central regions of all subbands of the first LL are occupied by doublets of filling factors (i.e. *n*. The repeating doublet of filling ratios for *n*=1 is notable in the data reported in [4] and in more accurate measurements on suspended samples [2,3].

However, the most convincing evidence supporting the correctness of the commensurability condition is the agreement of the resulting predictions with experimental observations in bilayer graphene [1]. The commensurability condition for bilayer graphene reproduces the observed experimental hierarchy perfectly, as shown in figure 3. Note finally, the FQHE hierarchy in bilayer systems with characteristic even denominators also holds for bilayer conventional 2DEG Hall set-ups; indeed, the

The structure of LLs in graphene is considerably different than that in conventional semiconductor 2DEG and is referred to as its relativistic version [6]. Despite nonlinear growth of relativistic LL energy (*n* in ordinary LL case) the hierarchy structure corresponding to FQHE is similar in both cases as the topological cyclotron effects are governed by the ‘bare kinetic energy’ the same as in conventional 2DEG. In the case of graphene, the further energy dressing with band effects does not influence the cyclotron orbit size within the braid group approach. However, the band effects in graphene including occurrence of Dirac points and valley pseudospin cause different fourfold splitting of LLs and the Berry phase *π* shift in monolayer graphene and also similar fourfold splitting and double Berry phase shift in bilayer graphene supplemented with an extra degeneration (for *n*=0 and *n*=1) of the LLL in the latter case, which influences the integer QHE in graphene [6] in terms of degeneracy and overall Berry phase-induced shift but not in its essential nature. The same holds for the FQHE effect in graphene (monolayer and bilayer)—the related hierarchy is shifted by the Berry phase and accommodated to different degeneracy of LLs in comparison to conventional 2DEG. Thus, one can expect in the bilayer conventional 2DEG the similar oddness as in the bilayer graphene modified by the different degeneracy of LLs and the absence of chiral Berry phase, because the fundamental difference between the monolayer and bilayer cyclotronic property is similar in both cases and displays the topological novelty of the additional two-dimensional sheet in comparison to the monolayer case in the same manner in graphene and in conventional 2DEG.

The bilayer graphene offers the possibility to control the correlated states by magnetic and electric fields which cause breaking of approximate spin-valley SU(4) symmetry in a more complicated manner than in monolayer graphene due to an extra orbital degeneracy (*n*=0,1) in the bilayer graphene configuration [16–18]. The eightfold subband splitting of the LLL level in bilayer graphene tuned by both fields results in a demonstration of the variety of fractional states. In [17], the pattern of sequence of FQH states is reported as observed in boron nitride supported bilayer graphene sample, different that the corresponding one observed in a bilayer suspended sample [1] and also neither the same as in monolayer graphene nor predicted in [13]. The FQH state patterns both for suspended and supported samples are associated with the specific ordering of spin-valley and orbital (*n*=0,1 in bilayer graphene) subbands, called the symmetry braking pattern [1,17]. It is supposed that the disagreement between these patterns may be induced by the different symmetry breaking order in graphene samples supported by the crystalline substrate and in free suspended samples. The pattern of FQH states on crystalline substrate is consistent across the entire sample (as verified by local measurements [17]) and also did not change with current annealing. The electron-hole asymmetric sequence of FQH states [17] can therefore be attributed to the intrinsic properties of bilayer graphene, rather than disorder or other local effects. The observation of an unconventional sequence of FQH states in bilayer graphene indicates the importance of its underlying symmetries and opens an avenue for exploring the nature and tunability of the FQHE. This observation agrees with the above-mentioned possibility of ordering in lifting of *n*=0,1 degeneracy of the LLL in bilayer graphene giving rise to various patterns of the FQHE hierarchy understandable in terms of topology, owing to essential differences in cyclotron commensurability between subbands with *n*=1 and *n*=0. The tunability of FQHE in bilayer graphene is also supported by Maher *et al*. [16] which reported inversion-symmetry breaking by vertical electrical polarization of the bilayer sample opening gap at the charge neutrality point. This again agrees with topology/symmetry changes causing various pattern of FQHE emergence. The layer polarization influences stability of ^{−1} [16]. The strong sensitivity of the hierarchy pattern (repeated in several samples to avoid local effects) supports topology/symmetry conditioning caused by distinction in the electron population in both layers at vertical polarization [18]. However, the state

## 5. Comments and conclusion

The essence of the presented cyclotron braid group approach is the interaction (obviously central for any correlated state, including FQHE). Only at the presence of the electron–electron interaction the cyclotron braid group on the plane at the magnetic field presence can be defined and linked with the commensurability condition, i.e. with the requirement of fitting of the cyclotron orbit size to interparticle separation allowing braid exchanges. Without interaction, the interparticle separation could not be fixed and the commensurability condition losses its sense and usability. FQHE is a result of the interaction and of specific topology of two-dimensional plane, it is not observed in three dimensions. The success in determination of hierarchy for FQHE using commensurability condition is linked with non-local and topological character of this approach. FQHE results from the non-perturbative effects of the Coulomb interaction in two dimensions, which give rise to gapped many-body states at special filling factors. These states have long-range entanglement and cannot be explained in a simple single-particle picture. Similarly, the CF theory, effective in the LLL of conventional 2DEG and in monolayer graphene, is in fact not strictly local using the auxiliary concept of flux tubes, or vortices, fixed to particles and allowing via this multiparticle effective trick to discriminate highly non-local effects. The commensurability approach basing on the topology braid group description of statistics in many body systems falls to the same class of non-local multiparticle models. It elucidates auxiliary fluxes assumed in CF theory and allows for the generalization of the topological approach beyond the limit of the CF model. In particular, the commensurability approach is effective also in higher LLs [9] in agreement with experimental observations. Using the energy minimization methods (exact diagonalization on finite models), the difference between FQH states at various filing fractions is referred to the different form of the effective Coulomb interaction (i.e. Coulomb interaction dressed by the form factor). The energy minimization gives energy gaps for fractions related to special multiparticle correlations of FQHE. These fractions (and their hierarchy) can be, however, identified by the non-local effective-multiparticle topological method of commensurability braid approach generalizing in this way the CF fermion approach.

The advantage of the commensurability condition is especially clearly visible in the case of bilayer graphene, where the states at fractions with even denominators cannot be explained by the CF model even in the LLL. Note, however, that Papić & Abanin [13] provides an alternative explanation of the *n*=1 LL is at zero energy in bilayer graphene (different from conventional 2DEG either single or bilayer). Therefore, Papić & Abanin [13] claims that it is possible that *n*=1 LL which is predicted for the

The method of braid group is rigorous. The braid group is the precisely defined object—*π*_{1} homotopy group of the configuration space of *N*-particle system located on some manifold, in the case of Hall systems, on the plane *R*^{2}. The configuration space of multiparticle system is not simply connected space—thus its *π*_{1} braid group is non-trivial and its particular form depends on the manifold. For *R*^{2}, it is the very well-known Artin group [7]. This Artin group, as any homotopy group, comprises all closed trajectories in the space—here, in the configuration multiparticle space. At the presence of the magnetic field perpendicular to the plane, these trajectories must be of cyclotron orbit type, even though not simple circles which is the case for free particles only. The plane area of the cyclotron orbit is not defined by the classically free particle radius, but instead it is defined by the archetype of correlated incompressible state IQHE when the flux of the field *B* piercing the system with the surface *S* results in the flux *BS*/*N*_{0} per particle, as for the completely filled LLL, *N*=*N*_{0}, i.e. number of particles is equalled to the LL degeneracy. Thus, the plane area with the surface *A*=*S*/*N*_{0} is the definition of the cyclotron orbit in the LLL at the interaction presence.

The braid group is the classical notion, and should not be referred to exchanges of quantum particles—the latter have not any trajectories. Exchanges of particles is associated rather with exchanges of arguments of the multiparticle wave function (like Laughlin function with exponent in Jastrow polynomial *q*=1 for the LLL), *Ψ*(*z*_{1},…,*z*_{N}). If one exchanges these arguments (taking into account that they display the classical particle positions on the plane) according to some braid group element, then the wave function *Ψ*(*z*_{1},…,*z*_{N}) attains the phase shift given by the one-dimensional unitary representation of this particular braid group element [12]. This is the transition from the classical braid group picture to the quantum multiparticle state—this state is assigned by the selected one-dimensional unitary representation of the braid group. The one-dimensional representation is usually not unequivocal—hence one encounters bosons, fermions and anyons. Moreover, one can implement in this way also CFs. They manifest themselves at fractional fillings, e.g. at *A*=*S*/*N*_{0}, are smaller than interparticle separation *S*/*N* because *N*=*N*_{0}/3. Nevertheless, the state at *R*^{3}) multi-loop trajectories can reach neighbours even at *π* (not *π*) exactly as revealed by the Laughlin function (with the exponent in the Jastrow polynomial *q*=3). The related state is the correlated incompressible one corresponding to FQHE.

The foremost observation is that only a perpendicular magnetic field produces the topological effect in two dimensions (modifies the corresponding braid group). No other interaction, including the crystal field resulting, e.g. in Dirac-type band structure in graphene, influences the braid group. This arises from the fact that cyclotron orbits in the LLL is defined as *S*/*N*_{0}, where the degeneracy *N*_{0} of each subband of each LL in graphene is exactly the same as in conventional semiconductor 2DEG. It means that only the ‘bare kinetic energy’ in LLs plays the role in topology of cyclotron orbits also in graphene and not a full ‘relativistic’ LL energy. Dressing with crystal field interaction including Barry phase does not change the topology of braid trajectories, but, what is essential in graphene, changes number of subbands in LLs (and shifts the energy ladder by the Berry phase contribution, including the extra degeneracy of the LLL in bilayer graphene).

Summarizing, the hierarchy of fractional filling for the FQHE as observed in monolayer and bilayer graphene has been successfully reproduced via the commensurability condition. Hierarchy evolution with increasing LL number has been elucidated and described. New opportunities for commensurability in higher LLs have been established, leading to the identification of unusual FQHE correlated states at some fractional fillings of LLs starting from the first level. They are referred as to FQHE(single-loop) correlated states because the related correlations are described by single-looped braids. The even-denominator main line of the FQHE hierarchy in bilayer graphene is found to be in agreement with experimental observations, including that corresponding to the noticed most pronounced correlated FQHE state at

## Data accessibility

There is no accompanying data.

## Authors' contributions

Contribution of co-authors can be estimated as follows: J.J. (2/3) L.J. (1/3). Both authors gave final approval for publication.

## Competing interests

Authors have no competing interests.

## Funding

Supported by NCN P.2011/02/A/ST3/00116.

- Received May 20, 2015.
- Accepted January 7, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.