Difference in hierarchy of FQHE between monolayer and bilayer graphene

Physics Letters A 379 (2015) 2130–2134

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Difference in hierarchy of FQHE between monolayer and bilayer graphene J. Jacak, L. Jacak ∗ Institute of Physics, Wrocław University of Technology, Wyb. Wyspia´ nskiego 27, 50-370 Wrocław, Poland

a r t i c l e

i n f o

Article history: Received 5 June 2015 Received in revised form 30 June 2015 Accepted 1 July 2015 Available online 3 July 2015 Communicated by V.M. Agranovich

a b s t r a c t 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. The difference in FQHE hierarchy for bilayer and monolayer graphene is outlined and explained. 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 ν = −1/2. © 2015 Elsevier B.V. All rights reserved.

Keywords: FQHE hierarchy FQHE in graphene Monolayer graphene Bilayer graphene Higher Landau levels Braid groups Composite fermions

Recent progress in Hall experiments on graphene has revealed many new features in longitudinal and transverse resistivity in Hall configuration 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 new filling fractions are observed in the first six subbands of LLs (Landau Levels) with n = 0 and n = 1 in monolayer graphene [4,5,2, 3], 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 ν = − 12 [1]. In this letter, we analyze the hierarchy of fractional fillings linked to strongly correlated multiparticle states in graphene using the topological commensurability approach developed earlier for ordinary 2DEG Hall systems [6–8]. Through this technique, we explain the structure of fractional fillings of LL subbands and demonstrate their 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 and bilayer graphene.

*

Corresponding author. E-mail address: [email protected] (L. Jacak).

http://dx.doi.org/10.1016/j.physleta.2015.07.001 0375-9601/© 2015 Elsevier B.V. All rights reserved.

1. Braid-commensurability condition 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 neighboring particles becomes possible in the presence of a perpendicular magnetic field and the appropriate braid group can be defined, which is necessary for the quantum statistics determination [6–8]. In 2D, all cyclotron orbits are planar, and in the case of uniformly distributed particles with the same velocities exposed to the perpendicular magnetic field, they either topologically admit exchanges of neighboring particles (expressed by elementary braids—generators of the braid group) or not, depending on the commensurability of cyclotron orbits with interparticle separation. If the planar size of the cyclotron orbit A equals the plane fraction per single particle (i.e., A = NS , where S is the surface area of the sample and N is the number of particles), a mutual exchange of particles is possible and the braid group can be defined. Otherwise, when NS > A the exchange of particles is impossible along ordinary single-loop cyclotron trajectories, as they are not sufficiently long to merge with neighbors on the plane. Furthermore, when NS < A, the interchange of 2D particles is also impossible along cyclotron trajectories because such an exchange does not conserve uniform particle distribution with constant interparticle spacing unless cyclotron orbit

J. Jacak, L. Jacak / Physics Letters A 379 (2015) 2130–2134

matches up with every second particles, every third particles, and so on (as typical for completely filled higher LLs with n = 1, 2, . . .). These three distinct commensurability situations are illustrated in panel L of Fig. 1. The discrepancy between cyclotron orbit size and interparticle spacing precludes definition of the braid group generators σi corresponding to the interchange of i-th and (i + 1)-th particles. Nevertheless, the braid group must be defined to establish a correlated multiparticle state with defined statistics of quantum particles. The statistics is governed by the one-dimensional unitary representation (1DUR) of this group [7]. If the particle classical positions defined by the arguments of the multiparticle wave function ( z1 , . . . , z N ) (where zi is the coordinate of i-th particle on the plane) change along a selected loop from the braid group, then this wave function acquires the phase shift e i α given by the 1DUR of this particular braid. In 2D these exchanges do not resolve themselves to the permutation alone as they would in 3D. In the case of the commensurability, as shown in panel L of Fig. 1 a), the full braid group can be defined and the statistics typical of a 2D system become available [6]. However, the generators σi of the full braid group cannot be defined if NS > A, as shown in panel L of Fig. 1 b). If one removes from the full braid group all impossible singleloop-related trajectories (corresponding to the generators σi that cannot be defined because the cyclotron orbits are too short to match the particles), one observes that the remaining braids with additional loops, i.e., multiloop, fit to particles in 2D that are separated by a greater distance than the singleloop cyclotron orbit can reach [8]. These braids are related to multiloop cyclotron orbits with p loops (i.e., p A = NS , where p is an odd integer) [8]. This property follows from the fact that multiloop cyclotron orbits on the plane must be larger than singleloop orbits for the same magnetic field. The external magnetic field flux passing through the multiloop 2D cyclotron orbit is the same as that passing through the singleloop 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 panel R of Fig. 1. Panel R of Fig. 1 (left) shows the cyclotron orbit for magnetic field B as accommodated to the quantum of the magnetic field flux (i.e., B A = hc ). The cyclotron orbit A coincides with NS (where S is e the sample area and N is the number of particles) in the case of the completely filled LLL (Lowest Landau Level). For a field that is larger by a factor of p (i.e., p B), the cyclotron orbit accommodated again to the flux quantum is smaller than the interparticle separation NS , as illustrated in the center graph in panel R of Fig. 1 for p = 3. If p-looped orbits are considered, then in 2D space the external flux p B A must be shared between p loops within the same surface A (i.e., B A for each loop). Thus, each loop accommodated to the flux quantum hc has an orbit with the surface A, resulting e in a total flux of p B A per particle, as illustrated in panel R of Fig. 1 (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 singleloop orbits do not fit. This property, attributed exclusively to the exact 2D topology, provides an explanation for the FQHE and related exotic Laughlin correlations [8]. 2. The FQHE in monolayer graphene When the magnetic field is sufficiently strong that ν ∈ (0, 1), one encounters fractionally filled 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; two lower subbands n = 0, 1 ↑ (↓) are completely filled with the valence holes). The degeneracy of each subband is

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Fig. 1. Schematic illustration of the commensurability (L panel) of a cyclotron orbit with interparticle separation: a) ideal fitting, b) overly short cyclotron radius, such that particles cannot be matched, c) overly large cyclotron radius, such that interparticle distances cannot be conserved. Schematic illustration of cyclotron orbit enhancement in 2D caused by the multiloop trajectory structure (R panel) (third dimension added for visual clarity).

N0 =

BS hc /e

ν ∈ (0, 1). The cyclotron orbits in the LLL are accommodated to the bare kinetic energy T = h¯ ωc (n + 12 ) eB with n = 0, where ωc = mc . The cyclotron orbits restrict the topoland N < N 0 for

ogy of all trajectories uniformly in 2D accommodated in size to pure kinetic energy (average velocity) of particles 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, despite the ‘relativistic’ LL energies caused by the crystal field. The difference between conventional 2DEG systems and graphene resolves in the regard of commensurability of cyclotron orbits and particle separation to a double of LL subband spin–valley structure in graphene compared with the conventional semiconductor 2DEG and to the Berry-phase-induced shift in the LL fillings. hc /e The cyclotron orbit size in the n = 0, 2 ↑ subband is B = NS 0 where S is the sample surface and N 0 is the subband degeneracy. Thus, the multiloop braid structure is necessary because this orbit size is smaller than the interparticle separation NS (be-

cause N < N 0 ). From the commensurability condition, q NS = N N0

1 q

0

S , N

one finds that ν = = (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 − 1q . As in the case of the ordinary 2DEG, one can generalize this simple series by assuming that the last loop of the multiloop 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(q−1l )±1 , ν = 1 − l(q−1l )±1 , where l = 1, 2, . . . (or is equal to another fractional filling factor) 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 l → ∞ in the above formula (i.e., zero flux taken away by the last loop, similar to ordinary fermions in the absence of a magnetic field, which is the case in the Hall metal archetype for ν = 12 in the conventional 2DEG), which provides the hierarchy for the 1 1 Hall metal states: ν = q− , ν = 1 − q− . To account for the Berry 1 1 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 NS is still lower than the interparticle separation N −SN 0 0 (because N − N 0 < N 0 ). As a result, the multiloop structure is repeated from the previous subband, resulting in the same hierarchy except for a shift ahead by 1. Let us now consider the filling of the next LL with n = 1. This LL also has four subbands, but in this level, the bare kinetic en= N3S . For ergy is 3h¯2ω B and the related cyclotron orbit size is 3hc eB 0

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J. Jacak, L. Jacak / Physics Letters A 379 (2015) 2130–2134

N ∈ (2N 0 , 3N 0 ], we address gradual filling of the n = 1, 1 ↑ subband. When only a small number of electrons fills this subband, one encounters the multiloop structure (corresponding to the in3S equality N < N −S2N 0 ). Thus, q N3S0 = N −S2N 0 (where q is an odd 0 integer) provides the main series for the FQHE (multiloop) in this 1 subband in the form ν = 2 + 3q . Similar to the earlier case, the complete hierarchy reads ν = 2 + 3l(q−l1)±1 , ν = 4 − 3l(q−l1)±1 (for subband holes), with the Hall metal hierarchy appearing in the limit of l → ∞. These series are pushed closer to the subband edges, whereas other commensurability conditions are possible in 3S the central part of the subband. When N = N −xS2N and x = 1, 2, 3, 0

0

we obtain ν = 73 , 83 , 3, respectively, corresponding to singleloop cyclotron orbits, which is similar to the case of the IQHE. Thus, for ν = 73 , 83 , one obtains a new Hall feature manifesting itself only in higher LLs (n > 0), where cyclotron orbits may be larger than interparticle separation distances and may fit to every second or every third particles. These correlated states are referred to as the FQHE (singleloop). The quantization of the transverse resistance R xy related to these fractional filling ratios of higher LLs, h

ν=

hc , eB

is the

same as for the FQHE, e2 ν , but the correlations of Laughlin type involve the exponent p = 1 in the Jastrow polynomial, given by 1DUR of singleloop braid exchanges as in the IQHE. The number of these new fractional filling ratios grows as 2n with the LL number n. 3S In the special case of N = N1−.5S , one obtains ν = 52 for the 2N 0 0 correlated state with the paired particles (pairing does not change the cyclotron radius but reduces the carrier number by a facN −2N 0 tor of two (i.e., ), which gives commensurability for pairs 2 3S = N −3S2N 0 ). The following subbands are filled with electrons unN0 der a similar scheme. For the n = 1, 1 ↓ subband, the cyclotron 3S size is N and the interparticle distances are measured with the 0

plaque dition

S , where N ∈ (3N 0 , 4N 0 ]. The commensurability conN −3N 0 q3S = N −S3N 0 results in the main series for the FQHE (mulN0 1 (i.e., = 3 + 3q ), from which the full hierarchy can be

ν tiloop) developed in a similar manner as described above. The condition 3S = N −xS3N with x = 1, 2, 3 results in fractions with single-loop N 0

0

correlations of the FQHE (singleloop) type for ν = 10 , 11 and the 3 3 IQHE for ν = 4, respectively, whereas a paired state can be realized at ν = 72 . 3. The FQHE in bilayer graphene

Bilayer graphene is not strictly two-dimensional, which changes the topological situation considerably. Two sheets of the graphene plane lie in close vicinity and electrons can jump between planes— a hopping constant 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 fulfill 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 doubleloop cyclotron orbits may have the same size in bilayer graphene as the singleloop 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 field that passes through such a doubleloop orbit is twice as large as the flux through a singleloop orbit. Each loop has in this case a separate individual surface—in contrast to the doubleloop orbit located on the purely 2D plane. Considering that a multiloop orbit in bilayer graphene may be partially located in each 2D sheet, the contribution of the one loop must be avoided, whereas

the remaining loops must share the same flux as that passing through a singleloop 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 singleloop 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) ehcB = ( p − 1) N2S = N −SN for ehcB = NS < N −SN , resulting in 0

0

0

0

1 the main line hierarchy ν = NN = p − = 12 , 14 , 16 , . . . , where N is 1 0 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 2D 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 multiloop orbits, the total number of loops is still p— thus, the generators of the corresponding cyclotron subgroup are ( p) p of the form b i = σi , resulting in the Laughlin correlations with the p exponent for the Jastrow polynomial. However, because of the distinct commensurability of orbits with interparticle separa1 tion in bilayer graphene, the related filling fractions are ν = p − 1 in the first particle-type subband of the 8-fold 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 − 12 for the va-

lence band hole mirror fraction with respect to 12 for electrons in the conduction band. For holes in the subband (these holes are not holes from the valence band but rather correspond to unfilled states in the nearly 1 filled particle-type subband), one can write ν = 1 − p − , whereas 1 generalization to the full hierarchy of the FQHE in this subband takes the form ν = l( p −l2)±1 , ν = 1 − l( p −l2)±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 3hc = N3S . The FQHE main series in the first of these subeB 0

bands of the LLL, n = 1, 2 ↑, has the form 1 ,2 6

1 ,4 12

1 ,.... 18

ν=

N N0

=2+

1 3( p −1)

=

2+ + + The generalization of this main series to the full FQHE hierarchy in the subband is as follows: for subband holes (i.e., for an incompletely filled electron subband), ν = 3 − 3( p1−1) ; for the full FQHE hierarchy in this subband,

ν =2+

l , 3l( p −2)±1

ν =3−

l 3l( p −2)±1

(with Hall metal hierarchy in the limit of l → ∞). Nevertheless, a new commensurability opportunity occurs in the n = 1, 2 ↑ subband of the LLL: N3 = N −x2N for x = 1, 2, 3, 0

which yields filling ratios

0

ν = 73 , 83 , 3, respectively. These ratios are

J. Jacak, L. Jacak / Physics Letters A 379 (2015) 2130–2134

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Table 1 FQHE hierarchy in monolayer graphene for the first particle subband in each of the first three LLs (n = 0, 1, 2), as determined by the commensurability condition. LL sb.

FQHE (singleloop), paired, IQHE

FQHE (multiloop) (q-odd, l = 1, 2, 3, . . .)

Hall metal

0, 2 ↑

1

1 l , , q l(q−1)±1 1 1 − q , 1 − l(q−1l )±1

1 , q−1

1, 1 ↑

7 8 , , 3 3

2, 1 ↑

31 32 33 34 , 5 , 5 , 5 , 5 ( 13 pair), 6, 7 2

( 52 pair), 2, 3

2+ 3− 6+ 7−

1 , 3q 1 , 3q

2+

1 , 5q 1 , 5q

6+

3−

7−

1 1 − q− 1

l 3l(q−1)±1 l 3l(q−1)±1

2 + 3(q1−1) , 3 − 3(q1−1)

l , 5l(q−1)±1 l 5l(q−1)±1

6 + 5(q1−1) , 7 − 5(q1−1)

Table 2 Hierarchy of the FQHE in bilayer graphene for the first particle subband in each of the first two LLs (n = 0, 1 for the extra degenerate LLL in bilayer graphene and n = 2 for the first LL above the LLL). LL sb.

FQHE (singleloop), paired, IQHE

FQHE (multiloop) (q-odd, l = 1, 2, 3, . . .)

0, 2 ↑

0, 1

(q−1) , l(q−2)±1 ,

1

l

1

1 − (q−1) , 1 − 1, 2 ↑

7 8 , , 3 3

( 52

pair), 2, 3

2+ 3−

2, 1 ↑

21 22 23 24 , 5 , 5 , 5 , 5

( 92 pair), 4, 5

Hall metal

1 , 3(q−1) 1 , 3(q−1)

1 , q−2

1 1 − q− 2

l l(q−2)±1

2+ 3−

l

3l(q−2)±1 l 3l(q−2)±1

,

4 + 5(q1−1) , 4 + 5l(q−l2)±1 , 5 − 5(q1−1) , 5 − 5l(q−l2)±1

2+ 3−

1 , 3(q−2) 1 3(q−2)

4 + 5(q1−2) , 5 − 5(q1−2)

related to singleloop cyclotron trajectories and thus with correlations similar to the case of the IQHE (though they do not involve integer filling, except for ν = 3). Moreover, for x = 1.5, one can consider the twice reduction in particle number ( N − 2N 0 )/2 to arise from the pairing, which provides perfect commensurability of cyclotron orbits of pairs with the separation of particle pairs occurring at ν = 52 . The last n = 1, 2 ↓ subband in the LLL in bilayer graphene is filled with electrons in a similar manner as in antecedent 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 previous subband without any other modification. However, the situation changes in the next LL (the first one above the LLL). In the 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 5hc = N5S0 . An analysis simeB ilar to that of the previous LL provides the main series and the full hierarchy of the FQHE (multiloop) in the n = 2, 1 ↑ subband: ν = 4 + 5( p1−1) , ν = 4 + 5l( p−l 2)±1 , respectively. (Inclusion of subband holes is accomplished by the replacement of 4+ by 5− in both expressions.) As before, the limit l → ∞ provides the Hall metal hierarchy. One difference compared with the previous LL is the presence of four (instead of two) satellite FQHE (singleloop) states symmetrically located around the central paired state. In the n = 2, 1 ↑ subband the satellite states occur at ν = 21 , 22 , 23 , 24 5 5 5 5

and with the central paired state at 92 . This hierarchy is repeated in all 4 subbands of the first LL. 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 eight-fold degeneracy of the LLL, i.e., in a doubling of the four-fold 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

Fig. 2. FQHE at T = 0.25 K in bilayer suspended graphene: magneto-resistance R xx (blue curve) and R xy (black curve) at the lateral voltage −27 V (after Ref. [1]). The fitting to the hierarchy as specified in Table 2 is shown in red.

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 [9]. The order inversion of n = 0, 1 to n = 1, 0 affects the filling ratio hierarchy. Assuming that the LLL subband with n = 1 is filled earlier than the n = 0 subband, we obtain the following hierarchy for the first n = 1, 2 ↑ subband: multiloop orbits for ν = 3l( p −l 1)±1 , ν = 1 − 3l( p −l 1)±1 ; singleloop orbits for ν = 13 , 23 and paired state for ν = 12 (along with Hall metal behavior in the limit of l → ∞). Taking the case of the next subband, corresponding to n = 0, 2, ↑, we obtain the hierarchy of filling in the following form: multiloop orbits for ν = 1 + l( p −l1)±1 ,

ν = 2 − l( p−l1)±1 and no singleloop orbits. 4. Comparison with experiment

In suspended ultrasmall graphene scrapings it has been possible to observe initially the FQHE at net filling ν = 1/3 and −1/3 [10]. Recent experimental progress has also enabled observation of the FQHE in graphene fabricated atop a crystal substrate of boron nitride (BN) under large magnetic fields of order of 40 T [4] (remarkably, in this study, FQHE features were observed at values of up to ν = 4). Experiments on monolayer graphene on BN substrates [4,5] and in the form of suspended small sheets [2,3] have enabled observation of Hall features at fractional fillings 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 deviates from this picture [2,3,5]. 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 modeled by auxiliary field flux quanta attached to CFs. However, the usefulness of the CF model breaks down in higher LLs because starting from the first LL the multiloop 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., { 73 , 83 }, { 10 , 11 }, 3 3

{ 13 , 14 }, { 16 , 17 }) that correspond to the singleloop commensu3 3 3 3 rability condition. These factors are not addressed in CF modeling but are sharply visible (better than other FQHE states [5]) in experiments as FQHE (singleloop) [2,3,5,4]. The number of centrally located filling rates for the FQHE (singleloop) grows subsequently with the LL number as 2n [4,2,3]. The commensurability condition for bilayer graphene reproduces the observed experimentally hierarchy [1] perfectly, as shown in Fig. 2. Note, that the characteristic even denominators fractional states in bilayer graphene were observed also in conventional 2DEG Hall systems, the ν = 12 state has been discovered there [11].

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In conclusion, 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 identified, leading to the new type FQHE correlated states beyond LLL. They are referred as to FQHE (singleloop) correlated states because the related correlations are described by singleloop braids. The even-denominator main line of the FQHE hierarchy in bilayer graphene is found in agreement with experimental observations, including that corresponding to the most pronounced correlated FQHE state in bilayer graphene at ν = −1/2. Acknowledgement Supported by NCN P.2011/02/A/ST3/00116. References [1] D.K. Ki, V.I. Falko, D.A. Abanin, A. Morpurgo, Observation of even denominator fractional quantum Hall effect in suspended bilayer graphene, Nano Lett. 14 (2014) 2135.

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