Explanation of an unexpected occurrence of ν=±12 fractional quantum Hall effect states in monolayer graphene

Recent experiment reveals the appearance of incompressible fractional quantum Hall effect states in monolayer graphene at and , substituting the compressible Hall metal states at these fillings in the lowest Landau level in a narrow magnetic field window depending on the sample parameters. Simultaneously, none such behavior has been observed either for or , . We propose an explanation of these observations in terms of homotopy of monolayer graphene in consistence with a general theory of correlated states in planar Hall systems.


Introduction
In the conventional two-dimensional (2D) Hall material, GaAs 2D electron system (2DES), at low temperatures (in mK scale) and at high magnetic field (of order of 10 T) the fractional quantum Hall effect (FQHE) [1,2] is observed in great detail revealing the mysterious discrete hierarchy of Landau level fillings. This hierarchy was the subject of intensive theoretical studies and the consistence with experiments has been achieved within the homotopy cyclotron braid commensurability approach [3] which generalized the former composite fermion (CF) model [4]. The latter well predicted the main line of the FQHE hierarchy in the lowest Landau level (LLL) of GaAs except for the so-called enigmatic states [5]. Inclusion of nesting of multiloop cyclotron orbits with next-nearest neighbors in multiparticle correlations allowed, however, for explanation also of enigmatic states beyond the CF model restricted to only nearest neighbor correlations. Both, the CF model and the homotopy model, predicted the compressible multiparticle states of Hall metal type at ν = 1 2 , 1 4 in GaAs, which agrees with experimental observations in GaAs [2].
Nevertheless, in the recent paper [6] it has been reported a puzzling appearance of incompressible FQHE states at ν = ± 1 2 and ± 1 4 , ± 3 4 in the first subband of the LLL in graphene monolayer at low temperatures (∼300 mK), whereas neither at ν = ± 3 2 nor at ± 5 4 , 7 4 . The even denominator FQHE states in the first subband of the LLL in monolayer graphene occurred only within some relatively narrow magnetic field windows (of a few T width around 28 T or 21 T, depending on a sample), i.e. they disappear beyond and beneath of this window borders-see. Figures 1 and 2. This behavior is astonishing because in the LLL, both in spin-up and spindown its subbands of GaAs 2DES, the Hall metal phase is observed at ν = 1 2 , 1 4 , 3 4 , 3 2 , 5 4 , 7 4 , which agrees with the conventional predictions of hierarchy for FQHE in the LLL [3,7] These unusual states at fractions with denominators 2 and 4 in the first subband of the LLL in monolayer graphene do not find any explanation in the framework of a conventional theory of FQHE in the LLL of monolayer graphene [6,8]. The authors of the experiment [6] suggest that the occurrence of even denominator FQHE states in the first subband of the LLL (ν ∈ (0, 1)) in monolayer graphene but not in the next subband of the LLL (ν ∈ (1, 2)), is connected with magnetic field induced competition or influence of various other correlated states which has been theoretically identified recently [9] at ν = 0 in monolayer graphene exposed to the magnetic fieldas presented in figure 3. This competition of magnetic-spin correlated phases mixed with valley pseudospin structure in monolayer graphene leads to phase diagram for ν = 0 as shown in figure 3(d). By comparison of the positions of magn etic field windows at which the state ν = ± 1 2 occurred (in various samples) versus sublattice splitting energy ∆ AB -see. Figure 3, one can notice that the so-called valley-coherent partially sublattice polarized phase (PSP) competing with the canted antiferromagnet (CAF) might have something in common with the appearance of even denominator FQH phases closer to ν = 0 (in (0, 1) range) but not in the more distant region (1,2). None arguments have been, however, drawn out to support the existence of even denominator FQH states in the first subband of the LLL in monolayer graphene from point of view of the conventional hierarchy of FQHE in this subband.
In the present letter we demonstrate, however, that incompressible states at ν = 1 2 , 1 4 can occur in the hierarchy for FQHE beyond the CF model. The later model precludes (a) False color plot of penetration field capacitance C P /c in sample A as a function of the magnetic field, B, and the nominal charge density, c is the average geometric capacitance of the two gates [6]. Fractional quantum Hall states appear as lines of high C P with a slope proportional to their quantized Hall conductivity. The dataset spans filling factors ν ∈ (−2, 2), encompassing the zero energy Landau level. (b) Traces taken at constant B = 28.3 T between filling factor ν = 0 and ν = 2 filling (indicated by black arrows in (a). C P is plotted as a function of relative filling factor ν = ν − ν, revealing a robust incompressible state at half-filling of the ν = [0, 1] LL (black line) and a fully compressible state at half filling of the ν = [1,2] LL (red line). Reproduced from [6] copyright © 2018, Springer Nature. With permission of Springer. FQHE at ν = 1 2 or 1 4 [7], whereas the homotopy theory of FQHE [3] offers the energy competition of various commensurability patterns at the same ν including nesting of multiloop cyclotron braids with nearest and next-nearest neighbors in Wigner-type web of electrons induced by Coulomb repulsion of electrons. Some homotopy patterns for multiparticle correlations in planar system in the presence of magnetic field result also in correlated incompressible FQHE states at ν = 1 2 , 1 4 besides Hall metal states at these fillings, and the realization of the compressible Hall metal phase is the matter of the energy competition between various possible multiparticle phase organizations. This energy competition is conditioned by the particular shape of multi-particle wave functions corre sponding to different homotopy patterns of correlations. The multiparticle wave functions for the homotopy patterns in GaAs and in graphene monolayer are different and this is probably the source of reported in [6] oddness with states at ν = ± 1 2 and ± 1 4 , ± 3 4 in graphene monolayer in comparison to GaAs.
The paper is organized as follows, in the following paragraph, the main points of the homotopy cyclotron braid group approach to FQHE hierarchy are outlined in comparison to the conventional CF model. Next, the method of construction of trial multiparticle wave functions for various correlation homotopy patters is described for both materials, GaAs and monolayer graphene. Next, the discussion of the states at ν = 1 2 , 1 4 in graphene monolayer is addressed to elucidate observations reported in the paper [6]. Some particular explanation is shifted to appendix.

Main points of the homotopy approach to FQHE
FQHE is observed at large magnetic field, B, for which the cyclotron orbit size, h eB ( h e is the magnetic field quantum) is smaller than the particle separation, S N (here, S is the planar sample surface, N is the number of electrons) i.e. when, Note that for B 0 at which, we deal with the integer quantum Hall effect (IQHE), because the degeneracy of LLs at the field B equals to, and for B = B 0 it is the same equation as equation (2) taken at N = N 0 , thus ν = N N0 = 1. However, for B 1/3 = 3B 0 (at S and N kept constant) equation (2) does not hold and the inequality (1) might be lifted to the equality by 3-fold enhancement the magnetic field flux quantum, h e → 3h e . An increase of the magnetic field flux quantum is proven for multiloop trajectories (the formal proof goes via the Bohr-Sommerfeld rule [3]), resulting in the magnetic flux quantum, Φ k = (2k+1)h e , for planar braids with additional k loops-see. Appendix. Braids define exchanges of identical indistinguishable particles in the classical multiparticle configuration space and must nest to at least neighboring particles uniformly distributed on the plane due to the Coulomb repulsion. Shorter braids cannot be defined. For planar electron systems exposed to the perpendicular magnetic field, braids are of finite range because of cyclotron effect, i.e. are of size, Φ k B . Ordinary braids (without any loops, i.e. with k = 0 for corresponding magnetic field flux quantum, Φ 0 = h e ) are too short to reach neighbors at B 1/3 , but in this case the larger braids with an additional loop, i.e. for k = 1, of size Φ1 B 1/3 , the braids with one additional loop perfectly fit to particle separation, which gives, corresponding to Laughlin wave-function hierarchy [10])see. Figure 4 for illustration.
The above presented cyclotron braid commensurability scheme can be generalized by inclusion of the nesting of braids also with next-nearest neighbors and for each loop of the multiloop braid separately, which leads to the commensurability condition [3,11], where q = 2k + 1 is the number of cyclotron loops (k is the number of loops in the braid), x i 1 denotes the order of next-nearest neighbors commensurate with ith loop of the multiloop orbit (x i = 1 denotes the nearest neighbor commensurability of ith loop), ± indicates possible congruent, +, or inverted, −, orientation of the subsequent loop with respect to the preceding one, (i.e. of the eight-digit shape orbit in the case of −, or, in other words, inverse to the external field flux). Due to energy preference discussed in [3,11] one can simplify (for GaAs 2DES case) the above condition assuming . . x q−1 = x and x q = y , and express it in the following form, which defines the hierarchy of FQHE in the LLL (when N 0 is given by equation (3)), with q = 2k + 1 odd integer and y x 1 positive integers 1, 2, . . .. The hierarchy (8) reproduces all experimentally observed FQHE filling ratios in GaAs in the LLL, including enigmatic states, ν = 4 11 , 5 13 , 3 8 , 3 10 , . . . [2,3]. The enigmatic states filling ratios are included to the hierarchy (8), but at x > 1 [3,12].
The former theory of the FQHE hierarchy in GaAs Hall system is the CF model basing on the heuristic assumption of auxiliary fictitious field flux quanta to be pinned somehow to electrons. Such complexes were called CFs [4] and the hierarchy of FQHE has been modeled by the IQHE in higher LLs completely filled in the magnetic field reduced by the averaged field of the auxiliary fluxes attached to CFs. The CF hierarchy is given by the formula [4], ν = 1 (q−1)y±1 , where q = 2k + 1 with k denoting number of flux quanta pinned to CFs, y = 1, 2, 3 . . .-positive integer, and ± indicates the orientation of the resultant magnetic field (the external magnetic field B reduced by averaged field of CF fluxes) conformal (+) or opposite (−) with respect to the external B field orientation. One can notice that the formula (8) reproduces the CF hierarchy at x = 1. Thus, the CF model is an effective illustration of multiloop cyclotron orbits and each flux quantum of the auxiliary field pinned to the CF indicates a sole loop in multiloop structure described above. The CF hierarchy agrees with the hierarchy (8) at x = 1, which means that the model construction of CFs cannot account for all types of homotopy patterns when the nesting with next-nearest neighbors with x > 1 of q − 1 first loops contribute to the commensurability condition (7). CFs neglect thus the possibility of a more complicated homotopy braid commensurability patters involving next-nearest neighbors.
The phase shift typical for the Laughlin function of q = (2k + 1)th order, i.e. e iqπ agrees with the scalar unitary representation of the cyclotron braid subgroup generated by multiloop braids σ q i where σ i , i = 1, . . . , N − 1 is the elementary braid defining the exchange of ith and i + 1th particle [13,14]. This unitary representation of the cyclotron subgroup is the projected unitary representation of the full braid group, σ i → e iπ (suitably to original electrons-fermions), i.e. σ q i → e iqπ . The homotopy cyclotron group approach is free of heuristic artificial assumptions of the CF concept in which the required Laughlin phase shift is produced via the Aharanov-Bohm-type effect.
The definition of the cyclotron braid generators for more complicated instances of the homotopy braid commensurability (i.e. with x > 1) as given by (7), together with the corresponding scalar unitary representations unequivocally define the polynomial part of the multiparticle eigen-functions for various homotopy classes. The uniqueness of this procedure results from the condition that the multiparticle wave function of interacting electrons in the LLL must be a holomorphic function unequivocally defined by its nodes, i.e. must be a regular polynomial multiplied by the exponential (thus non corresponding braid for exchange of particle blue one with red one-the braid trajectory is built of half-piece of the cyclotron orbit. (C) Schematic presentation of thee-loop cyclotron orbit (left) and corresponding braid for exchange of particle blue one with red onethe braid is built of half-piece of the cyclotron orbit, in this case, with one additional loop (the third dimension added for better visibility). For more detail of linkage of cyclotron orbits and braids see [15]. singular) factor of N-fold product of single-particle exponents e −|zi| 2 /4l 2 B (z i is the complex representation of ith particle position on the plane and l B = eB is the magnetic length). The form of exponential factor is the same as in the gas system and is maintained for an arbitrary state when interparticle interaction is switched-on (like for GaAs). In graphene the single-particle Landau states are, however, modified by graphene crystal field, which affect the envelope multiparticle factor besides the polynomial part, the same as in GaAs and conditioned by the selected cyclotron braid group acc. to the commensurability pattern. Scalar unitary representations of the cyclotron braid groups defining multiparticle correlations in corresponding homotopy phases in the LLL determine the shape of the related multiparticle holomorphic wave function (the polynomial part of the multiparticle wave function) in an unambiguous manner, as illustrated in [3]. This procedure is exact in contrary to the so-called projection on the LLL of higher LL wave functions in order to remove singularities in the CF model (being not unambiguously defined [7]). The envelope of the multiparticle wave function, invariant with respect to any braid exchanges of particles has the form e − N i=1 |xi| 2 /4L 2 B , but only in GaAs case (when rthe Hilbert space of the N-particle system is spanned by antisymmetrized products of single-particle gaseous Landau functions (at symmetic gauge)).
The hierarchy of Hall-metal states can be determined by the limit y → ∞ of the FQHE general hierarchy (8),

Explanation of even denominators in FQHE hierarchy in the LLL of monolayer graphene by homotopy-patterns
In graphene the crystal structure with two carbon atoms per the elementary cell leads to spin-valley four-fold structure of LLs in magnetic field [8]. The crystal field in graphene monolayer influences ordinary Landau quantization and instead of the single-particle Landau dispersion, E n = ω B (n + 1 2 ) with ω B = eB m , as in the electron gas, the LL dispersion in mono layer graphene is, E n = ω √ n with ω = 2vF lB , (v F is the Fermi velocity) [8]. This difference in the LL energy is caused by the relativistic-type electron spectrum near Dirac cones in corners of the hexagonal Brillouin zone of the graphene monolayer. The crystal field does not change, however, the homotopy of braid trajectories. The latter is governed in the presence of the perpendicular magnetic field by the commensurability of cyclotron orbits of bare electrons on the plane with Wignertype electron crystal lattice. The bare kinetical energy core of electrons at magnetic field presence in graphene monolayer is the same as in the gas, ω B (n + 1 2 ), and only due to the crystal field (electric-type interaction of electrons with the graphene crystal lattice) the dressed energy attains the form ω √ n. The homotopy classes are robust against the crystal field and the latter does not modify the FQHE hierarchy resulted from the cyclotron braid commensurability restrictions, the same as in 2DES. In the case od 2DES, the Wigner crystal is the triangle planar lattice-the classical lowest energy distribution of electrons on jellium at T = 0 K (when classical kinetical enery vanishes). The self-energies of particular homotopy classes depend, however, of the crystal field dressing electrons in graphene in different manner than in GaAs. Thus despite the same FQHE hierarchy the experimental its manifestation may differ in various materials due to energy competition between various homotopy phases admissible at the same filling rate. The self-energy depends on the shape of the multiparticle wave function defined in the LLL by the same symmetry (scalar unitary representations of particular cyclotron braid subgroups), however, in different subspace of multiparticle Hilbert space because of different envelope part of wave functions. The subspace spanned by combinations of single-particle Landau wave functions are different in graphene than in GaAs.
To account this effect qualitatively let us invoke the study of electron distribution in graphene monolayer at ν = 0 with respect to the magnetic field [9], which reveals a competition of various phases including spin and valley-pseudospin. Of particular interest is the charge density wave type distribution just in the similar window for the magnetic field as that one for which the astonishing even denominator FQHE states occur [6]-as imagined in figure 3. When the electron density wave with periodicity scale of order of elementary cell of graphene interferes with the ideal Wigner crystal distribution with larger spatial scale governed by the magnetic length, the correlations of next nearest neighbors may be energetically favored, at least close to ν = 0. Preferring by the wave function of some kind of correlation by commensurate density concentration resolves itself to the increase of number of loops linking favored next-nearest neighbors, because this enhances multiplicity of zeros (as in Laughlin function) for  (6), giving the same ν will not result finally in the lower energy the larger amount of x i = 1 occurs, as it was the case in GaAs, but inversely, for as most as possibly x i > 1 occur corresponding to the nextnearest neighbors. If the influence of the charge density wave stable at ν = 0 weakens for more distant ν , the privilege of next nearest-neighbors also diminishes. It will happen at lower magnetic field (as in the next subband of the LLL in graphene monolayer) where the Wigner crystal concerns only N/2 of electrons and the magnetic length (∼ 1 √ B ) grows resulting in the escape from the interference with of much smaller spatial scale charge density wave stable at ν = 0.
The scenario described above agrees with the experimental observations [6]. The Hall metal stable in the LLL of GaAs at ν = 1 2 , 1 4 (and 3 4 for LL band holes) and corresponding to the limit y → ∞ of the condition (8) but with x 1 = ... = x q−1 = x = 1 (nearest neighbors) is substituted in graphene by another homotopy patterns corresponding to ν = 1 2 , 1 4 with as much as possible loops commensurate with next-nearest neighbors. Especially frequently 1 2 and 1 4 occur as filling ratios for homotopy patterns with all finite x i (including y = x q ) for hierarchy family (q = 3, 5, 7), with x i 2 for i > 1 (equation (10) is equation (6) with equation (3) with ± incorporated in x q ). This situation is exemplified in table 1 and illustrated in figure 5. Note that homotopy patterns with several x i > 1 do not allow for the CF picture which is restricted only to x = 1 in hierarchy (8). These states are, however, incompressible (with stiffly fixed surface S required for commensurability condition) FQHE states as observed in the experiment [6]. This behavior is linked with the property that for the same filling ratio ν various homotopy patters (denoted by ( x i , i = 1, . . . , q)) are possible in general, and the eigenenergy decides which pattern is stable. The lower energy assigns the ground state at particular ν . The Hall metal state at ν = ± 1 2 or at ν = ± 1 4 , ± 3 4 (± reflects here the mirrored particle-hole states in band structure of the monolayer graphene, symmetric with respect to the zeroth energy at the center of the LLL four-fold degenerated in spin-valley degrees of freedom [8]) have the lower energy than other available (according to equation (10)) homotopy patters, unless these Hall states are overcome in energy gain by homotopy patterns privileging next-nearest neighbors. In the window for magnetic field as indicated in figures 1 and 2 such states are incompressible FQHE state with several x i > 1, as shown in table 1. Simultaneously, other fractional states are slightly changed in stability because the homotopy patterns with several x i > 1 substitute patters with x = 1 at the same ν (as in table 1 shown for ν = 1 3 , 2 5 , see figure 6). This small decrease of the stability of these states is also noticeable in the experiment [6]. All states corresponding to homotopy patterns with several x i > 1 are not of CF type.  table 1; the first pattern with the signature, x 1 = 1, x 2 = 1, x 3 = 1, corresponds to CF-type FQHE state at filling rate ν = 1 3 -commensurabilities of all three loops concern nearest neigboring electrons; the second pattern with the signature, x 1 = 1, x 2 = 1, x 3 = 2, corresponds also to CF-type FQHE state at filling rate ν = 2 5 -commensurabilities of two loops concern nearest neighbors, whereas of the third one concerns nextnearest neighbors of order 2; the third pattern with the signature, x 1 = 1, x 2 = 2, x 3 = 2, does not correspond to CF-type FQHE state at filling rate ν = 1 2 , that one which is observed in experiment [6]-commensurability of only one loop concerns nearest neighbors, whereas of the remaining two loops concern next-nearest neighbors of order 2. (B) Schematic visualisation of the commensurability of Wignertype electron distribution in graphene monolayer (the triangle Wigner 2D lattice of electrons indicated as blue balls)-three situations are shown, the first one is the sigleloop cyclotron orbit commensurability with nearest neighbors, which corresponds to IQHE at field B 0 , the second one concerns three-times larger field 3B 0 at which sigleloop orbit is too short to match nearest neigbors in the Wigner lattice-but the three-loop orbit perfectly fits to nearest neighbors, which corresponds to FQHE at ν = 1 3 , and the third case, which concerns the field 2B 0 at which singleloop orbit is also too short in comparison to electron separation, but the homotopy pattern with x 1 = 1, x 2 = 2, x 3 = 2 (see table 1) allows the braid commensurability with electron distribution including next-nearest neighbors for two loops-this state is of incompressible FQHE type at ν = 1 2 , beyond the CF model.
For lower magnetic field, when the magnetic length l B = eB grows, the modification of the Wigner crystal accommodated in the second subband of the LLL to the half of the electron amount, N/2, is of lower significance as the interference with charge density wave diminishes. In this range of ν the nearest neighbors (x = 1) are again favored and the CF-type homotopy patterns energetically prevail in energy over next-nearest neighbor nesting. Moreover, it is reasonably to expect that the charge density wave trace in the multiparticle wave functions in the range ν ∈ (1, 2) is weaker than in the region ν ∈ (0, 1) due to larger departure from ν = 0 at which the charge density wave is stable. The window with partially sublattice polarized (PSP) corresponding to the stable charge density wave does not overlap with smaller magn etic field corresponding to the second subband of the LLL in graphene monolayer [9,16] and CF-type FQHE hierarchy with x = 1 is restored in ν ∈ (1, 2) sector, which agrees with the observations [6].

Conclusions
It has been demonstrated that the FQHE incompressible states in the LLL can occur also at filling fractions with even denominators, including 2 and 4, as the commensurability condition admits such filling fraction though not of CF type. Because of the energy competition between various homotopy phases at the same filling fraction the occurrence in the experiment of particular hierarchy depends on the shape of multiparticle wave functions corresponding to symmetry imposed by the commensurability patterns including various divisions of correlation among nearest and next-nearest neighboring electrons and of material band structure, different for graphene in comparison to GaAs 2DES. The stability of CDW-type electron distribution (valley-coherent partially sublattice polarized phase) in monolayer graphene at ν = 0 suggests the energy preference of next-nearest correlations at similar magnetic field window in nearby ν = ± 1 2 , ± 1 4 , allowing the stability of incompressible FQHE states of not CF-type at these filling rates as experimentally observed in this material.

Appendix. Multi-loop cyclotron braids have larger size than single-loop ones-quantum of the magn etic field flux is not universally defined
Here we prove that the magnetic field flux quantum changes its value in various homotopy phases. In particular, for multiloop cyclotron braid orbits the larger field flux quantum defines larger dimension of the orbit, and it can reach particles too distant for single-loop orbits. This fact is the origin of the FQHE.
Let us consider the Bohr-Sommerfeld rule, which links the area of the 1D phase space ranged by the classical phase trajectory loop with the corresponding number of quantum states. The quasiclassical wave function in a 1D well, U(x), with turning points a and b has the form, 3 are shown, the first one is of CF-type (i.e. with x 1 = x 2 = x 3 = 1, see table 1), whereas the second pattern is not of CF-type ( x 1 = 1, x 2 = x 3 = x 4 = x 5 = 2, see table 1). The latter prefers next-nearest neighbors as in [6]. (B) Schematic visualisation of the commensurability of Wigner-type electron distribution in graphene monolayer-two different homotopy patterns for ν = 2 5 are shown, the first one is of CF-type (i.e. with x 1 = x 2 = 1, x 3 = 2), whereas the second pattern is not of CF-type ( x 1 = x 2 = 1, x 3 = x 4 = 2, x 5 = −2, see table 1, minus indicates the opposite orientation of the loop with respect to preceding one, marked by opposite arrow in the figure). The latter prefers next-nearest neighbors as in [6]. Various homotopy patterns for the same ν correspond to states with different activation energy. If envelope function of multiparticle states favors next-nearest neighbor correlations, then not CF states at ν = 1 3 , 2 5 are more stable than competitors of CF-type at these filling rates.
where p(x) = 2m(E − U(x)) (for simplicity, assuming vertical infinite borders of the well). Uniqueness of the wave function requires, which is the Bohr-Sommerfeld quantization rule (h is Planck constant, n is an integer; for non-vertical infinite borders, S xp = (n + 1 2 )h [17]). The above has been derived upon the condition that the trajectory is single-loop. For a different homotopy class and for a multi-loop trajectory one obtains, however, This is of particular importance when the Bohr-Sommerfeld rule is applied to an effective 1D phase-space (Y, P y ) of x, y components of the 2D kinematic momentum in the presence of a perpendicular magnetic field. The kinematic momentum components (at the Landau gauge, A = (0, Bx, 0)), do not commute, The pair of operators, Y = 1 eB P x and P y , can be treated as operators of canonically conjugated generalized position Y and momentum P y , because [Y, P y ] − = i . Thus, the 1D effective phase space, (Y,P y ), is actually the 2D space, (P x , P y ). The latter 2D kinematic momentum space is, on the other hand, the ordinary 2D space (x, y) renormalized by the factor 1 (eB) 2 and turned in plane by π/2, which is noticeable due to the Lorentz force, F = dP dt = e dr dt × B, which gives dP x = eBdy and dP y = −eBdx.
In 2D position space, trajectories (x, y) may belong to different homotopy classes and may be attributed to noncontractible additional loops (as in charged multiparticle planar systems at sufficiently strong magnetic field). Hence, in this homotopy-rich 2D case, from the generalized Bohr-Sommerfeld rule one obtains, which defines the generalized quantum of magnetic field flux, Φ k = ∆S x,y B = (2k + 1)h e , (A.8) ∆S x,y is the change of S x,y in equation (A.7) when n is changed by 1. Only for k = 0, i.e. for the homotopy class without additional loops, the flux quantum equals to Φ 0 = h e . The generalized magnetic field flux quanta Φ k define different sizes of multi-loop cyclotron orbits, Φ k /B. The IQHE corresponds to k = 0 (the homotopy class of single-loop cyclotron orbits) and the cyclotron orbit size for k = 0 equals to ∆S xy = h eB0 = S N = S N0 , which gives ν = N N0 = 1, ( N 0 = BSe h is the LL degeneracy taken here for B 0 , S is the sample surface size, N is the number of electrons, B 0 is the magnetic field for ν = 1). The FQHE-main line corresponds to k = 1, 2, . . . (the homotopy classes with q = (2k + 1)-loop cyclotron orbits or braids with k additional loops); e.g. for k = 1 (the simplest Laughlin state), the triple-loop cyclotron orbit has the size ∆S xy = 3h eB . This orbit for B = 3B 0 fits to interparticle separation S N -hence, from the commensurability condition, 3h eB = S N , one obtains, ν = N N0 = N BSe/h = 1 3 . Bohr-Sommerfeld quantization applied above to many particle systems is interaction independent, i.e. it holds for arbitrarily strongly interacting multiparticle systems. The sizes of the generalized magnetic flux quanta are thus also interaction independent for different homotopy classes, although the existence of nonhomotopic trajectories in (x, y) space is conditioned by the Coulomb interaction of 2D charged particles (via the cyclotron commensurability condition in Wigner-type crystal of electrons). In a gas system of noninteracting particles their mutual positions are arbitrary, which dismisses any correlations and related homotopies.