**3. IQHE and FQHE in conventional 2DEG, monolayer and bilayer graphene**

Before we proceed to the main aim of this section, it is worth discussing differences in the Landau level ladder between conventional 2DEG, monolayer and bilayer graphene samples (as well as their possible consequences). We first note that in both graphene structures, LLs are not distributed equidistantly on the energy axis, which was the case for typical quantum wells [36–39]. However, the latter is not reflected in the topological approach—an area encircled by a single-loop cyclotron orbit (*A*) is proportional to the bare kinetic energy of electrons (*Ek* ) and not the general energy determined by LLs. Since a crystal field cannot affect the value of *Ek* , the embraced surface of an orbit (representing a *π*1(Ω) element) varies in the same manner for monolayer and bilayer graphene as for conventional 2DEG. Explicitly, *A* is proportional to 2*n* + 1 (where *n* stands for an LL index). At the same time, this dependence carries another consequence—the commensurability conditions from higher LLs are different from those encountered in the lowest band (the single-loop cyclotron orbit for *n* ≠0 embraces exactly (2*n* + 1)*ϕ*0 and not a single quantum as in the LLL case). Thus, we expect that pyramids of fillings for levels indexed by *n* ≠0 are not counterparts of the well-known Jain's hierarchy, which has already been confirmed in experiments [40–42].

1DURs of the resulting subgroups can also be easily estimated (the scalar representation of a

Note that while constructing cyclotron subgroups of this type, one needs to keep an eye on the complexity of generators—highly structured braids are probably unfavorable and near corresponding fillings the Wigner crystal (and not the FQHE) phases may be in favor [29–31]. Filling factors obtained within the non-local theory of Hall systems are gathered in **Table 1**.

others (marked with red color), which cannot be predicted within the quasi-particle formula‐ tion (at least when unclear residual interactions are not implemented) [19, 32–35]. For this reason, the topological approach presented in this chapter is the first one to justify all incom‐ pressible (collective) states from the LLL, which appear in transport measurements [28].

Finally, it is also worth mentioning that the pyramid of fillings, established within the topological approach, is a generalized version of the well-known Jain's hierarchy (obtained when *α* =*q* −1 and *a* =1 are implemented into Eq. 21). Thus, *two artificial flux quanta pinned to electron only model (in a very convenient way) an additional loop, which evidences exactly one ϕ0 per particle and appears in the exchange trajectory (representing a generator of the system braid group)*. The great dependability of the CF theory in the LLL regime can be explained by highlighting the simplicity of cyclotron subgroups generators in the case of Jain's hierarchy—as it was already emphasized, highly structured braids (usually encountered for different configura‐ tions of loops) are probably unfavorable and may not result in pronounced plateaus in Hall

() ( ) ,1 1 *q i <sup>i</sup> b e i N* ® abg

Note that among these fillings one can find all famous and mysterious *v*, like <sup>3</sup>

**3. IQHE and FQHE in conventional 2DEG, monolayer and bilayer**

Before we proceed to the main aim of this section, it is worth discussing differences in the Landau level ladder between conventional 2DEG, monolayer and bilayer graphene samples (as well as their possible consequences). We first note that in both graphene structures, LLs are not distributed equidistantly on the energy axis, which was the case for typical quantum wells [36–39]. However, the latter is not reflected in the topological approach—an area encircled by a single-loop cyclotron orbit (*A*) is proportional to the bare kinetic energy of electrons (*Ek* ) and not the general energy determined by LLs. Since a crystal field cannot affect the value of *Ek* , the embraced surface of an orbit (representing a *π*1(Ω) element) varies in the same manner for monolayer and bilayer graphene as for conventional 2DEG. Explicitly, *A* is proportional to 2*n* + 1 (where *n* stands for an LL index). At the same time, this dependence carries another consequence—the commensurability conditions from higher LLs are different from those encountered in the lowest band (the single-loop cyclotron orbit for *n* ≠0 embraces

resistivity.

**graphene**

 q (Ω)),

<sup>8</sup> , <sup>4</sup> <sup>11</sup> , <sup>4</sup> <sup>13</sup> and

± ± ±¼ << - (22)

subgroup can be identified with reduced representation of the full *π*<sup>1</sup>

14 Recent Advances in Graphene Research

An important property of the Landau level ladder is its degeneracy (as a number of sublevels, rather than a number of one-particle states, in a single level). In typical 2DEG systems, the halfspin (usually marked with ↑ and ↓ ) stands for an exclusive degree of freedom and, thus, all Landau bands are doubly degenerated. In graphene materials, however, the existence of a socalled valley pseudospin (isospin)—due to the presence of identical Dirac cones in two nonequivalent corners of the Brillouin zone (usually marked with *K* ′ and *K*)—results in an additional degeneracy of the energy spectrum. As a result, for an arbitrarily picked range of factors *ν*1, *ν*<sup>2</sup> , the corresponding partially filled Landau level in monolayer and bilayer graphene samples is different than in conventional quantum-wells. We have already men‐ tioned that commensurability conditions (and so the hierarchies of fillings) for distinct *n* are not identical. We may, thus, expect that some incompressible states, which are marked with *v* and occur in one of these structures, may not be permissible for the other one.

It should be emphasized that a nonzero Berry phase causes the monolayer graphene LLL to be placed exactly in the Dirac point—where the valence band meets the conduction band in a gapless energy spectrum [36, 37]. The lowest Landau level is, thus, equally shared between free electrons and free holes (only two, not four, spin-valley branches accessible for one type of particles). Since it is natural to define the filling factor in terms of an electronic density measured from the charge neutrality point, ν is counted with respect to the bottom of a conduction band (a third sublevel) and not the LLL as in typical semiconductors [36, 37]. Generally, this also applies to bilayer graphene; however, not only the LLL is placed exactly in the Dirac point, but also the first Landau band. Obviously, the latter affects and determines the convenient impact factor definition, as well as the number of spin-valley branches of the 1LL available for free electrons.

Finally, it is easy to notice that the non-local theory of Hall systems predicts identical hierar‐ chies of LLL filling factors for typical semiconductor quantum-wells and monolayer graphene samples [31, 43–50]. The latter applies for both sublevels of the lowest band (also in graphene —we take into account only fillings accessible for free electrons). Additionally, collective Halllike states from the second (spin or spin-valley) branch are described with similar filling ratios (*ν*0,2) as those from the first one (*ν*0,1). The only difference is obvious and lies in a constant shift of all fractions, *ν*0,2 =1 + *ν*0,1. As we have already mentioned, the cyclotron subgroup model can be used to describe the FQHE in higher LLs of these structures too—the recipe for its appli‐ cation has been included in subsection 3.1.

The case of bilayer graphene is absolutely unique. The appearance of an additional sheet of carbon atoms (additional surface) leads to completely different commensurability conditions. This, for example, results in a surprising form of the basic set, where *ν* =1 / 2 corresponds to the most prominent incompressible state. We are going to describe this problem in detail in subsection 3.2.


Where *q* = the number of half loops in the exchange trajectory, *ν* = the hierarchy of fillings. Plus or minus signs denote the direction of a last loop in the multi-loop cyclotron path. Selected hole states are indicated in brackets. Experimentally observable ratios are highlighted—blue color for *ν* possible to explain within the CF model and red color for *ν* out of the Jain hierarchy. Additionally, results for different sublevels are separated by three lines.

**Table 2.** 1LL filling factors obtained from the non-local theory (typical semiconductor structures).

#### **3.1. Higher Landau levels: graphene and conventional 2DEG**

We consider a system subjected to external magnetic field, which leads to the partial filling of an arbitrary Landau band (*n* >0). To simplify further discussions, we introduce an additional parameter, *m*, that enumerates spin (or spin-valley) branches in each LL. The latter is twovalued (*m*∈{0, 1}) in the typical 2DEG case and four-valued (*m*∈{0, 1, 2, 3}) in the monolayer graphene case. We remind that the area embraced by a single-loop cyclotron orbit in the whole *n*'th Landau level equals to (2*<sup>n</sup>* <sup>+</sup> 1) *hc eB* = (2*n* + 1)*ϕ*<sup>0</sup> *<sup>B</sup>* (it takes exactly (2*n* + 1) flux quanta and its surface is considerably enhanced, when compared to the LLL orbit). This change of an encircled area modifies—and allows for the existence of novel—commensurability conditions, all of which are listed below.

Ultra-Quantum 2D Materials: Graphene, Bilayer Graphene, and Other Hall Systems—New Non-Local Quantum Theory of Hall Physics http://dx.doi.org/10.5772/64018 17


Where *q* = the number of half loops in the exchange trajectory, *ν* = the hierarchy of fillings. Plus or minus signs denote the direction of a last loop in the multi-loop cyclotron path. Selected hole states are indicated in brackets. Experimentally observable ratios are highlighted. Additionally, results for different sublevels (only three, accessible in measurement, are presented) are separated by three lines.

**Table 3.** 1LL filling factors obtained from the non-local theory (monolayer graphene).

Where *q* = the number of half loops in the exchange trajectory, *ν* = the hierarchy of fillings. Plus or minus signs denote

We consider a system subjected to external magnetic field, which leads to the partial filling of an arbitrary Landau band (*n* >0). To simplify further discussions, we introduce an additional parameter, *m*, that enumerates spin (or spin-valley) branches in each LL. The latter is twovalued (*m*∈{0, 1}) in the typical 2DEG case and four-valued (*m*∈{0, 1, 2, 3}) in the monolayer graphene case. We remind that the area embraced by a single-loop cyclotron orbit in the whole

(2*n* + 1)*ϕ*<sup>0</sup>

surface is considerably enhanced, when compared to the LLL orbit). This change of an encircled area modifies—and allows for the existence of novel—commensurability conditions, all of

*<sup>B</sup>* (it takes exactly (2*n* + 1) flux quanta and its

Experimentally observable ratios are highlighted—blue color for *ν* possible to explain within the CF model and red color for *ν* out of the Jain hierarchy. Additionally, results for different sublevels are separated by three lines.

the direction of a last loop in the multi-loop cyclotron path. Selected hole states are indicated in brackets.

**Table 2.** 1LL filling factors obtained from the non-local theory (typical semiconductor structures).

*eB* =

**3.1. Higher Landau levels: graphene and conventional 2DEG**

*n*'th Landau level equals to (2*<sup>n</sup>* <sup>+</sup> 1) *hc*

which are listed below.

16 Recent Advances in Graphene Research

**1.** We first consider a single-loop path (as a representative of a braid from the appropriate braid group). The simplest situation—in which this trajectory fits perfectly to the separa‐ tion of two neighboring particles—corresponds to 2*n* + 1 flux quanta attributed to a single electron from the partially filled level,

$$\begin{cases} \begin{array}{c} N\_0\\\hline N - (2n + m)N\_0 \end{array} = 2n + 1 \qquad \text{for conventional} \\\begin{array}{c} N\_0\\\hline N - (4n - 2 + m)N\_0 \end{array} = 2n + 1 \qquad \text{for graphene} \end{cases} \tag{23}$$

where a denominator of the left-hand side expression counts the number of electrons lying in the *n*'th LL (in the bottom expression a "−2" factor needs to be omitted, while investi‐ gating n=0). Furthermore, the whole fraction determines the number of *ϕ*0 per particle from the highest available level. Simultaneously, the right-hand side is also connected to the number of *ϕ*0, however, grasped by the single-loop cyclotron orbit. One can easily determine filling factors, which fulfil the above equation,

$$\begin{cases} \nu = \frac{N}{N\_0} = 2n + m + \frac{1}{2n+1} & \text{for conventional} \\ \nu = \frac{N}{N\_0} = 4n - 2 + m + \frac{1}{2n+1} & \text{for graphene} \end{cases} \tag{24}$$

If the system is described with these ν, then exchanges of nearest neighbors, *σ<sup>i</sup>* , are accessible and they generate *π*<sup>1</sup> (Ω). Something surprising should already be noticed—the full braid group is coupled to the IQHE only for the zeroth Landau band. For other levels, the corresponding filling is fractional (e.g. *ν* =7 / 3 is obtained, in both structures, when the *m*=0 branch of the 1LL is investigated). As a result, a collective Hall-like state is described with the fractional quantization of a transverse resistivity (just like for the ordinary FQHE state in the LLL), but the Laughlin correlations are described with a *q* =1 power in the Jastrow polynomial and loopless elements generate the system braid group (just like for the ordinary IQHE state in the LLL). We have, thus, stumbled across a novel phenomenon —the single-loop FQHE [51, 52]—which can be obtained only in higher Landau bands (*n* >0). We may also expect that this effect is very robust (simple, not structured, generators construction), which was confirmed in transport measurements of typical 2DEG struc‐ tures and monolayer graphene samples [20, 40–42, 53].

In higher LLs it is possible that the cyclotron orbit surface is greater than the separation of neighboring particles (classical—as arguments of *ΨN* or *<sup>M</sup> <sup>N</sup>* coordinates). Thus, for selected filling factors a single-loop path (as a representative of an element from the system braid group) can be generated as a double exchange of *x*-order neighbors (with integer *x*), (*σi*+1…*σi*+*x*−1…*σ<sup>i</sup>* <sup>−</sup>1*σi*+1 <sup>−</sup>1)2. This corresponds to the situation when exactly *ϕ*<sup>0</sup> / *x* is attributed to a single electron from the partially filled level,

#### Ultra-Quantum 2D Materials: Graphene, Bilayer Graphene, and Other Hall Systems—New Non-Local Quantum Theory of Hall Physics http://dx.doi.org/10.5772/64018 19

0

= + ï

ì

18 Recent Advances in Graphene Research

(2 )

<sup>ï</sup> - + <sup>í</sup>

0

<sup>ï</sup> = + <sup>ï</sup> - -+ <sup>î</sup>

determine filling factors, which fulfil the above equation,

<sup>ï</sup> = = -+ + <sup>ï</sup> <sup>+</sup> <sup>î</sup>

tures and monolayer graphene samples [20, 40–42, 53].

0

= = ++ ï <sup>ï</sup> <sup>+</sup> <sup>í</sup>

0

n

n

accessible and they generate *π*<sup>1</sup>

(*σi*+1…*σi*+*x*−1…*σ<sup>i</sup>*

<sup>−</sup>1*σi*+1

to a single electron from the partially filled level,

ì

(4 2 )

0

0

21

*<sup>N</sup> n for conventional N n mN*

*<sup>N</sup> n for graphene N n mN*

<sup>1</sup> <sup>2</sup> 2 1

2 1

If the system is described with these ν, then exchanges of nearest neighbors, *σ<sup>i</sup>*

full braid group is coupled to the IQHE only for the zeroth Landau band. For other levels, the corresponding filling is fractional (e.g. *ν* =7 / 3 is obtained, in both structures, when the *m*=0 branch of the 1LL is investigated). As a result, a collective Hall-like state is described with the fractional quantization of a transverse resistivity (just like for the ordinary FQHE state in the LLL), but the Laughlin correlations are described with a *q* =1 power in the Jastrow polynomial and loopless elements generate the system braid group (just like for the ordinary IQHE state in the LLL). We have, thus, stumbled across a novel phenomenon —the single-loop FQHE [51, 52]—which can be obtained only in higher Landau bands (*n* >0). We may also expect that this effect is very robust (simple, not structured, generators construction), which was confirmed in transport measurements of typical 2DEG struc‐

In higher LLs it is possible that the cyclotron orbit surface is greater than the separation of neighboring particles (classical—as arguments of *ΨN* or *<sup>M</sup> <sup>N</sup>* coordinates). Thus, for selected filling factors a single-loop path (as a representative of an element from the system braid group) can be generated as a double exchange of *x*-order neighbors (with integer *x*),

<sup>−</sup>1)2. This corresponds to the situation when exactly *ϕ*<sup>0</sup> / *x* is attributed

(Ω). Something surprising should already be noticed—the

*<sup>N</sup> n m for graphene N n*

*<sup>N</sup> n m for conventional N n*

<sup>1</sup> 4 2

21

where a denominator of the left-hand side expression counts the number of electrons lying in the *n*'th LL (in the bottom expression a "−2" factor needs to be omitted, while investi‐ gating n=0). Furthermore, the whole fraction determines the number of *ϕ*0 per particle from the highest available level. Simultaneously, the right-hand side is also connected to the number of *ϕ*0, however, grasped by the single-loop cyclotron orbit. One can easily

(23)

(24)

, are

$$\begin{cases} \begin{array}{c} N\_0 \\ N - (2n + m)N\_0 \end{array} = \frac{2n + 1}{x} \rightarrow \nu = 2n + m + \frac{x}{2n + 1} \qquad \text{for conventional} \\\frac{N\_0}{N - (4n - 2 + m)N\_0} = \frac{2n + 1}{x} \rightarrow \nu = 4n - 2 + m + \frac{x}{2n + 1} \qquad \text{for graphene} \end{cases} \tag{25}$$

Note that complete filling of the *n*'th LL is achieved when *x* =2*n* + 1 and the braid group is generated by the loopless exchanges of particles and their (2*n* + 1)-order neighbors.

**2.** In higher Landau bands also exists a possibility that the cyclotron orbit is too small to reach a neighboring (classical) carrier and, hence, the loopless exchanges are not allowed and need to be excluded from *π*1(Ω). Fortunately, we can consider—similarly as in the LLL case—more structured, multi-loop exchanges to provide generators of the braid group describing the system. Commensurability conditions, which allow to establish the corresponding filling factors (connected with subgroups generated by *bi* elements), resemble ones introduced for the zeroth Landau band. However, while possible ν for Halllike states, we need to remember about a different number of flux quanta grasped by the cyclotron orbit ((2*n* + 1)*ϕ*0 in the *n*'th LL). Additionally, we cannot take into account all particles in the system, but only those, which are placed in a partially filled level. Finally, we remind that if we consider a *q*-looped trajectory encircled in a 2D space, then all *q* loops must share the total magnetic field flux per particle, which is evidenced by a single-loop path.

To obtain a hierarchy—which resembles Jain's pyramid of fillings—we assume that *q* −1 loops experience exactly (2*n* + 1)*ϕ*0 per particle, while the last one embraces a reduced portion, (2*n* + 1)*ϕ*<sup>0</sup> / *x*, attributed to a single carrier. Thus, the appropriate commensura‐ bility conditions (with different, integer *x*) take the form of,

$$\begin{cases} \begin{array}{c} N\_0 \\ \hline N - (2n+m)N\_0 \end{array} = (2n+1) \cdot (q-1) \pm (2n+1) \frac{1}{\text{x}} \qquad \text{for conventional} \\\begin{array}{c} N\_0 \\ \hline N - (4n-2+m)N\_0 \end{array} = (2n+1) \cdot (q-1) \pm (2n+1) \frac{1}{\text{x}} \qquad \text{for graphene} \end{cases} \tag{26}$$

with filling factors, which satisfy the requirements above, belonging to the set,

$$\begin{cases} \nu = 2n + m + \frac{1}{\left(2n + 1\right) \cdot \left\lceil \left(q - 1\right) \pm 1 \right\rceil \cdot x} & \text{for conventional} \\\\ \nu = 4n - 2 + m + \frac{1}{\left(2n + 1\right) \cdot \left\lceil \left(q - 1\right) \pm 1 \right\rceil \cdot x} & \text{for graphene} \end{cases} \tag{27}$$

Note that—despite the general similarity to the CF hierarchy—a supplementary factor, 2*n* + 1, is included in the denominators of all fractions from Eq. 27 (as a consequence of an enhanced kinetic energy). The last winding of a *q*-looped trajectory, for these fillings, fits perfectly to the area embraced by *x* classical particles. As a consequence, exchanges of *x*'th order neighbors are allowed and, thus, elements

$$b\_i^{(q)} = \sigma\_i^{q-1} \sigma\_i \sigma\_{i+1} \dots \sigma\_{i+x-1}{}^{\pm 1} \dots \sigma\_i^{-1} \sigma\_{i+1}{}^{-1} \tag{28}$$

generate the cyclotron subgroup for the system.

**3.** Finally, the pairing of electrons—which occurs as a result of the Fermi sea instability can also be investigated. In this case the number of particles is reduced by half and it is reasonable to consider the IQHE formation (for pairs),

$$\begin{cases} \begin{array}{c} N\_0\\\hline N - (2n + m)N\_0 \end{array} = 2 \rightarrow \nu = 2n + m + \frac{1}{2} \qquad \text{for conventional} \\\begin{array}{c} N\_0\\\hline N - (4n - 2 + m)N\_0 \end{array} = 2 \rightarrow \nu = 4n - 2 + m + \frac{1}{2} \qquad \text{for graphene} \end{cases} \tag{29}$$

The equation above is constructed in such a manner that implementing a decreased number of particles— *N* − (2*n* + *m*)*N*<sup>0</sup> <sup>2</sup> or *N* − (4*n* − 2 + *m*)*N*<sup>0</sup> <sup>2</sup> —leads to the exact IQHE commen‐ surability condition.

We have presented hierarchies that gather all filling factors (from all LLs) in which particles can experience quantum Hall effects in typical semiconductor and monolayer graphene samples. We have also explicitly presented these results, for the first Landau band, in **Tables 2** and **3**.

#### **3.2. Bilayer graphene**

The FQHE in bilayer systems (not only bilayer graphene) is exceptional, as noticed earlier by Eisenstein [54]. The unparalleled basic set of fillings – with *ν* =1 / 2 being the most robust incompressible state – has its origin in the appearance of an additional surface [28, 55]. The supplementary sheet of carbon atoms, coupled to the primary one by a nonzero hopping integral [36–39], leads to the electron density located in both graphene planes. As a result, bilayer graphene samples are not strictly two-dimensional. Classically (trajectories, which represent elements of the system braid group, are classical), this means that particles can move freely between opposite layers of the structure. Thus, while considering whether the cyclotron orbit area is sufficiently large to enable the existence of particle exchanges in the system, graphene planes should not be investigated separately. For example, the *ν* =1 state—corre‐ sponding to the integer phenomenon—is realized when the single-loop cyclotron path encircles a surface equal to *S* / *N* , and not 2*S* / *N* . However, while examining states described by fractional fillings of the LLL, even more interesting feature is revealed. Since multi-looped trajectories can be partly located in both graphene layers, the surface (and flux) provided by the additional plane needs to be taken into account. It is expected that the most energetically efficient trajectory is realized, when only one loop embraces (utilizes) the supplementary surface and magnetic field flux. As a consequence, its dimensions are not raised—they are equal to those of a single-loop path. Simultaneously, remaining loops (*q* −1 in the case of a *q*looped trajectory) share the total per particle quantity, *BS* / *N* , associated with the primary layer. Latter results in a novel form of the commensurability conditions,

Note that—despite the general similarity to the CF hierarchy—a supplementary factor, 2*n* + 1, is included in the denominators of all fractions from Eq. 27 (as a consequence of an enhanced kinetic energy). The last winding of a *q*-looped trajectory, for these fillings, fits perfectly to the area embraced by *x* classical particles. As a consequence, exchanges of

11 1

**3.** Finally, the pairing of electrons—which occurs as a result of the Fermi sea instability can also be investigated. In this case the number of particles is reduced by half and it is

<sup>1</sup> <sup>22</sup>

The equation above is constructed in such a manner that implementing a decreased

*N* − (4*n* − 2 + *m*)*N*<sup>0</sup>

We have presented hierarchies that gather all filling factors (from all LLs) in which particles can experience quantum Hall effects in typical semiconductor and monolayer graphene samples. We have also explicitly presented these results, for the first Landau band, in **Tables**

The FQHE in bilayer systems (not only bilayer graphene) is exceptional, as noticed earlier by Eisenstein [54]. The unparalleled basic set of fillings – with *ν* =1 / 2 being the most robust incompressible state – has its origin in the appearance of an additional surface [28, 55]. The supplementary sheet of carbon atoms, coupled to the primary one by a nonzero hopping integral [36–39], leads to the electron density located in both graphene planes. As a result, bilayer graphene samples are not strictly two-dimensional. Classically (trajectories, which represent elements of the system braid group, are classical), this means that particles can move freely between opposite layers of the structure. Thus, while considering whether the cyclotron orbit area is sufficiently large to enable the existence of particle exchanges in the system, graphene planes should not be investigated separately. For example, the *ν* =1 state—corre‐ sponding to the integer phenomenon—is realized when the single-loop cyclotron path

<sup>1</sup> 2 42

*<sup>N</sup> n m for graphene N n mN*

*<sup>N</sup> n m for conventional N n mN*

 s s


<sup>2</sup> —leads to the exact IQHE commen‐

(29)

() 1 1 11

 s

*i i i i ix i i b*

*x*'th order neighbors are allowed and, thus, elements

*q q*

generate the cyclotron subgroup for the system.

0

0

<sup>ï</sup> - + <sup>í</sup>

number of particles—

surability condition.

20 Recent Advances in Graphene Research

**3.2. Bilayer graphene**

**2** and **3**.

s ss

reasonable to consider the IQHE formation (for pairs),

0

<sup>ì</sup> =®= + + <sup>ï</sup>

(2 ) 2

n

n

(4 2 ) 2

0

<sup>ï</sup> =®= -+ + <sup>ï</sup> - -+ <sup>î</sup>

*N* − (2*n* + *m*)*N*<sup>0</sup> <sup>2</sup> or

$$\frac{BS}{N} = \frac{1}{\nu} \phi\_o = \left(q - 2\right)\phi\_o \pm \frac{1}{\chi} \phi\_o \tag{30}$$

with the hierarchy of fillings being a modified version of the composite fermion pyramid,

$$\nu = \left( (q - 2) \pm \frac{1}{\chi} \right)^{-1} \tag{31}$$

Here we have assumed that one of *q* −2 loops evidences a diminished (per particle) portion, ± 1 *<sup>x</sup> ϕ*0. Thus, it fits perfectly to the separation of an arbitrary particle and its *x*-order neighbors (as *ΨN* arguments) and allows them to exchange. Additionally, we speculate that the latter loop can be overwrapped in the direction opposite to one enforced by an external magnetic field (a minus sign in Eq. 31).

Even-denominator filling factors form a basic set—they are expected to be associated with most prominent Hall-like states [22–24] However, ratios with odd denominators are also included in the above hierarchy. The latter can be seen in **Table 4**, which gathers results from the whole LLL (and 1LL). In this paper we assumed that spin-valley branches of zeroth and first Landau bands, accessible for free electrons, are filled alternately. In other words, these sublevels are placed on the energy axis in the following order, 0*K* ′ ↑, 1*K* ′ ↑, 0*K* ′ ↓, 1*K* ′ ↓. Thus, filling factors responsible for the FQHE plateaus—n the 2<*ν* <3 range—are defined by the hierarchy from Eq. 33 (with a constant "+2" shift).

Although an even number of loops, *q* −1, is explicitly included in the commensurability conditions, it is an odd number, *q*, that constitutes a multi-looped trajectory (which represents a square of the cyclotron subgroup generator). This results in the Laughlin correlations with an odd power in the Jastrow-like polynomial—the multi-particle wave function is antisym‐ metric even for *ν* =1 / 2 state.

In bilayer graphene, similarly as in typical 2DEG and monolayer graphene structures, kinetic energy of particles increases with Landau level index, *n*. Hence, the area encircled by a singleloop cyclotron orbit (representing a braid group element) is also enhanced, (2*n* + 1)*hc* /*eB*, and it embraces precisely (2*n* + 1) quanta of the magnetic field flux. This leads to commensurability conditions, which resemble ones already presented in this section and other authors' papers [10, 52, 55]. We investigate them very briefly,


Where *q* = the number of half loops in the exchange trajectory, *ν* = the hierarchy of fillings. Plus or minus signs denote the d irection of a last loop in the multi-loop cyclotron path. Selected hole states are indicated in brackets. Experimentally observable ratios are highlighted - a blue colour for *ν* possible to explain within the CF model and a red colour for *ν* out of the Jain hierarchy. Additionally, results for different sublevels are separated by three lines.

**Table 4.** LLL and 1LL filling factors obtained from the non-local theory (bilayer graphene).

**1.** First, it is possible that the system braid group is generated by loopless exchanges of *x*order neighbors. This corresponds to the situation where exactly (2*n* + 1)/ *x* flux quanta are attributed to a single particle,

loop cyclotron orbit (representing a braid group element) is also enhanced, (2*n* + 1)*hc* /*eB*, and it embraces precisely (2*n* + 1) quanta of the magnetic field flux. This leads to commensurability conditions, which resemble ones already presented in this section and other authors' papers

Where *q* = the number of half loops in the exchange trajectory, *ν* = the hierarchy of fillings. Plus or minus signs denote

the d irection of a last loop in the multi-loop cyclotron path. Selected hole states are indicated in brackets. Experimentally observable ratios are highlighted - a blue colour for *ν* possible to explain within the CF model and a red colour for *ν* out of the Jain hierarchy. Additionally, results for different sublevels are separated by three lines.

**Table 4.** LLL and 1LL filling factors obtained from the non-local theory (bilayer graphene).

[10, 52, 55]. We investigate them very briefly,

22 Recent Advances in Graphene Research

$$\frac{N\_0}{N - \varepsilon N\_0} = \frac{2n + 1}{\infty} \to \nu = \varepsilon + \frac{\infty}{2n + 1} \tag{32}$$

where ε counts completely filled Landau sublevels and *x* =1, 2, …, 2*n* + 1. Note that the IQHE is realized in the system when *x* equals 2*n* + 1 and the single-loop cyclotron orbit fits perfectly to the separation of a particle and its (2*n* + 1)-order neighbor. For other values of *x* electrons experience the so-called single-loop FQHE – the phenomenon allowed only in higher Landau bands (*n* >0).

**2.** The (*<sup>q</sup>* <sup>−</sup>1) / 2-loop exchanges of *x*-order neighbors (as arguments of *ΨN* or *<sup>M</sup> <sup>N</sup>* coordinates) can become generators of the system braid group (a cyclotron subgroup), when loopless ones are not accessible. This is achievable when *q*-looped path is arranged as follows: all *q* −2 loops experience precisely (2*n* + 1)*ϕ*<sup>0</sup> per particle, while the last loop evidences only a residual portion attributed to a single electron, (2*<sup>n</sup>* <sup>+</sup> 1) *<sup>x</sup> ϕ*0,

$$\frac{N\_0}{N - \varepsilon N\_0} = \left(2n + 1\right)\left(q - 2\right) \pm \frac{\left(2n + 1\right)}{\infty} \rightarrow \nu = \varepsilon + \frac{1}{\left(2n + 1\right)\left[q - 2 \pm 1/\ge\right]}\tag{33}$$

It should be emphasized that—in the above commensurability condition—we have not considered an additional per particle quantity, *BS <sup>N</sup>* , supplied by the second layer of the structure. The latter can be performed because a remaining loop of the trajectory utilizes the supplementary magnetic field flux—it can be, thus, omitted in Eq. 33.

**3.** Finally, the pairing of electrons—which occurs as a result of the Fermi sea instability can also be investigated. In this case the number of particles is reduced by half and it is reasonable to consider the IQHE formation (for pairs),

$$\frac{N\_0}{N - \varepsilon N\_0} = 2 \to \nu = \varepsilon + \frac{1}{2} \tag{34}$$

The above equation is constructed in such a manner that implementing a decreased number of particles, *N* − *εN*<sup>0</sup> <sup>2</sup> , leads to the exact IQHE commensurability condition.

We have presented hierarchies that gather all filling factors (from all LLs) in which particles can experience quantum Hall effects. We have also explicitly presented these results—for lowest and first Landau bands—in **Table 4**.

A careful reader probably has already noticed that an application of the non-local theory of Hall systems to structures with a greater (than two) number of layers is straightforward. The most important modification concerns the commensurability conditions for *q*-looped cyclotron trajectories (when only multi-loop exchanges are accessible and generate the braid group describing the system). Each additional layer of atoms supplies an additional surface (and magnetic field flux) that needs to be embraced by this path. The most energetically efficient trajectory seems to be achieved when every added plane is utilized by a single loop. Dimen‐ sions of these loops are not raised and they are not included, at least explicitly, in the com‐ mensurability conditions. As a result, only remaining *q* −*γ* ones (where *γ* stands for the number of supplementary layers) must share the total flux per particle, *BS* /(*N* −*εN*0), which is experienced by the single-loop path.
