**1. Introduction to the fractional quantum Hall effect**

In high magnetic fields and low temperatures, dips in a longitudinal resistivity (*ρ*xx→0) and plateaus in a transverse one (*ρ*xy = *h/ve*<sup>2</sup> ) appear for fractional fillings (*v*) of Landau levels (LLs) —this transport feature is called the fractional quantum Hall effect (FQHE). Mentioned minima show an activated behavior, vanishing exponentially as temperature goes to zero and indicat‐ ing the presence of a gap in the spectrum [1]. The latter phenomenon, despite its long history

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

[2], constantly receives a great amount of interest from the scientific society. Though many preliminary requirements for Hall-like states are commonly known, the comprehensive theory ofthiseffect—capableofexplainingallexperimentalfindings inanelegantway—is stillmissing. One of the obvious facts is that the FQHE is impossible to obtain within a single-particle picture without interactions—where a partial filling of an elongated-states band immediately results inanonzerovalue ofthe longitudinalresistivity.Amongthe explainednecessityof stronginterparticle correlations in the system, a two-dimensional (2D) topology and a quantized kinetic energy (flat bands—as in the case of LLs) are also compulsory for evidencing the FQHE. Additionally, due to the fragility of these incompressible states, a high purity of the sample needs to be ensured too.

Let us emphasize that when the lowest Landau level (LLL) is partially occupied (or when a collectivization is restricted to one, arbitrary LL), kinetic energy remains constant (*Ek* =ћ*ωc*, where *ω<sup>c</sup>* =*eB* / *mc* is a cyclotron angular frequency). This applies also to the background potential energy. The ground state is, thus, expected to minimize the Coulomb repulsion. Since for a wide collection of magnetic fields the FQHE is actually observed, the latter requirement refers to collective Hall-like states (and not Wignier-crystal states with localized electrons [3], as it may seem at first).

The initial step towards an explanation of the FQHE was taken by Laughlin [4], who proposed the exact solution for a basic set of fillings from the LLL, *ν* =1 / *q* (*q* – odd),

$$\Psi\_L = \prod\_{i$$

where *N* is the number of electrons, *l* <sup>0</sup> = *ћc* /*eB* stands for magnetic length and *z* = *x* + *iy* is a complex position. Note that when two arguments of this wave function are swapped, *Ψ<sup>L</sup>* gains an additional phase equal to *q*π. This detail of the solution (or more precisely—the quantum statistics of particles) is usually called the hallmark of Laughlin correlations and included in all theories of the FQHE—as a result of the Aharonov-Bohm effect [5] or the Berry phase acquired by a vortex [6–8]. However, the one-dimensional unitary representations (1DURs) of the full braid group (*π*1)—which define the particle statistics—are periodic with a periodicity of 2*π*(*eiqπ* = *eiπ* for odd *q*). In order to deal with this issue, we suggest to associate FQHE particles (so-called composite fermions) with the appropriately constructed braid subgroups of *π*1. This allows to differentiate them from ordinary fermions characterized with the full braid group. The latter idea is explained in detail in following sections and in authors' papers [9–11].

Unfortunately, despite its unbelievable accuracy confirmed by exact diagonalization methods (at least for a small amount of particles) [12], the Laughlin wave function was introduced as an ansatz or, if one prefers, as an inspired (educated) guess. Nonetheless, it might be educatory to recall his arguments [4, 13]. First, note that one can choose single-particle eigenstates within LLL (in a central gauge) to be eigenstates of a coordinate of the angular momentum (*l* ^ *z*),

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 5

[2], constantly receives a great amount of interest from the scientific society. Though many preliminary requirements for Hall-like states are commonly known, the comprehensive theory ofthiseffect—capableofexplainingallexperimentalfindings inanelegantway—is stillmissing. One of the obvious facts is that the FQHE is impossible to obtain within a single-particle picture without interactions—where a partial filling of an elongated-states band immediately results inanonzerovalue ofthe longitudinalresistivity.Amongthe explainednecessityof stronginterparticle correlations in the system, a two-dimensional (2D) topology and a quantized kinetic energy (flat bands—as in the case of LLs) are also compulsory for evidencing the FQHE. Additionally, due to the fragility of these incompressible states, a high purity of the sample

Let us emphasize that when the lowest Landau level (LLL) is partially occupied (or when a collectivization is restricted to one, arbitrary LL), kinetic energy remains constant (*Ek* =ћ*ωc*, where *ω<sup>c</sup>* =*eB* / *mc* is a cyclotron angular frequency). This applies also to the background potential energy. The ground state is, thus, expected to minimize the Coulomb repulsion. Since for a wide collection of magnetic fields the FQHE is actually observed, the latter requirement refers to collective Hall-like states (and not Wignier-crystal states with localized electrons [3],

The initial step towards an explanation of the FQHE was taken by Laughlin [4], who proposed

( ) <sup>2</sup> <sup>2</sup> <sup>1</sup> <sup>0</sup> <sup>1</sup> | | <sup>4</sup> *N*

<sup>å</sup> = - <sup>Õ</sup> (1)

<sup>0</sup> = *ћc* /*eB* stands for magnetic length and *z* = *x* + *iy* is a

^ *z*),

*<sup>i</sup> <sup>N</sup> <sup>i</sup> <sup>z</sup> <sup>q</sup> <sup>l</sup>*

complex position. Note that when two arguments of this wave function are swapped, *Ψ<sup>L</sup>* gains an additional phase equal to *q*π. This detail of the solution (or more precisely—the quantum statistics of particles) is usually called the hallmark of Laughlin correlations and included in all theories of the FQHE—as a result of the Aharonov-Bohm effect [5] or the Berry phase acquired by a vortex [6–8]. However, the one-dimensional unitary representations (1DURs) of the full braid group (*π*1)—which define the particle statistics—are periodic with a periodicity of 2*π*(*eiqπ* = *eiπ* for odd *q*). In order to deal with this issue, we suggest to associate FQHE particles (so-called composite fermions) with the appropriately constructed braid subgroups of *π*1. This allows to differentiate them from ordinary fermions characterized with the full braid group. The latter idea is explained in detail in following sections and in authors' papers [9–11].

Unfortunately, despite its unbelievable accuracy confirmed by exact diagonalization methods (at least for a small amount of particles) [12], the Laughlin wave function was introduced as an ansatz or, if one prefers, as an inspired (educated) guess. Nonetheless, it might be educatory to recall his arguments [4, 13]. First, note that one can choose single-particle eigenstates within

LLL (in a central gauge) to be eigenstates of a coordinate of the angular momentum (*l*

the exact solution for a basic set of fillings from the LLL, *ν* =1 / *q* (*q* – odd),

*i i L j j*

 *z ze* <sup>=</sup> - <

Y

where *N* is the number of electrons, *l*

needs to be ensured too.

4 Recent Advances in Graphene Research

as it may seem at first).

$$
\varphi\_m = \varpi^m e^{-\frac{|x|^2}{4\lambda\_0^2}} \tag{2}
$$

where *m* (integer) is an eigenvalue of *l* ^ *<sup>z</sup>*. It is easy to establish the average area covered by an arbitrary electron, *φm*|*π<sup>r</sup>* <sup>2</sup> |*φ<sup>m</sup>* =2*π*(*m* + 1)*l* 0 2 . Thus, the degeneracy of a whole Landau level equals,

$$N\_0 = m\_{\text{max}} + 1 = \frac{S}{2\pi l\_0^2} = \frac{SB}{hc/e} \tag{3}$$

where *S* is a sample surface and *hc*/*e* = *ϕ*0 is a magnetic field flux quantum. Additionally, *mmax* stands for a maximal value of *m*, which produces a state satisfying the *φm*|*π<sup>r</sup>* <sup>2</sup> |*φ<sup>m</sup>* ≤*S* relationship. As a result, the general state of a multi-particle system from the LLL takes the form of,

$$\Psi = f(z\_1, z\_2, \dots, z\_N)e^{-\frac{1}{4l\_0^2} \sum\_{i=1}^{\kappa} |z\_i|^2} \tag{4}$$

where *f* (*z*1, *z*2, …, *zN* ) is an ordinary polynomial. A degree of this polynomial, *D*, cannot exceed the maximal *l* ^ *<sup>Z</sup>* eigenvalue realizable in the system. In other words, *D* is restricted by an inverse of a filling factor times a number of particles (or a degeneracy), *<sup>D</sup>* <sup>≤</sup>*m*max <sup>≈</sup> *<sup>N</sup>*<sup>0</sup> <sup>=</sup>*<sup>ν</sup>* <sup>−</sup><sup>1</sup> *N* .

Consider now a two-body problem. The eigenfunction for two particles with a relative angular momentum *m* and a centre of mass angular momentum *M* is [13],

$$\boldsymbol{\wp}\_{m,M} = (\boldsymbol{z}\_1 - \boldsymbol{z}\_2)^m (\boldsymbol{z}\_1 + \boldsymbol{z}\_2)^M \boldsymbol{e}^{-\frac{1}{4\boldsymbol{z}\_0^2}(\|\boldsymbol{z}\|^2 + \|\boldsymbol{z}\|^2)} \tag{5}$$

As long as *m* and *M* are assumed non-negative integers, the above state lies entirely in the LLL (it is constructed from linear combinations of one-body functions, φ*m*) and, remarkably, appears to be the exact solution of a stationary Schrödinger equation with any central potential acting between two particles. Additionally, eigenvalues of the corresponding Hamiltonian are, simultaneously, eigenvalues of a Coulomb potential, *V* ^ <sup>=</sup>∑*<sup>i</sup>*<sup>&</sup>lt; *<sup>j</sup> <sup>N</sup> <sup>e</sup>* <sup>2</sup> / <sup>|</sup>*Zi* <sup>−</sup>*Zj* <sup>|</sup> (the fixed kinetic energy can be disregarded—it only adds even shifts to the energy levels),

$$\nu\_m = \frac{\left\langle \varphi\_{m,M} | \hat{V} | \varphi\_{m,M} \right\rangle}{\left\langle \varphi\_{m,M} | \varphi\_{m,M} \right\rangle} \tag{6}$$

where *vm* expectation values are so-called Haldane pseudopotentials, and they are independent from a central mass angular momentum eigenvalue. Hence, we obtained a discrete spectrum consisted of bound, repulsive potential states. This result is owed to the quenched kinetic energy, and it is possible to obtain only in two-dimensional spaces in the presence of a magnetic field. This finding should not be surprising—the high potential energy of interacting particles cannot be converted into the high kinetic energy (*Ek* is kept constant within one Landau level) and, hence, particles cannot "fly apart". Even classically, when a third (parallel to the magnetic field) dimension is lacking, the Lorentz force protects electrons from being pushed away from each other.

Let us now return to Laughlin's considerations, concerning the many-body ground-state wave function for filling factors, *v* = 1/*q* (*q* – odd). Though we are currently unable to derive it analytically for *v* < 1, the lowest energy solution for a complete filling of the LLL (*v* = 1) is well known. It can be written as a Slater determinant of all accessible (indexed with an angular momentum eigenvalue) one-particle states [14],

$$\Psi\_S = \begin{vmatrix} 1 & z\_1 & z\_1^2 & \dots & z\_1^{N-1} \\ \dots & \dots & \dots & \dots & \dots \\ 1 & z\_N & z\_1^2 & \dots & z\_N^{N-1} \end{vmatrix} e^{-\frac{1}{4z\_0^2} \sum\_{i=1}^N |z\_i|^2} = \prod\_{l$$

where *Ψ<sup>S</sup>* is the Slater function with the Vandermonde polynomial. Taking into account this conclusion, as well as bearing in mind the two-body eigenstate, it seems plausible that *<sup>f</sup>* (*z*1, *<sup>z</sup>*2, …, *zN* ) have a Jastrow-like form, ∏*<sup>i</sup>*<sup>&</sup>lt; *<sup>j</sup> <sup>N</sup>* (*zi* <sup>−</sup> *zj* )*<sup>q</sup>*. Another advantage of this choice lies in its tendency to keep electrons apart—there exists a considerable chance that it may reduce the Coulomb repulsion energy. Furthermore, the system is built from electrons, thus, only odd powers in the Jastrow factor are allowed. Finally, when we connect the *q* – solution with a *ν* =1 / *q* filling factor (this can supported by the extremal condition *D* =*m*max), we arrive at the Laughlin wave function. Generally, it should be emphasized that all presented arguments are insufficient to prove that *ΨL* is a correct, lowest energy eigenfunction of the multi-particle Hamiltonian. However, its excellent compatibility with numerical results [12] entirely justifies a widespread use of the Laughlin ansatz.

Before skipping to another topic it is worth describing two other features of the Laughlin solution. First, the form of this wave function guarantees that every pair of particles has a relative angular momentum greater or equal to *q*. Thus, it needs to be an exact eigenfunction of the so-called hard-core potential with neglected long range part of the Coulomb repulsion (*vmHC* =*vm* for *m*<*q* and *vmHC* =0 for *m*≥*q*) [15],

$$\sum\_{m=0}^{\infty} \sum\_{i$$

where the left-hand side equals to the hard-core potential operator (written in terms of projection operators, *Pm*(*ij*), that selects states in which particles *i* and *j* have relative angular momentum equal to *m*) times *Ψ<sup>L</sup>* . This implies that a finite amount of energy is needed to excite the system from its (Laughlin) ground state—there exists a gap in the spectrum (Δ). This gap is believed to be stable against perturbations, thus, all corrections arising from differences between the Haldane and the hard-core potential are expected to be small compared to Δ [13].

where *vm* expectation values are so-called Haldane pseudopotentials, and they are independent from a central mass angular momentum eigenvalue. Hence, we obtained a discrete spectrum consisted of bound, repulsive potential states. This result is owed to the quenched kinetic energy, and it is possible to obtain only in two-dimensional spaces in the presence of a magnetic field. This finding should not be surprising—the high potential energy of interacting particles cannot be converted into the high kinetic energy (*Ek* is kept constant within one Landau level) and, hence, particles cannot "fly apart". Even classically, when a third (parallel to the magnetic field) dimension is lacking, the Lorentz force protects electrons from being pushed away from

Let us now return to Laughlin's considerations, concerning the many-body ground-state wave function for filling factors, *v* = 1/*q* (*q* – odd). Though we are currently unable to derive it analytically for *v* < 1, the lowest energy solution for a complete filling of the LLL (*v* = 1) is well known. It can be written as a Slater determinant of all accessible (indexed with an angular

11 1 1 1 | | | | 4 4

where *Ψ<sup>S</sup>* is the Slater function with the Vandermonde polynomial. Taking into account this conclusion, as well as bearing in mind the two-body eigenstate, it seems plausible that

in its tendency to keep electrons apart—there exists a considerable chance that it may reduce the Coulomb repulsion energy. Furthermore, the system is built from electrons, thus, only odd powers in the Jastrow factor are allowed. Finally, when we connect the *q* – solution with a *ν* =1 / *q* filling factor (this can supported by the extremal condition *D* =*m*max), we arrive at the Laughlin wave function. Generally, it should be emphasized that all presented arguments are insufficient to prove that *ΨL* is a correct, lowest energy eigenfunction of the multi-particle Hamiltonian. However, its excellent compatibility with numerical results [12] entirely justifies

Before skipping to another topic it is worth describing two other features of the Laughlin solution. First, the form of this wave function guarantees that every pair of particles has a relative angular momentum greater or equal to *q*. Thus, it needs to be an exact eigenfunction of the so-called hard-core potential with neglected long range part of the Coulomb repulsion

> ( ) <sup>0</sup> <sup>0</sup> *<sup>N</sup> HC m ij mm L v P ij*

¥

Y

= < å å <sup>=</sup> (8)

*<sup>N</sup>* (*zi* <sup>−</sup> *zj*

*e z ze*


( ) 2 2 2 2 1 1 0 0

Õ (7)

)*<sup>q</sup>*. Another advantage of this choice lies

*N N i i i i*

*z z l l <sup>N</sup>*

each other.

momentum eigenvalue) one-particle states [14],

*<sup>f</sup>* (*z*1, *<sup>z</sup>*2, …, *zN* ) have a Jastrow-like form, ∏*<sup>i</sup>*<sup>&</sup>lt; *<sup>j</sup>*

a widespread use of the Laughlin ansatz.

(*vmHC* =*vm* for *m*<*q* and *vmHC* =0 for *m*≥*q*) [15],

1

Y

6 Recent Advances in Graphene Research

1

2 1

*N*



*S i j i j N*

= =

<sup>¼</sup> å å = ¼ ¼ ¼¼ ¼ = -

2 1

*NN N*

¼

*zz z*

*zz z*

Finally, one can prove—with the use of a beautiful analogy developed by Laughlin [4]—that the *Ψ<sup>L</sup>* state describes a uniform particle-density. Let us express the probability distribution in terms of the Boltzmann weight, |*Ψ<sup>L</sup>* | <sup>2</sup> =*e* <sup>−</sup>*Hq* =*e* −(−2*q*∑*<sup>i</sup>*<sup>&</sup>lt; *<sup>j</sup> <sup>N</sup>* ln(*zi* −*zj* )+∑*<sup>i</sup>*=1 *<sup>N</sup>* <sup>|</sup>*zi* |2 /2*l* 0 2 ) [14]. *Hq* turns out to be the potential energy of a one-component, two-dimensional classical plasma. Hence, we expect that electrons are distributed uniformly with a density *ρ* =1 / (2*πl* 0 2 *m*) in a state described by 1 / *q* filling of the LLL.

At this stage, readers should be warned that many incompressible Hall states—connected with filling factors falling out of the basic set (1/*q* with odd *q*)—have already been discovered. Inspired by these results, scientists have been searching for the thorough and microscopic theory of the FQHE ever since. We are not going to present all models or ideas introduced by researchers over the years—this is not the aim of this paper. However, before proceeding to the topological explanation, it is worth becoming acquainted (at least briefly) with today's most widely accepted theory—he theory of composite fermions (CFs). In this very short description, we are going to focus on substantial advantages, as well as built-in problems of the approach.

In the CF model [16, 17], it is assumed that the FQHE and the IQHE (integer effect) are deeply connected and can be unified. The latter phenomenon, however, is usually identified with noninteracting particles, while strong correlations are necessary for the appearance of a fractional one. Nonetheless, the mapping can still be obtained if we assume that Coulomb interactions can be utilized—weakly interacting quasi-fermions can appear in the system—for fractional *ν* (this is, actually, a first tricky part—no one proved that Landau formulation [18] can be used in 2D spaces, where the spectrum of a Coulomb potential energy is discrete). Now, we only need these novel particles to play the same role for the FQHE as electrons for the IQHE.

Jain, who proposed this approach, postulated that quasi-particles are appearing as complexes of electrons with an even number of magnetic field flux quanta (currently identified with vortices [17], though this disambiguation is not entirely clear and might be misleading) and called them "composite fermions". Note that the many-body wave function embraces an

additional Aharonov-Bohm phase, *<sup>e</sup>* <sup>ћ</sup>*<sup>c</sup>∮Adl* <sup>=</sup> *<sup>e</sup>* <sup>ћ</sup>*<sup>c</sup> ∫ Bds* <sup>=</sup> *<sup>e</sup>* <sup>ћ</sup>*<sup>c</sup> ϕ*<sup>0</sup> =2*π*, when an electron (as an argument of *Ψ<sup>N</sup>* ) is encircling a single quantum [5]. The latter affects the quantum statistics of CFs—when two quasi-particles are exchanging, the wave function acquires a phase factor,

$$e^{i\alpha} = e^{i\pi}e^{i(p-1)\pi} \tag{9}$$

where the first term corresponds to the statistics of ordinary fermions, while the second term —to the half of an Aharonov-Bohm phase for *p* − 1 flux quanta pinned to each electron (*p* is an odd number). We have already mentioned that this is usually presented as an explanation of the Laughlin correlations.

To understand how the assertion of *ϕ*0 results in the unification of IQHE and FQHE, it is useful to first consider non-interacting electrons in a completely filled Landau level, *ν* \* =*n* (with integer *n*). In this case an incompressible state is produced, thus, particles experience the integer (Hall) phenomenon. Furthermore, the evidenced magnetic field, | *<sup>B</sup>* \*| <sup>=</sup>*ϕ*0*<sup>N</sup>* / (*S<sup>ν</sup>* \* ), can be either positive or negative.

Now attach to every electron (e.g. by asserting an infinitely thin, massless solenoid) an even number of flux quanta pointing in +*z* direction. This converts electrons into composite fermions. Since these quasi-particles witness an unchanged magnetic field (*B*\* ), we may expect them to also experience the IQHE (despite a different general structure of the energy spectrum compared to ordinary fermions). Finally, we have transformed an incompressible state of electrons into an incompressible state of CFs.

In the last step we adiabatically spread flux quanta pinned to electrons, until they become a part of the external magnetic field with an enhanced (or diminished) strength,

$$B = (p - 1)\frac{\phi\_0 N}{S} \pm \frac{1}{\nu^\*} \frac{\phi\_0 N}{S} \tag{10}$$

connected with a fractional filling factor of the form (the obtained hierarchy contains all fillings from the basic set – *n* = 1 and *p* = *q* – but it includes many other ratios too),

$$\mathcal{V} = \frac{1}{\frac{1}{(p-1) \pm \frac{1}{\nu^\*}}} = \frac{n}{\frac{n}{n(p-1) \pm 1}}$$

The implemented assumption that a uniform magnetic field can be obtained from detaching localized *ϕ*0 was justified—in the original paper [16]—by the uniform density of an initial (IQHE) state. Note that the minus sign in the above equation arises from the possibility of a

negative initial field, *B*\* . Since , the resulting *B*-field is always positive and points in a + z direction. Eventually, we have successfully connected the FQHE of electrons (ν) with the IQHE of composite fermions (*ν* \* =*n*)—if and only if we assume that, during this smearing process, only quantitative changes occur in the energy spectrum. Thus, one can say that electrons effectively absorbed *p* − 1 flux quanta from the external magnetic field to transform into CFs, which evidence lower magnetic field and form integer quantum Hall state.

Popularity of the CF theory is owed to the fact that it is able to explain most of (experimentally observable) fractions from the LLL. It also gives a prescription for constructing ground-state (as well as excited-state) wave functions within the lowest Landau band. However, scientists have already stumbled upon a difficulty—the pyramids of fillings from higher levels are not counterparts of the Jain's hierarchy. It is, for example, impossible to justify the appearance of Hall plateaus in the vicinity of *ν* =5 / 2 and 7/2. To deal with this problem, residual interactions between composite particles were introduced to the considerations and they were supposed to turn some of them into higher order composite fermions with different Landau levels. This approach, unfortunately, was rather unsuccessful but, due to the lack of a better explanation, it is still (quite) widely used [19–21].

where the first term corresponds to the statistics of ordinary fermions, while the second term —to the half of an Aharonov-Bohm phase for *p* − 1 flux quanta pinned to each electron (*p* is an odd number). We have already mentioned that this is usually presented as an explanation of

To understand how the assertion of *ϕ*0 results in the unification of IQHE and FQHE, it is useful

integer *n*). In this case an incompressible state is produced, thus, particles experience the integer (Hall) phenomenon. Furthermore, the evidenced magnetic field, | *<sup>B</sup>* \*| <sup>=</sup>*ϕ*0*<sup>N</sup>* / (*S<sup>ν</sup>* \*

Now attach to every electron (e.g. by asserting an infinitely thin, massless solenoid) an even number of flux quanta pointing in +*z* direction. This converts electrons into composite

them to also experience the IQHE (despite a different general structure of the energy spectrum compared to ordinary fermions). Finally, we have transformed an incompressible state of

In the last step we adiabatically spread flux quanta pinned to electrons, until they become a

\*

n

connected with a fractional filling factor of the form (the obtained hierarchy contains all fillings

 f

*n*

=- ± (10)

= = **<sup>1</sup>** (11)

=*n*)—if and only if we assume that, during this smearing

. Since , the resulting *B*-field is always positive and points

( ) 0 0

<sup>1</sup> <sup>1</sup> *N N B p S S* f

\*

n

1 ( 1) ( 1)


The implemented assumption that a uniform magnetic field can be obtained from detaching localized *ϕ*0 was justified—in the original paper [16]—by the uniform density of an initial (IQHE) state. Note that the minus sign in the above equation arises from the possibility of a

in a + z direction. Eventually, we have successfully connected the FQHE of electrons (ν) with

process, only quantitative changes occur in the energy spectrum. Thus, one can say that electrons effectively absorbed *p* − 1 flux quanta from the external magnetic field to transform

Popularity of the CF theory is owed to the fact that it is able to explain most of (experimentally observable) fractions from the LLL. It also gives a prescription for constructing ground-state

into CFs, which evidence lower magnetic field and form integer quantum Hall state.

*n p <sup>p</sup>*

=*n* (with

), we may expect

), can

to first consider non-interacting electrons in a completely filled Landau level, *ν* \*

fermions. Since these quasi-particles witness an unchanged magnetic field (*B*\*

part of the external magnetic field with an enhanced (or diminished) strength,

from the basic set – *n* = 1 and *p* = *q* – but it includes many other ratios too),

*v*

1

the Laughlin correlations.

8 Recent Advances in Graphene Research

be either positive or negative.

negative initial field, *B*\*

the IQHE of composite fermions (*ν* \*

electrons into an incompressible state of CFs.

Moreover, recent experiments on graphene bilayers confirmed that this quasi-particle ap‐ proach, regardless of its undeniable advantages, cannot embrace the entire physics behind the FQHE [22–24]. This is due to the fact that the most robust plateau in a transverse resistance in samples consisted of two weakly coupled sheets of atoms—develops near a half filling (and not a one-third filling, as expected from the CF). Additionally, the improvement of sample quality led to the discovery of novel incompressible states that fall out of the Jain's pyramid of fillings, even in the lowest Landau level of standard GaAs/AlGaAs structures. Among them, there are filling factors with odd denominators (like 4/11) and with even denominators (like 3/8).

As a result of the hunt for a model which can find a solution for all mentioned problems, the topological approach to quantum Hall effects was formulated [9]. In this chapter, we briefly recall its basic assumptions and we demonstrate that it provides a topological explanation of Jain's approach (Section 2). Additionally, we demonstrate that it can be used to explain all incompressible Hall-like states observed in conventional 2DEG, monolayer, and bilayer graphene samples (Section 3). We compare our predictions with experimental findings (Section 4).
