**2. Non-local theory of Hall systems: the origin and generalization of composite fermions**

The non-local theory of Hall systems is entirely based on the well-known mathematical concept of braid groups [25]. The full braid group, *π*<sup>1</sup> (Ω), is a homotopy group of an *N*-particle system configuration space [26],

$$\Omega = \left(M^N \mid \Delta\right) / S\_N \tag{12}$$

where *M* is a manifold on which particles are placed, *SN* stands for the permutation group of *N* elements (its appearance results from the assumed indistinguishability of carriers) and Δ removes points (creates topological defects) for which positions of at least two classical particles, as *M <sup>N</sup>* coordinates, are the same. Hence, it is consisted of closed trajectories—though the quotient group structure allows initial and final orderings of particles to differ by a permutation—performed in Ω and organized in homotopy classes. In the case of simply connected manifold, *π*<sup>1</sup> (Ω) is generated by *σ<sup>i</sup>* elements which describe simple exchanges of *i*'th and (*i* + 1)'th carriers (classical—as *M <sup>N</sup>* coordinates). It is worth emphasizing that *quantum particles do not travel braid trajectories*. However, a relationship between topological properties (reflected in a braid group form) with quantum properties of an arbitrary system cannot be ignored. Note that in the standard quantization method, multi-particle state vectors, *Ψ<sup>N</sup>* , are selected as functions from Ω into complex numbers [26]. Thus, when arguments of *Ψ<sup>N</sup>* are encircling a closed path in their configuration space, the multi-particle wave function gains a phase equal to a one-dimensional unitary representation (1DUR) of the corresponding braid from *π*<sup>1</sup> (Ω). Additionally, in the Feynman path integral formulation of the propagator [27], an additional summation over classes of homotopical trajectories needs to be implemented for multiply connected Ω (one measure, *dλ*, in a whole path space cannot be defined). The weight factors emerging in this summation are defined by a one-dimensional unitary representation of *π*<sup>1</sup> (Ω). Finally, the braid group shape and its 1DUR determine a quantum statistics of particles in the system (the allowed types of particles are settled by the topology of a manifold and by certain external factors—e.g. anyons may exist only in 2D spaces).

Let us now move back to Hall systems. It seems reasonable to assume that in the presence of a strong magnetic field, trajectories representing elements of the system braid group are of cyclotron orbit type—although, in general, they are not ordinary circles for highly interacting particles. In the topological approach, we define the surface of an LLL cyclotron orbit (as a representative of a braid from *π*<sup>1</sup> (Ω) by the archetype of a correlated incompressible state—

the IQHE state. Hence, a plaque which encircles an area equal to *<sup>S</sup> <sup>N</sup>*<sup>0</sup> <sup>=</sup> *<sup>ϕ</sup>*<sup>0</sup> *<sup>B</sup>* <sup>=</sup> *hc eB* and embraces exactly one flux quantum can be identified with a cyclotron path in the lowest Landau band. In great magnetic fields (as for partial fillings of the LLL), this enclosed surface may be too small to allow an arbitrary particle (as an argument of *ΨN* or *<sup>M</sup> <sup>N</sup>* coordinate) to reach its nearest neighbor—the Coulomb repulsion protects a uniform distribution. In this situation *simple exchanges, σ<sup>i</sup> , are unenforceable and should be excluded from the braid group describing the system*. Note that it may be impossible to organize a *π*<sup>1</sup> (Ω)-subgroup (which satisfy group axioms) from remaining classes of trajectories. However, we have already established that the FQHE can be evidenced only for correlated many-body states and, thus, it requires a determined statistics of particles (and a determined system braid group). Seeing that this phenomenon is actually observed in transport measurements, the *π*<sup>1</sup> (Ω) reduction needs to result in the emergence of a new group (*a cyclotron subgroup*), at least for selected fillings. Generators *bi* of a latter subgroup stand for novel, multi-loop exchanges of carriers (loopless, *σ<sup>i</sup>* , cannot be defined). Finally, each pair—a subgroup of the full braid group and a condition, a magnetic field strength, upon which this subgroup characterizes the multi-particle system—represents (and can be used to identify) a FQHE state in the LLL [28].

As we have already mentioned, cyclotron subgroups are always generated with multi-loop exchanges. Thus, trajectories (circled in *M*-space) representing *bi* elements need to be open they need to contain an integer number of closed loops, *n*, and exactly one half-loop. Simul‐ taneously, cyclotron orbits (representatives of braids) are closed by the definition. As a result, connected manifold, *π*<sup>1</sup>

10 Recent Advances in Graphene Research

representative of a braid from *π*<sup>1</sup>

Note that it may be impossible to organize a *π*<sup>1</sup>

actually observed in transport measurements, the *π*<sup>1</sup>

(and can be used to identify) a FQHE state in the LLL [28].

from *π*<sup>1</sup>

of *π*<sup>1</sup>

*exchanges, σ<sup>i</sup>*

(Ω) is generated by *σ<sup>i</sup>*

and by certain external factors—e.g. anyons may exist only in 2D spaces).

the IQHE state. Hence, a plaque which encircles an area equal to *<sup>S</sup>*

*i*'th and (*i* + 1)'th carriers (classical—as *M <sup>N</sup>* coordinates). It is worth emphasizing that *quantum particles do not travel braid trajectories*. However, a relationship between topological properties (reflected in a braid group form) with quantum properties of an arbitrary system cannot be ignored. Note that in the standard quantization method, multi-particle state vectors, *Ψ<sup>N</sup>* , are selected as functions from Ω into complex numbers [26]. Thus, when arguments of *Ψ<sup>N</sup>* are encircling a closed path in their configuration space, the multi-particle wave function gains a phase equal to a one-dimensional unitary representation (1DUR) of the corresponding braid

(Ω). Additionally, in the Feynman path integral formulation of the propagator [27], an

(Ω). Finally, the braid group shape and its 1DUR determine a quantum statistics of particles in the system (the allowed types of particles are settled by the topology of a manifold

(Ω) by the archetype of a correlated incompressible state—

*<sup>N</sup>*<sup>0</sup> <sup>=</sup> *<sup>ϕ</sup>*<sup>0</sup> *<sup>B</sup>* <sup>=</sup> *hc*

(Ω)-subgroup (which satisfy group axioms)

(Ω) reduction needs to result in the

*eB* and embraces

of

, cannot be

additional summation over classes of homotopical trajectories needs to be implemented for multiply connected Ω (one measure, *dλ*, in a whole path space cannot be defined). The weight factors emerging in this summation are defined by a one-dimensional unitary representation

Let us now move back to Hall systems. It seems reasonable to assume that in the presence of a strong magnetic field, trajectories representing elements of the system braid group are of cyclotron orbit type—although, in general, they are not ordinary circles for highly interacting particles. In the topological approach, we define the surface of an LLL cyclotron orbit (as a

exactly one flux quantum can be identified with a cyclotron path in the lowest Landau band. In great magnetic fields (as for partial fillings of the LLL), this enclosed surface may be too small to allow an arbitrary particle (as an argument of *ΨN* or *<sup>M</sup> <sup>N</sup>* coordinate) to reach its nearest neighbor—the Coulomb repulsion protects a uniform distribution. In this situation *simple*

from remaining classes of trajectories. However, we have already established that the FQHE can be evidenced only for correlated many-body states and, thus, it requires a determined statistics of particles (and a determined system braid group). Seeing that this phenomenon is

emergence of a new group (*a cyclotron subgroup*), at least for selected fillings. Generators *bi*

defined). Finally, each pair—a subgroup of the full braid group and a condition, a magnetic field strength, upon which this subgroup characterizes the multi-particle system—represents

As we have already mentioned, cyclotron subgroups are always generated with multi-loop exchanges. Thus, trajectories (circled in *M*-space) representing *bi* elements need to be open they need to contain an integer number of closed loops, *n*, and exactly one half-loop. Simul‐ taneously, cyclotron orbits (representatives of braids) are closed by the definition. As a result,

a latter subgroup stand for novel, multi-loop exchanges of carriers (loopless, *σ<sup>i</sup>*

*, are unenforceable and should be excluded from the braid group describing the system*.

elements which describe simple exchanges of

the simplest non-trivial cyclotron paths are generated by *bi* <sup>2</sup> elements and, hence, they consist an odd number of loops, 2(*n* + 1 / 2)=2*n* + 1. We are going to demonstrate why multi-loop exchanges of particles (as arguments of *ΨN* or *<sup>M</sup> <sup>N</sup>* coordinates) can be permissible in the system, even when loopless ones are not. For simplicity we restrict (at first) our considerations to 1 / *q* (*q* – odd) fillings of the lowest Landau level. In this case, a single-loop path of a carrier experience *q* flux quanta per particle in the system and—since an LLL orbit embraces exactly one *ϕ*0 – its surface is not large enough to reach a neighboring carrier (*σ<sup>i</sup>* exchanges are unfeasible). In 3D manifolds this also applies to multi-loop trajectories (meaning that effective dimensions of loops are not raised; *σ<sup>i</sup>* exchanges are, however, always accessible because particles can move freely in the direction parallel to the magnetic field)—each loop increases the total surface encircled by the path, as a circumvolution adds a new surface to the coil. Therefore, the latter also increases the magnetic field flux evidenced by the path—an individual loop experience *q* flux quanta per particle too. In 2D manifolds, however, an additional loop cannot enhance path's area and the evidenced flux remains unchanged (the whole trajectory, which represents an element of the system braid group, experiences *q* flux quanta per particle). As a consequence, all loops must share the total *BS* / *N* per particle quantity, which passes through a single-loop path—each loop receives a diminished portion of *BS* / *N* and experiences a lower effective magnetic field, *B* \* . Additionally, if the number of loops in a trajectory coincides with the inverse of an LLL filling factor, *ν* <sup>−</sup><sup>1</sup> =*q*, then every loop evidences exactly one *ϕ*0 per particle in the system (*BS* / (*qN* )= *BS* / *N*<sup>0</sup> =*ϕ*0) and an effective magnetic field defined by the relation,

$$\frac{B^\*S}{N} = \frac{1}{q}\frac{BS}{N} \to B^\* = \frac{B}{q} \tag{13}$$

Since on the LLL cyclotron orbit falls a single magnetic field flux quantum, the surface encircled by an arbitrary loop (from the *q*-loop trajectory) is large enough to reach a neighboring particle (as an argument of *ΨN* or *<sup>M</sup> <sup>N</sup>* coordinate). *Although the loopless exchanges are not permitted in the system, we have just demonstrated that ( <sup>q</sup> <sup>−</sup> <sup>1</sup> <sup>2</sup> )-loop ones are accessible and they generate the cyclotron subgroup of π1(Ω)*. The explicit form of *bi* elements for a 1 / *q* filling of the lowest Landau level is presented below,

$$b\_i^{(q)} = \sigma\_i^q, \quad 1 < i < N - 1 \tag{14}$$

with 1DURs of the form (*θ* =*π* for composite fermions—we stick to this generally confusing name for history reasons),

$$b\_i^{(q)} \to e^{iq\theta}, \quad 1 < i < N - 1 \tag{15}$$

Note that the even-denominator rule (*q*– odd) follows immediately from the requirement of an open trajectory (as an *M*-space representative) for *bi* . Additionally, Laughlin correlations seem to follow from this cyclotron subgroup formalism, rather than from quasi-particles formulation with auxiliary objects pinned to electrons [10].

Consider now a filling factor, which does not belong to the basic set of fillings (*ν* ≠1 / *q*). In this situation, it is impossible for every loop from the *q*-loop trajectory to embrace exactly one flux quantum attributed to a single particle. We can, however, assume that all *q* −1 loops experience

a single *ϕ*0 per particle and only last loop evidences a residual (per particle) portion, ± 1 *<sup>x</sup> ϕ*0. Thus, the total magnetic field flux per particle, *BS* / *N* , can be written as follows,

$$\frac{BS}{N} = \frac{1}{\nu} \phi\_0 = \left(q - 1\right)\phi\_0 \pm \frac{1}{\chi} \phi\_0 \tag{16}$$

A fortunate situation occurs when *x* is an integer number. In this case the last loop can embrace an entire flux quantum, as the LLL cyclotron orbit does, when it fits perfectly to the separation of an electron (as an argument of *ΨN* or *<sup>M</sup> <sup>N</sup>* coordinate) and its arbitrary *x*'th order neighbor. As a result, *the closing loop (as a representative of a winding of a cyclotron subgroup braid) defines the type of accessible exchanges, which are not consisted of q loopless σ<sup>i</sup> (these are not accessible for the system), but are integral exchanges of multi-loop type*. Thus, for filling factors included in the Jainlike hierarchy [16, 17],

$$\nu = \left( (q - \mathbf{l}) \pm \frac{1}{\mathbf{x}} \right)^{-1} \tag{17}$$

a cyclotron subgroup of the full braid group can be defined and it is generated by elements,

$$b\_i^{(q)} = \sigma\_i^{q-1} \sigma\_i \sigma\_{i+1} \dots \sigma\_{i+x-1}{}^{\pm 1} \dots \sigma\_i^{-1} \sigma\_{i+1}{}^{-1} \tag{18}$$

The quantum statistics follows immediately from the 1DUR (*θ* =*π* for composite fermions),

$$b\_i^{(q)} \to e^{i(q-1\pm 1)\theta}, \quad 1 < i < N-1 \tag{19}$$

In the above considerations, we assumed that the last loop of a *q*-looped path (as a represen‐ tative of a trajectory circled in the Ω space) can be overwrapped in a direction opposite to one enforced by the external magnetic field. The latter results in an appearance of the minus sign in Eqs. (16)–(19).

It is reasonable to consider even more complex commensurability conditions (of an area encircled by each loop and a surface attributed to a single particle or a group of *x* particles)—

#### 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 13

where *q* −1 loops, which constitute a simplest non-trivial cyclotron path, are divided into sets characterized by different integer numbers, *a* ≤*b* ≤*c*… (different evidenced fractions of flux quantum per particle, <sup>1</sup> *<sup>a</sup> ϕ*<sup>0</sup> ≥ 1 *<sup>b</sup> ϕ*<sup>0</sup> ≥ 1 *<sup>c</sup> ϕ*0… ). These sets need to be of even multitudes, *α* ≥*β* ≥*γ* …, to keep a rational character of the exchange braid,

Note that the even-denominator rule (*q*– odd) follows immediately from the requirement of

seem to follow from this cyclotron subgroup formalism, rather than from quasi-particles

Consider now a filling factor, which does not belong to the basic set of fillings (*ν* ≠1 / *q*). In this situation, it is impossible for every loop from the *q*-loop trajectory to embrace exactly one flux quantum attributed to a single particle. We can, however, assume that all *q* −1 loops experience

a single *ϕ*0 per particle and only last loop evidences a residual (per particle) portion, ±

0 00 ( ) 1 1 <sup>1</sup> *BS <sup>q</sup> N x* f

 ff

A fortunate situation occurs when *x* is an integer number. In this case the last loop can embrace an entire flux quantum, as the LLL cyclotron orbit does, when it fits perfectly to the separation of an electron (as an argument of *ΨN* or *<sup>M</sup> <sup>N</sup>* coordinate) and its arbitrary *x*'th order neighbor. As a result, *the closing loop (as a representative of a winding of a cyclotron subgroup braid) defines the*

*system), but are integral exchanges of multi-loop type*. Thus, for filling factors included in the Jain-

<sup>1</sup> ( 1) *<sup>q</sup> <sup>x</sup>*

() 1 1 11

( ) ( 1 1) ,1 1 *q iq <sup>i</sup> b e iN* ® << - - ± q

 s

*i i i i ix i i b*

a cyclotron subgroup of the full braid group can be defined and it is generated by elements,

The quantum statistics follows immediately from the 1DUR (*θ* =*π* for composite fermions),

In the above considerations, we assumed that the last loop of a *q*-looped path (as a represen‐ tative of a trajectory circled in the Ω space) can be overwrapped in a direction opposite to one enforced by the external magnetic field. The latter results in an appearance of the minus sign

It is reasonable to consider even more complex commensurability conditions (of an area encircled by each loop and a surface attributed to a single particle or a group of *x* particles)—

11 1

 s s

æ ö = -± ç ÷

1


= =- ± (16)

è ø (17)

(19)


Thus, the total magnetic field flux per particle, *BS* / *N* , can be written as follows,

n

*type of accessible exchanges, which are not consisted of q loopless σ<sup>i</sup>*

*q q*

s ss

n

like hierarchy [16, 17],

in Eqs. (16)–(19).

. Additionally, Laughlin correlations

 *(these are not accessible for the*

1 *<sup>x</sup> ϕ*0.

an open trajectory (as an *M*-space representative) for *bi*

12 Recent Advances in Graphene Research

formulation with auxiliary objects pinned to electrons [10].

$$b\_i^{(q)} = \left(\sigma\_i \sigma\_{i+1} \dots \sigma\_{i+a-1} \dots \sigma\_i^{-1} \sigma\_{i+1}\right)^a \cdot \left(\sigma\_i \sigma\_{i+1} \dots \sigma\_{i+b-1} \stackrel{\pm 1}{\dots} \sigma\_i^{-1} \sigma\_{i+1}\right)^\beta \cdot \dots \tag{20}$$


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, filling factors of monolayer graphene Hall states (which develop in transport measurements) are underlined.

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

The elements above generate cyclotron subgroups for filling factors from the generalized hierarchy of the form,

$$\mathbf{v} = (\alpha \mid a \pm \beta \mid b \pm \gamma \mid c \pm \cdots)^{-1} \tag{21}$$

1DURs of the resulting subgroups can also be easily estimated (the scalar representation of a subgroup can be identified with reduced representation of the full *π*<sup>1</sup> (Ω)),

$$b\_i^{(q)} \to e^{\ell(a \pm \beta \pm \gamma \pm \ldots)\theta}, \quad 1 < i < N - 1 \tag{22}$$

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**. Note that among these fillings one can find all famous and mysterious *v*, like <sup>3</sup> <sup>8</sup> , <sup>4</sup> <sup>11</sup> , <sup>4</sup> <sup>13</sup> and 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 resistivity.
