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Fractional quantum Hall effect revisited J. Jacak, P. Łydżba n, L. Jacak Faculty of Fundamental Problems of Technology, Wroclaw University of Technology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland

art ic l e i nf o

a b s t r a c t

Article history: Received 17 April 2015 Accepted 21 July 2015

The topology-based explanation of the fractional quantum Hall effect (FQHE) is summarized. The cyclotron braid subgroups crucial for this approach are introduced in order to identify the origin of the Laughlin correlations in 2D (two-dimensional) Hall systems. Flux-tubes and vortices for composite fermions in their standard constructions are explained in terms of cyclotron braids. The derivation of the hierarchy of the FQHE is proposed by mapping onto the integer effect within the topology-based approach. The experimental observations of the FQHE supporting the cyclotron braid picture are reviewed with a special attention paid to recent experiments with a suspended graphene. The triggering role of a carrier mobility for organization of the fractional state in Hall conﬁguration is emphasized. The prerequisites for the FQHE are indicated including topological conditions substantially increasing the previously accepted set of physical necessities. The explanation of numerical studies by exact diagonalizations of the fractional Chern insulator states is formulated in terms of the topology condition applied to the Berry ﬁeld ﬂux quantization. Some new ideas withz regard to the synthetic fractional states in the optical lattices are also formulated. & 2015 Published by Elsevier B.V.

Keywords: Quantum Hall effects Braid groups Cyclotron braid subgroups Topological insulators Graphene

1. Introduction Observation of the fractional quantum Hall effect (FQHE) was one of the most important discoveries of the 20th century. The experiment carried out by Tsui et al. revealed plateaus in the longitudinal resistance appearing concomitantly with dips in the transverse one for 2DEG partially ﬁlling the lowest Landau level (LLL) upon strong magnetic ﬁelds and temperatures below 4 K [1]. An origin of the elder integer quantum Hall effect (IQHE) was explained shortly after its discovery within a single particle approach including topology arguments [2]. Actually, the ﬁrst explanation was even simpler—it is assumed that for completely ﬁlled Landau levels (LLs) an electron cannot scatter between different one particle states and a current cannot ﬂow in the direction of a voltage ( Rxx = 0). In opposition, the fractional quantum Hall effect is a collective phenomenon being a manifestation of strong interparticle correlations and, despite the intensive research, its nature is still not fully understood. The basic prerequisite for the FQHE formation is the ﬂat band with quenched kinetic energy, as in the almost degenerated LLL in the presence of interaction (and massively degenerated without interactions). Reducing of the kinetic energy role allows for the subtle interaction effects resulting in the organization of correlated multiparticle n

Corresponding author. E-mail addresses: [email protected] (J. Jacak), [email protected] (P. Łydżba), [email protected] (L. Jacak).

states. An important role is played by a very special 2D topology— there is no evidence of the FQHE in three-dimensional (3D) samples. The ﬁrst step towards the description of correlations in the LLL was taken by Laughlin. He proposed a wave function for 1 (q-odd) q

ﬁlling factors formed with a Jastrow polynomial and a Gaussian factor [2] N

ΨL (z1, …zN ) =

∏

N

2 /4l2)

(zi − zj )qe− ∑i = 1(| zi |

i, j = 1, i > j

. (1)

where zi = xi + iyi is a complex position of ith particle on a plane,

l=

=c eB

is a magnetic length. The representation of the Coulomb

repulsion in the form of Haldane pseudopotential revealed that this Laughlin function (LF) describes the exact ground state for N charged particles placed on a plane, if one neglects the long-range part of the Coulomb forces [4–6]. Division of the interaction for near- and long-range parts is expressed by its projection onto the relative angular momenta of particle pairs: values greater than q 2 correspond to the long-range tail, while values lower than q 2—to the near range part of the ﬁeld. It has already been proved that the long-range tail inﬂuences only slightly the exact ground state obtained only with a short-range part included. Note that the LF is actually a generalized Slater function [7] with a p power introduced in the Vandermonde polynomial

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2

1.5

T ~ 35 mK

7 11

Rxx (kΩ)

1.0

2 3

5 4 13 11

0.5

0.0

3 8 10 10 19 21

6

7

8

1 3

2 5

1 2

3 5

4 5 13 17

9

10

11

2 7

3 10 6 17 12

13

14

(

— ν = (q − 1) ±

MAGNETIC FIELD [T] Fig. 1. Observation of the FQHE in a GaAs/AlGaAs quantum well with an electron 2 2 density of 1011 1/cm 2. Rxx for > ν > at the temperature equal T ∼35 mK is 3 7 7 4 presented. The Hall resistance Rxy in the region of ν = and ν = is marked with 11 11 a dotted line (after Ref. [3]). Fractions outside the standard CF hierarchy are indicated in color. (For interpretation of the references to color in this ﬁgure caption, the reader is referred to the web version of this paper.) N

ΨS (z1, …zN ) =

∏ i, j = 1, i > j

N

2 /4l2)

(zi − zj ) e− ∑i = 1(| zi |

is the number of ﬂux quanta of an auxiliary magnetic ﬁeld) resulting from interactions, in analogy to the three-dimensional Landau quasiparticles in solids [9]. The proper phase shift is obtained due to the Aharonov–Bohm effect, since an exchanging composite particle sees the ﬂux quanta placed on the opposite fermion. The success of the CF construction lies in the possibility of reduction of the FQHE of electrons to the well-understood IQHE of composite fermions experiencing a lower effective magnetic ﬁeld (the localized magnetic ﬂux quanta screen an external ﬁeld, leading to q − (q − 1) = 1 quanta per composite particle). However, assertion of ﬂux tubes—as a result of the Coulomb repulsion forces —should not be used in a 2D space without proper explanation (the matrix element of the interaction is not a continuous function of distance). As a consequence, this simplistic, one-particle theory seems to model with artiﬁcial objects, rather than explain, the real complicated behavior of the particles. Nevertheless, the CF idea allows for estimating the main line of the FQHE hierarchy

, (2)

The main difference between these two functions is disclosed when two particles are exchanging on a plane—the phase shift obtained by the Laughlin function is p times greater (pπ) than by the Slater function (π). It should be mentioned that exchanges of particles located on locally 2D manifolds are considerably different than those for particles located on manifolds of higher dimensions. In the latter case the exchanges correspond to simple particle permutations and are expressed with algebraic properties of the multi-argument wave function. For 2D spaces the exchanges are related to the full braid group elements, different from simple particle permutations. The full braid group is actually a homotopy group (π1) of the conﬁguration space of the N indistinguishable particle system, so a group of classes of homotopic trajectories. Finally, association of the algebraic properties of multi-argument functions with exchanges of particles described by these functions may be misleading and exchanges of function arguments must be referred to the braid group distinct than the permutation group (Fig. 2). However, the mentioned difference is considered to be a hallmark of Laughlin correlations, even though the unitary factor eiqπ = − 1 equals to eiπ = − 1. So, this property does not allow us to distinguish correlated particles from ordinary fermions. Despite the fact that neglecting the topology and relying only on properties of the LF is insufﬁcient for a complete explanation of the FQHE origin, some quite successful theories where introduced. One of these theories is the so-called composite fermion (CF) model, proposed by Jain [8]. It assumes that these new particles are just electrons dressed with q 1 magnetic ﬁeld ﬂux-quanta (q

Fig. 2. The geometrical presentation of si—the generator of the full braid group of R2 space and σi−1—its inverse; in 2D σi2 ≠ e .

1 −1 n

)

=

n (q − 1) n ± 1

(q—odd integer, n—integer) [8]—

corresponding to the complete ﬁlling of the nth LL in the screened magnetic ﬁeld. This resultant effective ﬁeld can be oriented along or opposite to the original one, thus 7 is appearing in the ﬁlling factor expression. The compatibility of the hierarchy with the experiment suggests that despite all problems and ambiguities mentioned above, it models some more fundamental properties of 2D systems in strong magnetic ﬁelds. For a relatively long time these properties were not recognized. A progress was achieved recently [10] in terms of cyclotron braid subgroups, which will be also presented in more details in the present paper. The competitive construction of CFs was formulated shortly afterwards by Read [11,12]. This formulation is based on the conception of collective ﬂuid-like objects called vortices, which are characterized with q-vorticity and are similar to the well-known constructions present in superﬂuid systems. The vortices are pinned to bare fermions—resulting complexes are also called composite particles (not only composite fermions, but also composite bosons), since they reproduce the Laughlin correlations [11]. However, the vortex is expressed with a fragment of the Jastrow N polynomial— V (z ) = ∏ j = 1 (zj − z )q . Thus, all particles contribute to the vortex deﬁnition and the vorticity coincides with the q-power in the LF. Therefore, it is not surprising that fermions dressed with vortices (when the argument z is assumed zi) reproduce the Laughlin correlations, since the vortex notion arises immediately from a simple decomposition of the LF with the vorticity taken from the Jastrow polynomial known in advance. Thus, this competitive conception do not explain the FQHE origin (it can be rather understand as a different representation of the LF) and its signiﬁcance is rather of illustrative type. Both types of composite particles, with vortices or with ﬂux tubes, are thus phenomenological in nature, and the question arises as of what is a more fundamental reason of Laughlin correlations in 2D charged systems upon sufﬁciently strong magnetic ﬁeld and of how are they linked to speciﬁc 2D topology. The role of topology in the strongly correlated state creation was noticed [13– 15] in the context of exceptional topological properties of a plane and locally 2D manifolds like sphere or torus. This unique topology of planar systems is linked with an exceptionally rich structure of their braid groups in comparison to ones of higher dimensional spaces (Rd, d > 2) [16]. As it was already mentioned, the full braid group is a group of multi-particle closed trajectory classes, disjoint and topologically nonequivalent (trajectories from different classes cannot be continuously deformed one into another). In the case of 2D spaces the full braid group is inﬁnite, while for manifolds of higher dimensions it is ﬁnite and equal to SN—the permutation group of N elements [16]. This property makes 2D systems exceptional in geometry–topology sense, which inherently lies in

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foundations of the FQHE. In the present paper we revisit the topological approach to Hall systems and recover Laughlin correlations by employing geometry properties of the cyclotron braids [17,18] in the framework of formal braid group approach and without invoking to any auxiliary elements. We will demonstrate that CFs are not composites of fermions with ﬂux-tubes or vortices and their statistic properties are not caused by any auxiliary objects, but are the rightful 2D particles characterized by appropriate one dimensional unitary representations (1DURs) of the cyclotron subgroups. We notice that the explanation of the effective ﬁeld creation at fractional ﬁllings of the LLL in terms of the cyclotron braid groups would also be helpful for the utilization of Chern–Simons ﬁeld constructions [19], which were widely spread in modeling of CFs and anyons within mathematical effective approach. These ﬁelds take advantage of properties of singular gauge transformations which ﬁx in mathematical sense the ﬁeld ﬂuxes or vortices to original particles undergoing these transformations. In must be, however, noted that these gauge transformations are not canonical ones and are as artiﬁcial as objects they create [19], i.e., they allow for calculus manipulation with the ﬂux-tubes or vortices but do not supply any explanation of them. They are introduced upon Chern– Simons ﬁeld approach ‘by hands’ though in an elegant mathematical manner.

2. Cyclotron trajectories too-short for interchanges in a strong magnetic ﬁeld 2.1. Two-dimensional manifolds without an external magnetic ﬁeld Matching the topological properties with quantum system properties can be achieved by quantization according to the Feynman path integral method. In the case of not simply-connected conﬁguration spaces, like for all multi-particle systems, additional summation over classes of homotopic trajectories [13,14,20] with the proper 1DUR of the full braid group deﬁning weight factors must be included in the Feynman integral deﬁnition [14,16,21]. In other words, if trajectories from the conﬁguration space are located in separated homotopy classes that are distinguished by non-equivalent closed loops (from π1) attached to an open trajectory λ a, b , then a single measure in the trajectory space dλ cannot be deﬁned and an additional summation over homotopy classes should be included in the path integral, but with appropriate (1DUR) weights-unitary factors [20]. Finally, the propagator—the integral for a transition from the point a at the time moment t = t1 to the point b at t = t2 [13,14]

Ia, t1→ b, t 2 =

∑ eiαl ∫ l ∈ π1

l

dλ l eiS [λ(a, b) ], (3)

where π1 stands for the full braid group, while l enumerates π1 group elements. Additionally, λl indicates an open trajectory λ (a, b) between a and b with lth loop added. Note that different 1DURs of the full braid group give rise to distinct types of quantum particles corresponding to the same classical ones. In this manner one can get fermions and bosons corresponding to 1DURs of the permutation group SN, σi → eiπ and σi → ei0 , respectively (si are generators of the full braid group in 2D). For the far more rich braid groups in 2D, however, one can obtain the inﬁnite number of possible anyons (including bosons an fermions) related to 1DURs, σi → eiΘ with Θ ∈ [0, 2π ) [13–16]. Note that it is impossible to associate CFs with the 1DURs of the full braid group, because 1DURs are periodic with a period of 2π , whereas the CFs require the statistics phase shift qπ , q-odd integer. In order to solve this problem, we propose to associate CFs with

3

the appropriately constructed braid subgroups instead of the π1 and in this way to distinguish them from ordinary fermions. 2.2. Two-dimensional manifolds in a strong magnetic ﬁeld The full braid group contains all accessible closed multi-particle classical trajectories, i.e., braids organized in homotopy classes. Initial and ﬁnal orderings of indistinguishable particles (traversing a closed classical loop) may differ by permutation. However, an inclusion of a strong magnetic ﬁeld may substantially change trajectories—a classical cyclotron motion may conﬁne a variety of accessible braids if magnetic ﬁeld is strong enough and the manifold is 2D. When the minimal particle separation (locked with the Coulomb repulsion forces) is greater than twice the cyclotron radius, the exchanges of particles along single-looped cyclotron trajectories are precluded, because the cyclotron orbits are too short. This situation occurs for fractional ﬁllings of the LLL. It is worth to mention that the interaction cannot enhance cyclotron orbit size in the uniform multiparticle system. In the braid picture, particles must interchange to deﬁne the statistics and to allow for the creation of the collective correlated state (like the FQHE). Therefore, in order to restore exchanges, the cyclotron radius must be enhanced somehow. This can be achieved by screening the external magnetic ﬁeld, like in the construction of the CFs with pinned ﬂux quanta [8], or by diluting a local charge, as for the CFs with q-vortices [11]. Both these constructions are, however, artiﬁcial and rather model another (fundamental) effect leading to the enlargement of cyclotron orbits in two-dimensional manifolds. We suppose that the natural way to obtain this enhancement is to exclude inaccessible braids from the π1. The remaining braids can be organized in classes forming the cyclotron subgroup of the full braid group. In the proceeding section we will demonstrate that a cyclotron subgroup is consisted of multi-looped braids and that they allow for the effective enlargement of cyclotron orbits, and thus for the restoring of particle exchanges in a natural way. Additionally, in the presence of magnetic ﬁeld, the summation in the Feynman propagator must be conﬁned to the elements of this subgroup (actually, for a ﬁxed magnetic ﬁeld and cyclotron loop orientation, of this semigroup, but with the same 1DURs as a whole subgroup).

3. Cyclotron braid subgroups The multi-looped braids, that are still accessible after the external magnetic ﬁeld is applied, form the cyclotron braid subgroup, which is generated with the following elements:

bi(q) = σiq

(q = 3, 5…),

i = 1, …, N − 1,

(4)

where each q corresponds to a different type of a cyclotron subgroup and si are the generators of the full braid group (loopless/ simple exchanges of neighboring particles). Furthermore, the group elements bi(q) represent the interchanges of ith and (i + 1) th particles with additional q − 1 loops, what becomes clear after 2 analyzing the deﬁnition of the simple interchange si (Fig. 3). Note that bi(q) generates a subgroup of the full braid group as they are expressed with the full braid group generators. The one-dimensional unitary representations of the full braid group conﬁned to the proper cyclotron subgroup are identical to 1DURs deﬁned directly from this subgroup

bi(q) → eiqα ,

i = 1, …, N − 1,

(5)

where q is an odd integer and α ∈ ( − π , π ]. These elements do not depend on the particle index i by a virtue of the si generators property [16,22]

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Fig. 3. The generator si of the full braid group and the corresponding relative trajectory of ith and (i + 1) th particles (upper). The generator of the cyclotron braid subgroup, 3−1 q−1 (here bi(q) = σiq (here q¼ 3), corresponding to additional = 1) loops (lower). 2R0 is the inter-particle separation, Rc is the cyclotron radius and a 3D view is added for a 2 2 better visualization.

σi σi + 1σi = σi + 1σi σi + 1

(6)

that is fulﬁlled for all indices from the interval 1 ≤ i ≤ N − 1. We argue that these 1DURs, enumerated by (q, α) pairs, describe

composite anyons (CFs if α = π ). Thus in order to distinguish various types of composite particles one has to consider 1DURs of the cyclotron braid subgroups and not of the full braid group as usually.

Fig. 4. Cyclotron trajectories of individual particles must be closed, therefore they need to be constructed with double exchanges (represented with double exchange braids) 1 1 for both simple generators ν ¼1 (a) and generators with additional loops ν = (the picture shows the exemplary ﬁlling factor of ν = ) (b). p

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In agreement with the general rules of quantization [15,21], when the particle traverses (in a classical terms) a closed loop in the conﬁguration space, the N-particle wave function must transform according to the 1DUR of the full braid group element, which corresponds to this trajectory. In this reasoning, the wave function acquires a phase shift after neighboring particles exchange positions (as arguments of the wave function). Using 1DURs from Eq. (5), the Aharonov–Bohm phase of CFs ﬁctitious ﬂuxes is replaced with the phase shift derived from the additional loops (if α ¼ π, as for CFs, each loop adds a 2π phase change, Fig. 3), which is equal to one required by the Laughlin correlations. Let us emphasize that the real particles do not traverse the braid trajectories, since the quantum particles do not traverse trajectories at all, but the exchanges of arguments of the N-particle wave function can be represented by braid group elements. Each additional loop of a relative trajectory for the particle pair exchange (as deﬁned by the generators bi(p) ) reproduces a supplementary loop in the individual trajectories of both interchanging particles. The relative trajectory is characterized by twice the radius of the individual particle trajectories. In the quantum BS hc language one can conclude only on the number, N / e (B—an external magnetic ﬁeld, S—a sample surface, N—a total number of particles, hc —a magnetic ﬁeld ﬂux quantum), of ﬂux quanta per e

single particle in the system. When the lowest Landau level is completely ﬁlled, a cyclotron orbit matches perfectly to the interparticle distance and so it is performed as a double (simple/ loopless) exchange of neighboring particles. In this case, when there is exactly one ﬂux quantum per electron in the system, one can deﬁne the cyclotron orbit (for all ﬁllings ν in the LLL) as a trajectory which embrace precisely one hc /e . When the number BS hc / N e

takes the value of p—the odd denominator ﬁlling factor is 1 —the p

equal to

cyclotron orbit still embrace one quantum, so it

cannot be implemented as a double (loopless) exchange. However, this number coincides with the number of individual particle closed loops resulting from a cyclotron subgroup generator perp−1 formed twice ( p = 2 2 + 1 = 2n + 1, where n = 1, 2… indicates how many additional loops are in a new generator deﬁnition)—cf. Fig. 4. Note that the individual trajectories for both interchanging particles are closed, only when their relative trajectory is also closed. In this way, they are associated with double exchanges of particle pairs (the braid trajectory is open)—cf. Fig. 5. Finally, if the interchange is simple—without any additional loops, the corresponding individual particle closed trajectories are also simple, i.e., single-looped. When the interchange of particles is multi-looped, as associated with the p-type cyclotron subgroup, the relative trajectory and individual trajectories of the double interchange p−1 have 2 2 + 1 = p closed loops [18,10]. One may, thus, notice that for ν =

1 p

each supplementary loop of an exchange braid (two

loops of closed path) is connected with two additional ﬂux quanta piercing the individual particle closed trajectories. The cyclotron orbit enlargement mechanism (and all properties

5

mentioned above) arises from the speciﬁc character of multilooped planar trajectories, that is very different from the character of multi-looped trajectories of 3D systems. It is important to emphasize that in 3D spaces any trajectory with many loops can be represented as a coil. So, every additional loop adds a new portion of external magnetic ﬁeld ﬂux piercing the trajectory, as a circumvolution adds a new surface to a coil. In 2D case, multi-looped trajectories do not enhance the surface of the system and therefore do not enhance the total ﬂux. Thus, all loops must share the same total BS /N , which results in diminishing ﬂux-portion per a single hc

loop (now equal to one, since p e divided into p loops equals to hc 1 e ).

Finally, the multi-looped trajectory is a proper effective (but multi-looped) cyclotron trajectory with elongated radius allowing for the exchanges of neighboring particles. The additional loops in 2D take away the ﬂux-portions (equal to q 1 quanta at ν = q1 , q odd), hence one can imagine that the last

loop experiences a lower magnetic ﬁeld, just like in a Jain's simplistic model with ﬂuxes screening the external ﬁeld B. This means that CFs just model the 2D rightful particles corresponding to the 1DUR of the full braid group reduced to the cyclotron subgroup due to the presence of an external magnetic ﬁeld (and by too short single-looped cyclotron trajectories). However, the original name ’composite fermions’ can be still used for the history reason. Moreover, one can also deﬁne ’composite anyons’ for particles associated with fractional 1DURs of the cyclotron subgroup (when α has a fractional value) connected rather with ordinary anyons, when there is no external magnetic ﬁeld.

4. Mapping of the FQHE onto the IQHE: the hierarchy out of ﬁllings

1 q

It is worth to emphasize that the agreement between the number of external magnetic ﬁeld ﬂux quanta per particle and a number of loops in a (closed) cyclotron trajectory is held only for 1 q

(q—odd) ﬁllings of the LLL. In fact, this accordance allows for a formulation of Jain's model of CFs, where each additional loop is represented as an additional ﬂux-tube attached to a fermion. Nonetheless, out of this basic ﬁllings the number of loops (always an integer) cannot be equal to the portion of non-integer ﬂux quanta per particle. So, one can assume that ﬁrst q 1 loops take a full ﬂux quantum each, while a fractional leftover portion falls on a last loop and the external magnetic ﬁeld is taken away by all loops together. In the conventional CF model, one deals with the conception, that even for ﬁllings out of the basic set ν = q1 , an integer number q 1 of rigid ﬂux quanta is attached to particles. In particular, the resultant effective ﬁeld experienced by CFs can be oriented op1 positely to the B ﬁled as for ν > 2 and q ¼3 [8,23]. This image leads to the hierarchy obtained via mapping of the FQHE of electrons onto the IQHE of CFs ( νCF = n) [8]

⎛ 1 ⎞−1 n ν = ⎜ (q − 1) ± ) ⎟ = ⎝ n ⎠ (q − 1) n ± 1

(7)

It is obvious that any ﬂux-tubes pinned to the CFs do not exist, but they provide a convenient model for additional loops appearing in the cyclotron trajectory in a classical cyclotron braid picture, when simple loopless orbits become to short for particle exchanges. Exceptionally for ν = q1 these loops would be imagined as of q 1

Fig. 5. The exchanges of neighboring particles are unforceable for equidistantly distributed 2D particles when the cyclotron radius Rc is smaller than a half of the particle separation distance.

ﬂux quanta attached to particles and oppositely oriented to external ﬁeld, but out of these ﬁllings, not. Nevertheless, in order to match the cyclotron subgroup model to Jain's composite fermion conception for ﬁlling factors out of a basic set, one can consider a

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situation when all loops, except from the last one, of the multilooped trajectory embrace the integer number of ﬂux quanta. In this approach only the last loops take away a remaining fractional portion. Note that in this case, the ﬂux quantization rule is not fulﬁlled with regard to the last loop (the cyclotron orbit is deﬁned with a one ﬂux quantum as for ν ¼1). However, when the remaining ﬂux hc is equal to the nth part of the entire ﬂux quantum, ± ne , then one could expect an organization of the collective correlation (every hc hc nth particle would exchange, since en n = 1 e ). The sign of the residual ﬂux passing through the last loop can be positive, or even negative. The latter case might manifest itself in the opposite direction of the last loop (contrary to the direction forced by the external magnetic ﬁeld). Finally, this would result in something like an eight number-shape structure of the eventual individual cyclotron trajectory. Such a picture (given by Jain's FQHE hierarchy n ν = n (q − 1) ± 1 ) satisﬁes, however, the requirements of the ﬂux quantization and explains the heuristic assumption of Jain's composite fermion construction. To verify the presented concept, one can conduct a very simple experiment based on a cyclotron focusing of a beam of 2D ballistic carriers passing through the nanometer-scale source slot [24]. This beam is bend to the left or to the right (there are two possibilities of the cyclotron movement direction for the last loop) due to the external magnetic ﬁeld presence. The resonant signal is observed in one of the target slots (left or right), when its separation from the nano-scale source is equal to the cyclotron orbit diameter [24,23]. An asymmetry should be observed when one is lowering 1 (or rising) the magnetic ﬁeld strength near (when passing) ν = 2 ﬁlling factor. One should already capture a very interesting issue arising from the cyclotron subgroup deﬁnition of CFs. In the latter model the composite fermions are not ordinary fermions dressed with Coulomb repulsion forces, but are separated, rightful 2D quantum particles. Thus, they cannot be mixed with ordinary fermions, like ordinary fermions cannot be mixed with ordinary bosons or other anyons. It is especially essential when one conducts numerical variational interaction minimization or diagonalization, because only antisymmetric functions that describe CFs—that transform themselves according to appropriate 1DUR of the cyclotron subgroup—should be taken into account. Such functions create a subspace of the Hilbert space of antisymmetric functions and a minimization done on the whole domain may lead to the incorrect results. 4.1. Additional loops vs. the cyclotron radius enhancement The braid group approach does not describe the speciﬁed classical trajectories of particles placed on 2D manifolds—it only determines the homotopy classes of trajectories that are available upon some topological constraints. If one suspects that some physical factors may restrict a collection of available classes of trajectories, then one needs to ﬁnd a proper condition upon which it is possible to determine whether a particular family is accessible for a system, or not. For N particles placed on a 2D manifold upon a strong magnetic ﬁeld such a condition can be based on a cyclotron radius. It should be emphasized that a cyclotron radius is properly deﬁned only for non-interacting systems, but some kind of focusing of a particle motion can still be consider in the presence of Coulomb interactions—especially isotropic—like in planar homogeneous systems. Additionally, non-interacting fermions are reported for a completely ﬁlled LLL, when there is exactly one external magnetic ﬁeld ﬂux quantum per particle, hc . Thus, a cycloe

tron radius in the lowest Landau level can be deﬁned as a trajectory, which surface embraces one hc . Note that in the case of the e

degenerated LLL all particles are characterized with the same cyclotron radius, even though the velocity is not properly determined (the operators of velocity coordinates do not commute). However, all particles have the same kinetic energy, so it is reasonable to assume that they also have the same average velocity and so the cyclotron radius. To avoid interpretation problems let us deﬁne the cyclotron radius as the radius of a classical cyclotron orbit with the velocity taken equal to its average value in the completely ﬁlled LLL (ν ¼1). Finally, in order to determine whether a homotopy class of trajectories is available for particles in the system, it is sufﬁcient to compare the cyclotron orbit surface with a surface attributed to a single particle, which is deﬁned from a density and protected by a short-range part of Coulomb interactions (symmetry-breaking is not considered here). Summing up, the potentially permissible trajectory must ensure that an arbitrary particle can reach to the neighboring fermion (and the exchanges of particles are possible), so the trajectory must ﬁt perfectly to the minimal distance between 2D particles. The rigorous requirement leads to the observation that in too strong magnetic ﬁelds certain trajectory classes are banned for two-dimensional systems, as shown in Fig. 5. If for a particular magnetic ﬁeld strength the cyclotron radius ﬁts to the interparticle separation distance, then a cyclotron trajectory permits neighboring particles to exchange. Nonetheless, if one increases the external ﬁeld magnitude, then the cyclotron radius will decrease and the condition established above will be fulﬁlled causing the trajectory of the simplest two-particle exchange impossible. This exchange corresponds to the loopless generator of the full braid group and, since it is infeasible, it should be excluded from the group. This situation constitutes a formal reduction of the full braid group to a cyclotron subgroup with new generators describing 2D multi-looped braids. To note that these allowed trajectories have larger effective cyclotron orbits, let us argue it as follows. The trajectories corresponding to exchanges are open (some arbitrary ordering of particles is assumed), while the cyclotron orbits are closed paths, otherwise it would be impossible to deﬁne the piercing ﬂux of the external magnetic ﬁeld. The smallest closed trajectory is a double exchange–in a simplistic scheme two semicircles (loopless generator) create a full circle. In fact, for multilooped exchanges ( bi(q) = σiq ) each particle traverses a closed path with q 1 additional loops. The q 1 number is always even, since only closed loops can be added to the subgroup generator (one by one), to ensure that it induces an open exchange trajectory. Additionally, to closed trajectories must be added twice the number of bi(q) additional loops, two in the case of ν ¼1/3. So, the simplest exchange with one additional loop results in three-looped closed (and also cyclotron) trajectory of individual particles. This explains why the simplest and the most stable FQHE ﬁlling factor is equal to ν ¼ 1/3. This fact explains why closed braid trajectories are always oddlooped (1, 3, 5, etc. for respectively ν = 1, ν = 1/3, ν = 1/5, etc.)— because it is divided into half give proper exchange trajectories. So, for ﬁlling factors from the basic set ν = q1 the cyclotron trajectories are q-looped (q 1 additional loops) and related exchange trajectories are q -looped. To understand that this two-dimensional 2

multi-looped closed trajectories are the proper cyclotron trajectories with enhanced cyclotron radius, which restore the exchanges of particles in the system it is worth to carefully trace the following reasoning (already brieﬂy presented in the previous section). Note that each loop from a closed trajectory in a 3D space —just like each circumvolution in a solenoid—adds a new surface to the trajectory and the ﬂux quanta piercing this surface are added to the total BS /N . Situation is completely different for a 2D manifold, since an additional loop cannot add any new surface and

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Fig. 6. A schematic presentation of effective growth of cyclotron radius for multilooped cyclotron trajectory in comparison to singlelooped trajectory for two-dimensional spaces (length A − B means uniform separation between particles in the system and arrows denote the ﬂux quanta).

the total ﬂux quanta cannot be raised. In exemplary case ν = 1/3 the cyclotron radius is too short for exchanges. In a corresponding ﬁeld there are 3 ﬂux quanta— 3hc /e —per particle, while the cyclotron radius is still deﬁned by one ﬂux quantum. However, if one considers a greater 3-looped trajectory that matches to the interparticle separation and corresponds to 3hc /e , then the total amount of ﬂux quanta that falls on a trajectory remains unchanged. Finally, these 3 quanta need to be shared between all 3 loops resulting in the situation that through every loop passes one hc /e . The diameters of all loops ﬁt to the minimal distance between neighboring particles, which means that for the multilooped trajectory the cyclotron radius is greater than for a singlelooped one. This feature is illustrated in Fig. 6. It should be obvious that the cyclotron trajectory is proportional to the bare (without crystal ﬁeld) kinetic energy, so its value is changing with the Landau level index n

(2n + 1) hc eB

(8) hc

where the cyclotron area in the LLL ( eB =

S ) N0

can be derived from

the LL degeneracy N0. Finally, in higher Landau levels, along the necessity of introduction of multi-looped cyclotron trajectories when the arbitrary electron cannot reach its nearest neighbor

q

(2n + 1) hc (2n + 1) S S =q = eB N0 N − (m) N0

(9)

also exist the possibility that the cyclotron trajectory may be too long to ﬁt the interparticle separation, however, its radius must still be elongated to provide the exchanges with the second, third, … or even (2n þ1)th particle

q

(2n + 1) hc (2n + 1) S x· S =q = eB N0 N − (m) N0

(10)

where x = 2, 3, …2n + 1, m stands for all ﬁlled spin branches of Landau levels and q is the number of loops in the cyclotron trajectory. Additionally, when q¼ 1 the perfect matching, without additional loops, is considered. From the above equations the basic hierarchy

νe =

N x (2n + 1)·q·m + x , = =m+ N0 (2n + 1)·q (2n + 1)·q x νh = m + 1 − , (2n + 1)·q

(11)

and the generalized (Jain like) hierarchy of the FQHE ﬁlling factors can be established

x· y , (2n + 1)·(y ·(q − 1) ± 1) x· y νh = m + 1 − , (2n + 1)·(y ·(q − 1) ± 1) νe = m +

(12)

where y = 1, 2, 3…—in the standard Jain's theory, this parameter stands for a number of completely ﬁlled Landau (sub)levels of composite fermions. We assumed that the electron–hole symmetry holds. There is no doubt that the Coulomb repulsion forces play a very important role in a creation of collective Hall-like states [4–6]. However, in 2D interactions cannot be incorporated in a form of the particle dressing and the quasiparticle formation, what was a common procedure in solids. This concept is improper, because in a 2D Hall regime the Coulomb repulsion does not have a continuous spectrum with respect to a particle separation (expressed with the operators of projection on a relative angular momentum of particle pair) [4,5]. Therefore, the mass operator in two-dimensional multiparticle systems upon strong magnetic ﬁelds also experiences a non-continuous character and so it is impossible to introduce a Landau quasiparticle deﬁnition (as a pole of the retarded single-particle Green function). The Coulomb interaction can be operationally included within the Chern–Simons (Ch–S) ﬁeld theory [19,25], which formulates an effective description of the local gauge transformation of particles. In the case of Hall systems this suits to model particles with ﬂux-tubes or vortices such as CFs [23]. It has been demonstrated [5,26] that the short-range part of the Coulomb interaction stabilizes CFs against the action of the Ch–S ﬁeld (its antihermitian term [26,27]), which mixes states with distinct angular momenta within LL [26], in disagreement with the CF model [23,26]. The Coulomb interaction removes the degeneracy of these states and results in energy gaps which stabilize the CF picture, especially effectively in the lowest LL. For higher LLs, the CFs are not so useful also due to possible mixing between the LLs induced by the interaction [28]. The interaction also stabilizes the composite fermions formulated within the cyclotron subgroup model (similarly as they stabilize the angular momentum orbits in the presence of nonzero Ch–S ﬁeld) [17]. The Coulomb repulsion forces ensure that the condition, that particles do not approach into each other closer than a minimal distance, is rigidly kept. Otherwise (for non-interacting systems), it would be fulﬁlled only in average and some other trajectories, in addition to those for a ﬁxed particle separation (multi-loop at ν = 1p ), could appear (could be allowed for particles in the system), simultaneously precluding the cyclotron subgroup construction.

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5. Read's composite fermions in cyclotron braid terms In Read's model [11,12] composite particles (CPs) are postulated to be complexes of ordinary fermions and collective ﬂuid-like objects called vortices attached to this particles. Each vortex is characterized with a vorticity ’q’ and it can be deﬁned as follows [11]: N

V (z ) =

∏ (zj − z )q j=1

(13)

where z is a complex position of a centre of a vortex and zj are the complex positions of particles occurring in the 2D system. The above function suggests that a vortex placed on a 2D charged system will generate a charge depletion around its center. The resulting local positive charge (due to the background jellium) will attract the nearest ith electron that can build up with a vortex a bounded state ( z → zi )—a composite particle state. Note that after attaching vortices (with the odd vorticity q) to all particles in the system, one will arrive with the exact copy of the Jastrow factor of the LF [2] N

U (z ) =

∏ (zj − zi )q i>j

(14)

In particular, for q ¼1 the above function is the Vandermonde determinant (a polynomial part of the Slater function), N ∏i > j (zj − zi ), associated with the ordinary single-looped cyclotron motion of N two-dimensional non-interacting fermions completely ﬁlling lowest Landau level. Since the vortex deﬁnition results from the decomposition of the Laughlin function, thus it contains pieces of information beyond just the statistic phase shift —for example, the radial dependence of q-fold zeros attached to particles through the Jastrow polynomial. Simultaneously, the one-dimensional unitary representations of cyclotron braid groups cannot deﬁne the shape of the wave function. This shape can only be deﬁned after analyzing the energy competition between various wave functions with the same statistics symmetry. However, the vorticity q in the vortex construction is taken equal to the Laughlin function exponent known in advance, so it requires a motivation within the cyclotron structure. Some additional features of vortices (and also composite particles) are listed below [11]:

When the arbitrary particle zj is traversing a closed loop around

the vortex center, then the gain in phase is equal to 2πq . It was already mentioned and explained that a vortex induces a depletion of the local charge density around its center, what results in a locally positive charge −qνe (due to the background jellium). This leads to the screening of the charge of the bounded electron. Note that for ν = q1 ﬁllings, the electronic charge is completely screened (since −qνe = − e ) and an effective charge of a whole composite particle is zero. The phase shift obtained after exchanging of neighboring composite particles (electrons with pinned vortices) has several sources. First contribution is due to the charge depletion around the vortex center, which results in a non-zero phase shift gain 1 πνq2 (equal to the qπ for ν = q ﬁlling factor) for interchanging vortices, that restore Laughlin correlations. Another contribution results from a nonzero statistics of the original particles. Finally, composite particles are bosons for odd q and fermions for even q.Thus, composite particles corresponding to ν = q1 ﬁllings of LLL are always chargeless (so they do not feel the external magnetic ﬁeld). When q is an odd number this complexes are bosons, that in low temperatures can condense to the Bose–Einstein condensate—an incompressible state that can

reproduce the FQHE. Additionally, when q is an even number one deals with the Fermi sea of CFs that allows for the formation of Hall metal states [29–31]. It should be also highlighted that Read's conception of composite particles works, because of a reduced effective charge of the electron–vortex complex, which leads to the increase of the cyclotron radius (CPs are ‘less susceptible’ to the external ﬁeld), which is necessary to provide exchanges of neighboring carriers in the 2D system. The exceptional character of Read's conception is clearly visible for the completely ﬁlled LLL, ν ¼1. When vortices, attached to electrons in the system, are characterized with the vorticity equal to one, then the polynomial in LF is exactly the Vandermonde factor (being the Jastrow factor with power q¼ 1). In the latter case, the corresponding Laughlin state is expressed with the Slater function of N noninteracting fermions, but can also be effectively described by the Bose–Einstein condensate of complexes of fermions with vortices with q¼ 1 (all the action of the magnetic ﬁeld on ordinary fermions is replaced with this Bose condensation). The Coulomb interaction is not essential in this particular situation, because for complete ﬁlling ν ¼1 the Haldane pseudopotential [4,5] (and so the short range part of the Coulomb interaction, being crucial in determining of the Laughlin wave function) is equal to zero ( q − 2 = 1 − 2 = − 1 < 0). Therefore, the Slater function of noninteracting particles is the eigen-state of the interacting system at ν ¼1. The shift of the position of the vortex center with respect to the position of the bounded electron is equal to zero only when the kinetic energy of the complex is zero. This shift (deformation of the bounded state) leads to the Fermi sea instability due to the attraction of composite fermions and results in BCS-like pairing at e.g., ν ¼ 5/2 [12,32]. This state can be described by a new function in the form of Pfafﬁan (which modiﬁes LF) [12,32]. Note that this state is still of the same statistical symmetry as that for the particular sort of braid-composite fermions (deﬁned by 1DUR of the corresponding cyclotron subgroup). Let us also remark that all properties of vortices or ﬂux-tubes in CF constructions can be grasped together by a formal local gauge transformation [27] of the original fermions to composite particles. The original carriers are described with the ﬁeld operator Ψ (x), while the complexes with the annihilation and creation operators

Φ (x) = e−J (x) Ψ (x),

Θ (x) = Ψ +(x) e J (x)

(15) |z|2

where J (x) = q ∫ d2x′ρ (x′) log (z − z′) − 2 , and e J corresponds to a 4l nonunitary, in general, transformation that describes the attachment of Read's vortices (or Jain's ﬂux-tubes if J (x) is taken without last term) to the bare fermions. Also Ψ (x) and Ψ +(x) stand for the original fermion annihilation and creation operators, respectively. If one restricts J (x) (what is equivalent to the restricting of log) to its imaginary part, one arrives at the hermitian Ch–S ﬁeld (that actually describes the attachment of Jain's ﬂux-tubes) [33]. Note that initially not conjugated ﬁeld operators Φ (x) and Θ (x) + ( Φ+(x) = Θ (x) e J (x)+ J (x) ≠ Θ (x)) are perfectly conjugated after the restriction. In Read's model they describe operators of composite bosons for odd q and composite fermions for even q within the mean ﬁeld approach [27] (the omitted real part of J turns out to vanish in the mean ﬁeld, as the real part of log is canceled by the Gaussian; simultaneously, the hermitian Ch–S ﬁeld is canceled by the external magnetic ﬁeld). Finally, the form of the transformation q e ∑j

log (z − z j )

N

=

∏ (z − zj )q j

(16)

where the density of particles is taken in the form of N ρ (x) = Ψ +(x) Ψ (x)⟹ ∑ j = 1 δ (z − zj )), coincides with the deﬁnition

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of Read's vortex, so one can expect that it reproduces all properties of vortices. This gauge transformation allows for the interpretation of the Laughing state as a Bose–Einstein condensate of composite bosons, at ν = q1 , q—odd [11,27], and as a compressible fermion sea, at q—even [30,31]. Note that p = q − 1 is the number of ﬂux-tubes pinned to the CFs in Jain's approach and the remaining phase shift is obtained thanks to an ordinary fermion wave function. Assuming that the CFs are deﬁned by the 1DURs of the cyclotron subgroup, the hermitian term of this gauge transformation should be omitted, because it deﬁnes CFs when starting from ordinary fermions, which are already taken into account in terms of cyclotron braids.

6. Monolayer graphene in a strong magnetic ﬁeld Graphene is a 2D single layer of carbon atoms (of graphite) organized in a hexagonal structure as depicted in Fig. 7. The presence of two carbon atoms in a unit cell results in the existence of two trigonal shaped sublattices. The appropriate effective Hamiltonian can be derived from the tight binding model. To simplify the equations the Bravais lattice vectors were chosen to be those of the A sublattice and δi (i ∈ 1, 2, 3) stands for the vector pointing to the ith nearest neighbor on site B [34]. Additionally, it should be

9

mentioned that a form of the diagonal elements of the effective Hamiltonian (ﬁrst matrix) results from the common procedure to set the energy of the orbital 2pz to zero [35]

⎡ 0 γ ⁎⎤ ⎡ ⎤ k TB ⎥ − t′|γk |2 ⎢ 1 0⎥, = − t⎢ HML ⎢⎣ γk 0 ⎥⎦ ⎢⎣0 1⎥⎦

(17)

The effective wave functions are of spinor type

⎡α λ ⎤ k Ψkλ = ⎢ λ⎥, ⎢⎣ βk ⎥⎦

(18)

where t ¼2.46 eV is the hopping energy between nearest neighbors (NN), t′ = 0.2t is the hopping energy (as a matter of fact it is a real hopping energy combined with an overlapping correction) 3 between next-nearest neighbors (NNN), γk = ∑ j = 1 eik·δj is the sum of the NN phase factors and λ = 1 denotes the conduction band, whereas λ = − 1 the valence band. The real trial wave function is equal to Φ k (r) = α k Φ k(A) + βk Φ k(B), so components of the spinor eigenstates are the probability amplitudes of the Bloch wave functions on the different sublattices A or B. Thus, the effective wave function is similar to the wave function of a system of particles owning spin degree of freedom. For this reason, it is assumed that when the electronic density is completely located on the A sublattice, the system is described with the pseudospin ”up” state and

kx a1

a2

b1 3

0.142nm

2

a3

K'

K M

b2

ky

1

B A

Fig. 7. (a) The hexagonal structure of graphene with two sublattices A and B in the real space (left hand side picture) and in the reciprocal space with marked points of the highest symmetry like K or K′ (right hand side picture). The a1, a2 are Bravais lattice vectors and b1, b2 are vectors of the reciprocal lattice, (b) the band structure of graphene 3 3 with energy described with the relation E±(k ) = ± t 3 + f (k ) − t′f (k ) , where f (k ) = 2 cos ( 3 ky a) + 4 cos ( ky a) cos ( kx a) , t ¼ 2.7 eV is the hopping energy to the NN 2 2 having different pseudospin degree of freedom ( δ i ) and t′ = 0.2t is the hopping energy to the NNN located on the same sublattice.

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when it is located solely on the B sublattice—the pseudospin ”down” state. Note that components α k , βk also depend on the value of λ = ± 1 parameter. Finally, the energy spectrum (plotted in Fig. 7b)

E±(k) = ± t f (k) + 1 − t′ (f (k) + 1)

(19)

⎛ 3 ⎞ ⎛3 ⎞ ⎛ 3 ⎞ f (k) = 4 cos ⎜ k y a⎟ cos ⎜ k x a⎟ + 4 cos2 ⎜ k y a⎟ , ⎝ 2 ⎠ ⎝2 ⎠ ⎝ 2 ⎠

(20)

where a is the distance between nearest atoms (0.142 nm). Note that the energy is particle-hole symmetric only when the NNN hopping (and so the overlapping parameter) is excluded from the above equation t′ = 0). Furthermore, the valence band and the conduction band touch in two inequivalent corners of the Brillouin zone K and K′, where the value of the energy is zero. This statement can be easily proved, since E ¼0 only if the following expression vanishes

⎛ 3 ⎞ ⎛3 ⎞ ⎛ 3 ⎞ f (k) + 1 = 4 cos ⎜ k y a⎟ cos ⎜ k x a⎟ + 4 cos2 ⎜ k y a⎟ + 1 ⎝ 2 ⎠ ⎝2 ⎠ ⎝ 2 ⎠ (21)

= 0. One can choose (cf. Fig. 7) kx ¼0

⎛ 3 ⎞ ⎛ 3 ⎞ 4 cos ⎜ k y a⎟ + 4 cos2 ⎜ k y a⎟ + 1 = 0. ⎝ 2 ⎠ ⎝ 2 ⎠

(22)

This expression can be simpliﬁed to the quadratic equation using the substitution x = cos ( form

3 2

k y a). The ultimate solution has the

⎛ 3 ⎞ 1 4π cos ⎜ k y a⎟ = − , ↔ ky = ± 2 ⎝ 2 ⎠ 3 3a

K=

4π ^ ey 3 3a

and

K′ = −

4π ^ ey . 3 3a

(23)

(24)

Consequently, in order to analyze the low energy excitations, one may restrict the consideration to the vicinity of this corners denoted by K and K′. This restriction is performed when a new parameter is introduced (plus stands for the K′ symmetry point)

q=k±K

(25)

and the energy spectrum (and so the Hamiltonian) is expanded around the contact points with only ﬁrst order terms in |qa| taken into account. This yields to the linear energy-momentum dispersion

Eq, λ = λ

h vf |q| , 2π

(26)

and the effective ‘low energy’ Hamiltonian being formally equivalent to one for the relativistic massless (Dirac) fermions

(E = ±

m02 vF4 + p2vF2 with m0 = 0 and the light velocity replaced

with the Fermi velocity − vF =

Hqζ, low = ζ

h vf σ·q, 2π

3taπ ) h

(27)

where σ = [σx, σy ] is a vector consisted of Pauli matrices and corresponds to the pseudospin structure, q = ⎡⎣qx , qy ⎤⎦ is the quasimomentum vector introduced earlier. The parameter ζ = ± 1 describes a new degree of freedom for graphene particles called valley pseudospin or isospin and the states that differ only with this parameter are degenerated. Note that isospin arises from the

existence of two inequivalent corners of a Brillouin zone K and K′. It should be mentioned that the above equation is proper only when the K′ spinor components are swapped (with respect to the components of the K spinor). In some articles authors implement wave functions with four spinor components instead of two, thus avoiding the necessity of separate considerations of the corners [34]. For the reason mentioned above, the K and K′ points are often referred to as “the Dirac points” and their vicinity as “the Dirac cones”. The different characterization of graphene fermions in comparison to fermions present in classical heterostructures leads to numerous consequences [36–39]. For example, electron states (with non-negative E) can mix with hole states (negative E) in the time evolution, since the momentum uncertainty induces the energy uncertainty. If one expresses the wavenumber with the product of its magnitude and the direction versor n, then the effective low energy Hamiltonian becomes

Hqζ, low =

h qσn 2π

(28)

The operator σn projects the pseudospin onto the direction of the quasimomentum and, obviously, its eigenstates are also the eigenstates of the Hamiltonian with eigenvalues þ 1 for electrons and 1 for holes [35]. The resulting effect is astounding—the massless Dirac particles are chiral; the pseudospin is always parallel to the momentum for particles (the conduction band) and always antiparallel for antiparticles (the valence band). This leads to the Klein paradox (the ideal tunneling of particles) and high mobilities (the classical scattering processes cannot break the sublattice A-B symmetry—so they cannot ﬂip the pseudospin) in graphene structures, what was conﬁrmed with many experiments [36,38–40]. The most important feature arising from the relativistic type of the graphene particles is the extraordinary setup of the Landau levels. This setup can be calculated from the ‘low energy’ effective Hamiltonian, if one treats the quasimomentum q with a standard substitution [41,34,42]

h e q=p→p− A 2π c

(29)

and one chooses the Landau gauge of the form A = [0, − Bz x], where A is the vector potential of the magnetic ﬁeld ( B = rot (A)). It needs to be clariﬁed that q is actually a wavenumber, not a quasimomentum. However, the latter one takes the form of p = = q . For this reason, q is often referred to as a quasimomentum and we will hold to this somewhat inadequate naming. All Landau levels (except from the lowest one) derived in that manner consist of four sublevels originating from the valley degree of freedom (K , K′) and the free electron spin ( ↑ , ↓ ) while the electronic density is shared equally between both sublattices A and B. However, wave function components of the nth LL labeled with parameters K , A and K′, B are of the (n − 1) th harmonic oscillator wave function type ϕn − 1, while those labeled with parameters K, B and K′, A , are of the nth harmonic oscillator wave function type ϕn . Already from this property one can conclude that the lowest level is unique. In fact, the LLL is also quadruply degenerated, but the valley isospin coincides with the sublattice pseudospin and the states can be labeled with parameters from set (K , K′)—but also from (A,B)— and a free electron spin. Finally, the energy eigenvalues [39]

EL = ±

2h|e|Bz h Dirac vf n = ± n ω 2π c 2π

(30)

where n is the LL index. Bear in mind that for the linear energy– momentum dependence the LLs are not distributed equidistantly (oppositely to standard heterostructures), and so the cyclotron

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energy is changing (decreasing) with the LL index. The gap between the n¼ 0 and n¼1 LL reaches very high values (1000 K in 10 T ﬁeld) and this allows us to observe the integer quantum Hall effect even in room temperatures [38,39]. Furthermore, the energy ﬁeld dependence its also different, because ωc ∼ B1/2, and not ∼B as expected. Fig. 8 illustrates the (anomalous) integer quantum Hall effect in a graphene sample. The plateaus in the longitudinal conductance are correlated with ﬁlling factors

⎛ 1⎞ ν = 4n + 2 = 4 ⎜n + ⎟, ⎝ 2⎠

(31)

where 4 results from the spin and isospin degeneracy (the spin degeneracy can be lifted with a strong magnetic ﬁled—Zeeman effect, while the isospin degeneracy—with strong interactions between particles in the system). The appearance of an additional element— 1 —is a consequence of sharing of states between the 2

1

valence and the conduction band inside the LLL ( 2 ·2π is also referred to as a Berry phase—a phase acquired by the spinor eigenvector when a quasimomentum is rotated through 2π in the reciprocal space) [36,40,37–39,45]. In the end, let us add some comments about the bilayer graphene. It is composed of two coupled layers of carbon atoms arranged in the Bernal stacking (where B1 atom from the upper layer sits on the top of the A2 atom from the bottom layer) [46]. Additionally, bilayers are quite different from monolayers, because (despite the similar gapless spectrum) the interlayer coupling leads to the appearance of the band mass and the parabolic energy dispersion expressed with a low energy effective Hamiltonian [39]

Hqlow =

⎡ 0 (qx + iqy )2 ⎤ (vf )2 ⎢ ⎥, ⎥⎦ γ1 ⎢⎣(qx − iqy )2 0

(32)

where γ1 stands for the A2 − B1 hopping integral. Recently, the experiments involving bilayer graphene samples are quite popular due to exceptional hierarchy of Landau levels, N ¼0 and N ¼ 1 LL

11

both have vanishing energy (another type of degeneracy) [39]. This feature leads to unexplored regimes of the FQHE, where mixing of two LLs is possible and a partially ﬁlled lowest Landau level occurs also at low densities of carriers. Although the stability of the FQHE in a bilayer graphene is an order of magnitude smaller compared to a monolayer one, some successful experiments were carried out (i.e., the separate measurements of transverse and longitudinal resistance allowing observations of the even denominator fractions [47] or local electronic compressibility measurements with scanning probe method [48]). Fig. 8 shows the integer quantum Hall effect measurement in a bilayer graphene. The structure of the transverse conductivity can be explained with the fourfold degeneracy of all Landau levels similar to the monolayer graphene case (due to the free electron spin and the valley pseudospin). The lowest Landau level is mixed with the ﬁrst Landau level leading to the eightfold degeneracy and an increased dimension of the conductivity plateau for low concentrations of carriers (near n ¼0). As in the case of the monolayer graphene, when n ¼0 the layer (sublattice) pseudospin coincides with a valley one in opposition to the higher LL. The bilayer with the monolayer graphene IQHE comparison is presented in Fig. 9. 6.1. The FQHE in graphene vs. the cyclotron braid group approach The quantum Hall effect experiments in the graphene samples are exceptional, since one can ﬁx the magnetic ﬁeld strength and modify the density of fermions with a lateral gate voltage (up to 10 V [50]), what is not practised in standard semiconductors (although it can be more or less accomplished with the use of ﬁeld effect transistors [51]). The possibility of obtaining low concentrations of carriers in high magnetic ﬁelds provides the conditions for exceeding the stability threshold of the FQHE and favoring the Wigner crystallization of fermions. The insulating state near the Dirac point appears. This transition may not be as sharp as expected, because Wigner crystallized triangular sublattices

Fig. 8. (a) IQHE in the monolayer graphene as a function of the electron concentration (controlled with lateral gate voltage). The peak for the concentration equal to n¼ 0 conﬁrms the existence of the Landau level for a zero energy (Dirac point). Plateaus sxy correspond to half multiplicities of 4e2/h , what implies that the valley and spin degeneracy is not lifted. The particle–hole symmetric oscillations are observed. The inset presents the IQHE in bilayer graphene [43]. (b) IQHE for the bilayer graphene sample as a function of the electron concentration. The transverse conductivity measurements are presented in two different magnetic ﬁeld strengths, 12 T and 20 T. Positions of σxy plateaus suggest that the spin and valley degeneracy is not lifted. The electron–hole symmetrical oscillations of the longitudinal resistivity are observed [44].

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Fig. 9. The LL sequencing for the monolayer graphene and the bilayer graphene samples—comparison [49].

Fig. 10. (a) The gradual transition to the insulating state caused by the increase in a magnetic ﬁeld strength in the vicinity of the Dirac point and (b) the competition between 1 Hall states and the Wigner crystallization for the ν = − ﬁlling factor. When the mobility is enhanced (due to annealing), a plateau indicating the FQHE state appears [50]. 3

may exhibit an interference (the hopping of carriers may occur)— Fig. 10. Note that in a typical 2DEG the magnetic ﬁeld corresponding to the ﬁlling factor ν ≃ 1/9 also destabilize the FQHE [52]. These observations stay in an agreement with the cyclotron subgroups model, since traversing multilooped trajectories with many additional loops is just energetically unfavorable. In graphene structures high mobilities (250,000 cm2 V 1 s 1) and small concentrations [53,54] result in a mean free path of electrons greatly exceeding sample dimensions (it is also the case in typical semiconductor devices, since higher mobilities—a record of 30 106 cm2 V 1 s 1 [53]—can compensate lower densities). In a consequence, a single electron in a magnetic ﬁeld (integer ﬁlling factor) can traverse an edge to edge trajectory in a web of equidistantly distributed particles (protected with the Coulomb repulsion). Its pathway actually consists of simple cyclotronic exchanges, so one can argue that it corresponds to the realization of more complicated braids (other than the generators of the full braid group)—cf. Fig. 11. Taking into account fractional ﬁllings 1 ν = p of the lowest Landau level with p-looped cyclotronic orbits, the resulting and required mean free path should be p times

longer than sample dimensions, what was already conﬁrmed for graphene [54] and classical heterostructures [53]. This observation proves that the Coulomb interactions, the basis of Jain's and Read's theories [23,52], are crucial but not sufﬁcient for the FQHE formation. It seems that also a high carrier mobility plays a triggering role, since in some measurements the plateaus in the transverse conductivity are appearing only after the annealing of the sample [55]. It seems that this fact goes beyond explanation abilities of the CF theory, which focuses only on the interactions itself and do not take into account the extraordinary topological properties of the 2D space. The wave packet idea corresponding to the semiclassical approximation becomes important when one analyzes the collective states of a multiparticle system. Note that when the LLL is ﬁlled with non-interacting particles the group velocity of any packet is equal to zero. However, the interactions lift the Landau level degeneracy providing nonzero group velocity and introducing the wave packet dynamics. Interactions naturally prefer localization (the Wigner crystallization), but at the same time the collectivization seeks for a minimalization of kinetic energy, so it prefers

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Fig. 11. The complicated braid: b1(q) b2(q)…, where bi(p) are generators of the cyclotron braid subgroup (here p¼ 3) [17,18]. This braid group element arranges edge-to-edge cyclotron hopping trajectory due to collisions of every second particles. Since the cyclotron radius enhancement is obtained with the multi-looped trajectory structure, completing of p − 1 -looped braids is required before collisions. Finally, performing these additional loops naturally need sufﬁciently longer mean free path. 2

the quasiclassical movement of a wave packet along periodic closed trajectories (the magnetic ﬁeld presence [56]), which must embrace the external magnetic ﬁeld ﬂux quanta. Even though the real dynamics of wave packets is beyond the description ability of the simpliﬁed, classical picture of 2D multiparticle charged system in a magnetic ﬁeld presence, some general conclusions may be drawn. Note that in high magnetic ﬁelds the classical cyclotron radius must be enhanced to provide the exchanges. So, the multilooped trajectories (exchanges) are introduced. If one assumes that the ﬂux quantization is held (as indicated by the great success of the CF theory), then the real wave packet dynamics should also be multilooped. Finally, the wave packet pathways should be represented with appropriate multilooped braids [18]. Also, the carrier mobility is connected with wave packet dynamics (through drift velocity, classical Hall effect and various scattering phenomena) and its high value is required when the real wave packets are forced, like for fractional ﬁllings of the LLL, to traverse multilooped trajectories. Some speciﬁc character of the FQHE in graphene could be also connected with SU(4) symmetry of massless fermions [57,34]. However, the proper multicomponent model of the FQHE is described with the Halperin wave function, which is actually a generalized Laughlin function Nj

4 (4) ΨmSU1, … , m 4, ni, j =

∏∏ j

kj < l j

4

×

Ni

(z kj − zl j )mj e j

N 4 ∑kjj= 1 | z kjj |2 /4 j =1

−∑

j

Nj

∏ ∏ ∏ (zkii − zkjj )ni,j , i

ki

kj

(33)

where mj must be odd integers, whereas nij may be also even integers. Therefore, the topological interpretation is analogous to the single component electron liquid case. Fig. 12 shows the exemplary experiment of the FQHE in the monolayer graphene sample. Observed fractions are placed within the range ν = ± 2, ± 6, where ν ¼2 corresponds to the completely ﬁlled LLL (only electron sublevels) and ν ¼6 corresponds to the situation when also the ﬁrst LL is completely ﬁlled. The inset

Fig. 12. Integer and fractional quantum Hall effect features in graphene, a and b. Magnetoresistance (left axis) and Hall resistance (right axis) in the n¼0 and n¼ 1 Landau levels at B¼ 35 T and temperature ∼0.3 K; all integer ﬁlling factors for IQHE plateaux are indicated; inset demonstrates Shubnikov–de Haas oscillations at magnetic ﬁeld of order of 1 T [57].

presents the Shubnikov–de Haas oscillations, robust for ﬁelds B ≈ 1 − 2 T. These experiments were not selected randomly, but in a view of fractional quantum Hall features. It appears that in higher LLs there are problems with developing (experimentally) plateaus in the Hall resistance. This feature is quite interesting, since numerical calculations within Jain's theory predicts that these states should be even more robust in comparison to the traditional semiconductor heterostructures (since, for example, in the graphene the N ¼ 1 LL reveals similar character to N ¼0 LL in the opposition to the typical semiconductors) [58]. Note that all fractions present in the experiment can be easily obtained within the cyclotron subgroup model, both in the lowest Landau level

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⎧ 1 1 for m = 0, = ⎪ ⎪ 1·(3 − 1) + 1 3 y νe = m + =⎨ 1 4 (y·(q − 1) ± 1) ⎪ 1+ for m = 1, = ⎪ 1·(3 − 1) + 1 3 ⎩

bilayer graphene that are recently of a great interest—Fig. 13. It is worth to mention that the FQHE is asymmetrical in these structures; it is easier to observe the longitudinal resistivity dips for antiparticles (holes) than for particles (electrons).

⎧ 1 2 = ⎪1 − ⎪ 1·(3 − 1) + 1 3 y νh = m + 1 − =⎨ 1 5 (y·(q − 1) ± 1) ⎪ 2− = ⎪ 1·(3 − 1) + 1 3 ⎩

6.2. Development of a Hall-type experiment in conventional semiconductors

for m = 0, for m = 1,

(34) Since the ﬁrst observation of the FQHE in semiconductor heterostructures [1], the sample manufacturing techniques have been greatly improved, resulting in higher mobilities and longer mean free paths of carriers for a wide range of concentrations [53]. In the original experiment the value of a mobility was about 9 104 cm2 V 1 s 1 and was quickly increased to 5 106 cm2 V 1 s 1 [59]. Today's record 31 106 cm2 V 1 s 1 was set for the electron density of 109 − 1011 1/cm2 and it scales as μ ∼ n0.7 [53]. With the increase in a mobility some new plateaus in the Hall 5 3 resistivity appeared, e.g., for the ﬁlling factors ν = 8 , ν = 13 , and

and in the ﬁrst LL

νe = m +

y 3·(y ·(q − 1) ± 1)

⎧ 1 7 for m = 2, = ⎪ 2+ 3 1 3 1 1 3 ·( ·( − ) − ) ⎪ ⎪ 1 10 = for m = 3, ⎪ 3+ 3·(1·(3 − 1) − 1) 3 ⎪ νh =⎨ 1 13 ⎪ 4+ = for m = 4, ⎪ 3·(1·(3 − 1) − 1) 3 ⎪ 1 16 ⎪ 5+ = for m = 5, ⎪ 3·(1·(3 − 1) − 1) 3 ⎩ y =m+1− 3·(y ·(q − 1) ± 1) ⎧ 1 8 = for m = 2, ⎪3 − 3·(1·(3 − 1) − 1) 3 ⎪ ⎪ 1 11 = for m = 3, ⎪4 − 3·(1·(3 − 1) − 1) 3 ⎪ =⎨ 1 14 ⎪5 − = for m = 4, ⎪ 3·(1·(3 − 1) − 1) 3 ⎪ 1 17 ⎪6 − = for m = 5, ⎪ 3·(1·(3 − 1) − 1) 3 ⎩

4

(35)

Finally, let us also present the illustrative measurements for the

ν = 11 . The observation of these ﬁllings (marked with color in Fig. 1) cannot be explained with the use of the standard composite fermion theory (although the idea of CF interaction was implemented to explain some of these troublesome ratios [52]). However, they can be easily derived from the cyclotron subgroup model without any additional and somehow artiﬁcial assumptions. Also some interesting and sensitive to high mobility re-entrant 5 insulating phases between two IQHE plateaus in Rxy near ν = 2 fraction were reported [61]—Fig. 14. These phases are believed to be connected with two kinds of collective phenomenons: with the integer quantum Hall effect experienced by particles in all completely ﬁlled Landau levels and a stripe or a bubble insulating state (characterized with an anisotropic electron density, which forms charge density waves) present in the partially ﬁlled one. To sum up, the high mobility requirements mentioned above indicate the important role of the quasiclassical wave packet

Fig. 13. The FQHE in the bilayer graphene sample on the hole (antiparticle) site for a constant density of carriers and a varying magnetic ﬁled strength from 0 to 14 T. Temperature was reduced to 0.25 K [48].

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Fig. 14. FQHE in the second Landau level for high mobility samples of 31 × 106 cm2 V−1 s−1. Re-entrant insulating behavior near fractions 5/2 and 7/2 are reported. Rxy oscillates (with the slowly increasing magnetic ﬁeld strength) between the expected FQHE plateaus and the nearby IQHE plateaus [60].

15

developed in order to grasp the essential topology of various multiparticle structures [63,65,70]. The mappings of the Brillouin zone into the state related objects can be in that manner classiﬁed by disjoint classes corresponding to topologically nonequivalent band organizations protected by energy gaps conditioned by various physical factors and leading to distinct incompressible states (if gaps are related with a system volume as typically in a 2D case, where in relation to quantum Hall effects, the area of a system enters via the external ﬁeld total ﬂux) in analogy to their prototype in the form of the IQHE. The distinctive character of a 2D space is linked here with the magnetic ﬁeld ﬂux quantization, which is the quasiclassical property of trajectory loops being 2D object as a rule. Topological notions allow for deﬁnition of a new crystal state called topological insulator. Despite a local similarity between the gapped states of the ordinary and the topological insulator, the global arrangement of the band, noticeable only non-locally (on the Brillouin zone as a whole), induces different overall behavior of the system. In the latter case one deals with insulating state inside the sample, whereas with a conducting non-dissipative state on the sample edge, protected topologically. This surprising phenomenon was conﬁrmed experimentally, which was a strong stimulus to the rapid and enormous great growth of the interest. The topological insulators from a band organization point of view must be characterized by ﬂat bands which meet in summits of locally cone shaped valleys resembling Dirac points in graphene. These Dirac points change the topology and allow Chern-type invariant to attain nonzero value, indicating the emergence of the different global state. Spin degrees of freedom are of a high signiﬁcance resulting in spin-type topological insulators (the spin IQHE). 7.1. Hall effect without magnetic ﬁeld

dynamics for the FQHE creation, which can be pretty well captured within the cyclotron subgroup model, but it seems to be rather beyond the explanation ability of the single-particle composite fermion theory.

7. Topological Chern insulators Investigation of the IQHE and the FQHE opened a broad area of topologically conditioned effects [62]. Especially deeply developed is the present understanding of the IQHE treated in single-particle topology terms, related with a novel view on the standard band structure [63]. This is based on the observation that the Hall states protected by Landau quantization gaps are not connected with the symmetry breaking as many other condensed matter phases in scenario of ordinary phase transitions, but rather with some topological invariants associated to a particular geometry and a matter organization [64,65]. These invariants are better and better currently recognized in terms of homotopy groups related to specially deﬁned multidimensional transformations of physically conditioned objects like Green functions and their derivatives [66,67], previously developed for a description of a topology of textures in multicomponent condensed matter states with a rich matrix order parameter, including superﬂuid He3 or liquid crystals [68]. The role of various factors protecting gaps separating ﬂat bands is of particular interest in view of a magnetic ﬁeld breaking time reversion or other effects, like a spin–orbit interaction or a special type of a closed loop traversing inside an elementary cell with complex hopping constants [69]. Generalization of the familiar in mathematics Chern invariants, which can characterize and make possible to classify geometrically non-equivalent objects (like manifolds with different numbers of defects—holes [70]), is

Commonly accepted deﬁnition of the topological insulator emphasizes the metallic character of edge (surface) states and extended bulk insulating states that are also robust against disorder. This is an extraordinary behavior, especially when a surface is cut in a topological insulator sample. Thus, edge states seem to be connected to the bulk states and can be viewed as these extended bulk states terminating at the boundary. For this reason, the bulk and the edge properties of the topological insulators are equally important and mutually dependent. In the same time, the development of the interpretation of the IQHE revealed a spectacular emergence of a non-dissipative charge current ﬂowing around the edges of any ﬁnite IQHE sample. However, the IQHE was observed only in the presence of an externally applied magnetic ﬁeld. In 1988 Haldane presented a model of a condensed matter phase that exhibits the IQHE without the need of a macroscopic magnetic ﬁeld [71]. The general idea of this effect can be sketched by writing a model Hamiltonian for the system of spinless particles occupying a honeycomb-type planar lattice with one state |n> per site

^ H=

∑ < n, m >

|n > < m| +

∑ ⪡ n, m ⪢

⎤ ⎡ ⎢ξn |n > < m| + h. c .⎥, ⎥⎦ ⎢⎣

(36)

where

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Fig. 15. (a) The honeycomb structure for the Haldane model [71]; e1, e 2 are Bravais lattice vectors, nonequivalent site positions are indicated by α = ± 1. (b) The phase diagram for the model Hamiltonian (36), when Chern number C equals ±1, the ground state corresponds to the topological insulator, whereas for C¼ 0 to the ordinary insulator [65].

properties (expressed by the Chern number) for various values t, η as shown in Fig. 15. The difference between a quantum Hall state and an ordinary insulator is a matter of topology [72]. A 2D band structure consists of a mapping of the momentum k deﬁned on a torus of the Brillouin zone to the Bloch Hamiltonian H (k). Gapped band structures can be classiﬁed topologically by considering the classes of these mappings that cannot be continuously deformed into one another without closing the energy gap. These classes are distinguished by integer topological invariants—the Chern numbers. The Chern numbers were introduced in the theory of ﬁber bundles [73] and they can be understood physically in terms of the Berry phase [74] associated with the Bloch wave functions |um (k)>. When k traverses a closed loop in the reciprocal space, the Bloch function acquires a Berry phase given by the line integral of Am = i < um |∇k |um >, or by the surface integral of the Berry ﬁeld Fm = ∇ × Am . The Chern invariant is the total Berry ﬁeld ﬂux for the Brillouin zone

7.2. Fractional Chern insulators

1 C= 2π

analogously to the ordinary FQHE. Again arises a question of how single particle trajectories ranged by Berry ﬁeld ﬂux quantization to the elementary cell can match every third cell (for 1/3 ﬁlling). The concept of CFs with auxiliary ﬁeld ﬂux-tubes is not useful here, due to the absence of the magnetic ﬁeld. The trajectories must be, however, enlarged somehow to allow organization of the collective state with the determined statistics. Again the multilooped braid subgroup might be the solution of the problem with too short orbits, especially in this case without any other conceptual competition. This supports the topological attitude to fractional correlated 2D states, robust against details of quantized ﬁeld ﬂux, it also explains why such states occur only at speciﬁc fractions of ﬁllings, surprisingly coinciding for the FQHE and for fractional Chern insulators. To be more speciﬁc with this regard, the recent analyzes are worth emphasizing [85,86]. In the paper [85] a class of model Hamiltonians on 2D lattice (giving nearly ﬂat bands with nontrivial topology) is constructed. This property is regarded as a required prerequisite for organization of the FQHE-like states. For ﬂat band the kinetic energy is frozen what allows the interaction to dominate and create strongly correlated state. This has been theoretically veriﬁed in the paper [86] for a checkerboard lattice including a nearest and a next-nearest interaction in model Hamiltonian

∫

d2kFm

∈ Z,

(37)

where C is an integer for reasons analogous to ones appearing in the quantization of the Dirac magnetic monopole. The Chern number is a topological invariant—it cannot change its value when the Hamiltonian varies smoothly and it explains the quantization of conductivity in the IQHE [72]. For a better understanding lets us introduce a simple analogy. Rather than maps from the Brillouin zone to a Hilbert space, one can consider maps from two to three dimensions, which describe surfaces. 2D surfaces can be topologically classiﬁed by their genus g, which counts the number of holes. For instance, g ¼0 for a sphere, while g ¼1 for a torus. A theorem in mathematics states that the integral of the Gaussian curvature over a closed surface is a quantized topological invariant, and its value is related to g. The Chern number is an integral of a related curvature. Change of the Chern number requires closing the insulating gap. This happens for the Hamiltonian (36) in the Dirac-like points when the locally cone shaped valleys of conduction and valence bands touch together. This allows for the change of the Chern number and for the metallic boundary states protected by still insulating phase inside the sample [65,63]. The systems that behave like the one described by Haldane are now called Chern insulators. The time reversal symmetry in these systems is broken like in the IQHE, but it is broken by the presence of a net magnetic moment in each unit cell rather than by an external magnetic ﬁeld. The Chern insulators were never found experimentally as of yet.

Recently grew up a new ﬁeld related to the IQHE and its fractional version, namely Chern insulators and fractional Chern insulators, respectively. This ﬁeld is linked with development of the Haldane model of the IQHE effect without Landau levels, but with a time-braking imaginary part of hopping factors for carries on planar lattice. The quantization of carrier orbit along elementary cell is given in this case by the Berry ﬁeld ﬂux quantization which can be expressed with Chern numbers. Even though neither Chern insulator nor its fractional state has been observed, the theoretical studies are currently extensively developing [75–84]. From exact diagonalization of corresponding Hamiltonians, provided they give sufﬁciently ﬂat bends, it follows that the interaction causes collective state analogous to the FQHE, though without any magnetic ﬁeld and with the dynamics assigned by nontrivial Chern number. What is especially challenging, the fractional Chern insulator is predicted for 1 (p—odd integer) ﬁllings of planar crystal lattice, p

H = − H0 + U

∑ < i, j >

ni nj + V

∑

ni nj ,

⪡ i, j ⪢

(38)

where H0 is a two-band checkerboard lattice model with nonzero Chern number implemented by complex hopping factors of the

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type

H0 = − t

∑

eiψ ij (ci+cj + h. c . ) + H1,

< i, j >

(39)

and H1 describes ordinary real-number-assigned hopping between next and next–next nearest neighbors [85]. Some attributes characteristic for FQHE have been indicated within this model for fractional ﬁllings 1/3 and 1/5 (in the latter case repulsion V was required above a certain threshold value, unless U enhances strongly) [86]. This observation supports an idea of multilooped structure of quasiclassical wave packets which in this way can reach equidistantly separated 2D particles (in every third cell for ν ¼1/3) due to repulsion interaction, quite similarly as in the described above 2D systems in the magnetic ﬁeld. As one can see, any links can be here drawn toward CF model with auxiliary magnetic ﬁeld ﬂux quanta attached to hypothetical composite particles, but the multilooped requirements in order to enhance orbits still hold. The role of cyclotron orbit quantization is substituted here by the orbit quantization due to Chern number invariant conservation (i.e., the quantization of the Berry ﬁeld ﬂux instead of the magnetic ﬁeld ﬂux). 7.3. Topological states in optical lattices Finally one can mention also recent search for synthetic fractional states in so-called optical lattices, created by laser beams with trapped atoms instead of electrons. In these systems the magnetic ﬁeld induced cyclotron movement is imitated by the precession of a trapped atom position controlled by time modulated laser signals [87]. None fractional state is, however, demonstrated as of yet. The way to obtain the fractional correlated state in optical lattice is probably linked to the multilooped trajectories allowing for matching carriers too distantly separated for single-looped synthetic trajectories. Realization of such proposal might be a direct conﬁrmation of the presented above topological concepts regarding fractional states of 2D systems. Trapping of cold atoms in artiﬁcially arranged optical lattices opened a new avenue to experimental and related theoretical investigations of correlated 2D multiparticle systems including fermions, bosons and mixtures. Though trapped particles are neutral atoms as a rule, the methods of inclusion of a strong interaction were proposed [88,89]. The methods of magnetic ﬁeld imitations were also developed via rotation of unit cells in optical lattices. Special attention is paid to studies of topological states including also the FQHE [90,89,91,92,88]. The scheme of arrangement of optical lattice with rotation modeling magnetic ﬁeld is presented in Fig. 16. It should be noted that experimental observation of the FQHE in optical lattices is not achieved as of yet. Similarly other collective states are not arranged in practical setups, though theoretical investigations are quite advanced. Therefore, one can conjecture that the analogy of this experimental novel systems of atoms trapped in optical lattices even with magnetic cyclotron rotation implemented by phase shifts induced rotation is not full in comparison to real multiparticle quantum systems like graphene. This would be related with classical in fact arrangement of the precession rotation imitating the perpendicular magnetic ﬁeld action. In the case when these pseudo-cyclotron orbits are too short for particle interchanges, as at fractional ﬁllings, then the multilooped braids in 2D ought to be important along the scheme presented in previous chapters. In the model quantum type calculus this is arranged automatically, but not in the optical lattice setup. Thus despite many theoretical results supporting the existence of the FQHE for model Hamiltonians in optical lattices, the experimental observation failed. Probably a further development of the method

Fig. 16. The sketch of the method of creation of optical lattice with artiﬁcial magnetic ﬁeld implemented. The three laser beams cross each other at 120o in the plane; phase modulators are placed in the paths of the two beams (a); the illustration of the on-site rotation [87]; the entire lattice takes the motions of a fast oscillation and a slow precession with the frequency Ω, as schematically plotted with the red solid lines in one lattice site; after taking the time average of the fast oscillation, atoms feel that each site is rotating around its own center with the precession frequency Ω, which is plotted with the red dot-dashed line around each site. (For interpretation of the references to color in this ﬁgure caption, the reader is referred to the web version of this paper.)

of magnetic ﬁeld imitation by means of phase induced rotation is required here in order to realize in practice the effect of multilooped kinetics of 2D quasiclassical wave packets.

8. Conclusions In summary, we intend to explain the topological origin of the FQHE which manifests itself only at specially chosen LL ﬁlling fractions. This strongly correlated charged 2D system state requires ﬂat band when the kinetic energy does not compete with the interaction. The standard theories of the FQHE do not explain its occurrence at specially selected ﬁlling fractions and in the case of CFs approach the auxiliary ﬁctitious elements are introduced in order to illustrate the Laughlin correlations. By means of braid group approach it is possible, however, to select appropriate braid subgroups of the full braid group of which 1DURs deﬁne effective particles in the correlated states referred to the FQHE. The argumentation is linked with the simple observation that at fractional ﬁllings of the LLL the cyclotron trajectories which build braids in equidistantly distributed 2D charged systems are too short to ﬁt the particle separation. This precludes particle exchanges—necessary for organization of the collective multiparticle state with the determined statistics according to 1DURs of a related braid subgroup. One can observe that only at these ﬁlling fractions at which FQHE occurs the multi-looped braid structure recover exchanges along enhanced cyclotron trajectories. This enhancement is an exclusive property of 2D system when additional loops of cyclotron trajectory cannot add a surface, but must share the same total external magnetic ﬁeld ﬂux, which leads to effective enlargement of cyclotron orbits. For 1/q ﬁllings of the LLL the braid trajectories must be q-looped and then corresponding cyclotron orbits match neighboring particles without any artiﬁcial constructions. This unavoidable property of braids recovers Laughlin correlations in the natural way and simultaneously explains the underlying spirit and structure of CFs both with auxiliary ﬂux-tubes and vortices. The ﬂux tubes attached to CFs do not actually exist and they only model the result of additional cyclotron loops. The multilooped quasiclassical wave packet orbits with larger radii strongly favor higher mobility of carriers which is proportional to the mean free path of carriers. This property has been conﬁrmed experimentally in the suspended graphene. Direct experiment demonstrated that

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via reducing of structure imperfections by the annealing of the graphene sample one can trigger the formation of the FQHE state in the insulating system before annealing. This indicated that the mean free path and thus the mobility of carriers conditions forming of the FQHE state. This observation, together with the occurrence of the FQHE in graphene at lower ﬁelds for lower concentration of carriers, support the topological cyclotron braid approach. Simultaneously, recently developed theoretical studies of the fractional Chern insulators apparently go beyond the explanation ability of the standard CF concept (lack of a magnetic ﬁeld) but still admit the multilooped braid group explanation. Especially interesting would be veriﬁcation of this idea in the optical lattices with the synthetic fractional state.

Acknowledgments The support from the NCN Project UMO-2011/02/A/ST3/00116 is acknowledged.

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