Skip to main content. Advertisement Hide. Commensurability condition and fractional quantum Hall effect hierarchy in higher Landau levels. Authors Authors and affiliations J. Jacak L. Condensed Matter First Online: 16 September This process is experimental and the keywords may be updated as the learning algorithm improves. This is a preview of subscription content, log in to check access. Pan, H. Tsui, L.

Pfeiffer, K. Baldwin, and K.

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West, Phys. Eisenstein, M. Lilly, K. Cooper, L. Pfeiffer, and K. West, Physica E 6 , 29 Dolev, Y. Gross, R. Sabo, I. Gurman, M. Heiblum, V. Umansky, and D. Mahalu, Phys. Willett, Rep. Pan, K. Baldwin, K. West, L. Pfeiffer, and D. Tsui, Phys. Xia, W. Pan, C. Vicente, E. Adams, N. Sullivan, H. Jain, Phys. CrossRef Google Scholar. As it was proved [12] , these weight factors form a one-dimensional unitary representation 1DUR of the related braid group.

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. For the far more rich braid groups in 2D one encounters, however, the infinite number of possible so-called anyons including bosons an fermions related to 1DURs, , are here generators of the full braid group in 2D, cf.

Figure 2 [11] [23] - [25]. We will develop the topological approach to Hall systems and recover Laughlin correlations by employing geometry properties of the so-called cyclotron braids [13] [14] in the framework of formal braid group approach and without invoking to any auxiliary elements inherent to the CF concepts. We will demonstrate that particles with statistics properties familiar in the CF model are 2D quantum particles characterized by appropriate 1DURs of the cyclotron braid subgroups.

One-dimensional unitary representations 1DURs of the full braid group [11] , i.

## Jacak Janusz Gonczarek Ryszard Jacak Lucjan Jozwiak Ireneusz

Then an additional summation over these classes with an appropriate unitary factor the weight of the particular trajectory class should be included in the path integral for transition from the point a at the time moment to the point b at [23] [24] :. The factors form a 1DUR of the full braid group and the distinct representations correspond to the distinct types of quantum particles [12] [24].

The closed loops from the full braid group describe exchanges of identical particles, thus, the full braid group 1DURs indicate the statistics of particles [23] -[25]. The full braid group for 2D manifold has infinite number of 1DURs, , with. The corresponding particles are called anyons, including bosons for and fermions for. In order to solve this problem, we propose to associate CFs with the appropriately constructed braid subgroups instead of the full braid group and in this way to distinguish CFs from the ordinary fermions.

Figure 2. The geometrical presentation of the generator of the full braid group for R 2 and its inverse left ; in 2D right. The full braid group contains all accessible closed multi-particle classical trajectories, i. When the separation of particles is greater than twice the cyclotron radius, which situation occurs at fractional LLL fillings, the exchanges of particles along single-looped cyclotron trajectories are precluded, because the cyclotron orbits are too short for particle interchanges in this case.

Interaction cannot enhance cyclotron orbit size in the uniform multiparticle system. Particles must, however, interchange in the braid picture for the reason of defining the statistics and creation of the collective correlated state. Therefore, in order to allow exchanges again, the cyclotron radius must somehow be enhanced. This can be achieved by screening the external field, like in the construction of the CFs with flux tubes oppositely oriented with respect to the external field [22].

We suppose that the natural way to enhance the range of cyclotronic movement is to exclude inaccessible braids from the full braid group. We will show that remaining braids would be sufficiently large in size for particle exchanges realization [13] [14]. We will demonstrate below that at high magnetic fields in 2D charged N-particle systems, the multi-looped braids allow for the effective enlargement of cyclotron orbits, thus restoring particle exchanges in a natural way.

The multi-looped braids form the cyclotron braid subgroups which are generated by the following generators:. Figure 3 , e. Figure 3. It is clear that generate a subgroup of the full braid group as they are expressed by the full braid group generators. The 1DURs of the full group confined to the cyclotron subgroup they do not depend on i as 1DURs of the full braid group do not depend on i by virtue of the generators property, , , [11] are 1DURs of the cyclotron subgroup:. Thus in order to distinguish various types of composite particles one has to consider 1DURs of cyclotron braid subgroups.

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In agreement with the general rules of quantization [25] , the N-particle wave function must transform according to the 1DUR of an appropriate element of the braid group, when the particles traverse, in classical terms, a closed loop in the configuration space corresponding to this particular braid element.

In this way the wave function acquires an appropriate phase shift due to particle interchanges i. Figure 3 right. Let us emphasize that the real particles do not traverse the braid trajectories, as quantum particles do not have any trajectories, but the exchanges of arguments of the N-particle wave function can be represented by braid group elements; in 2D an exchange of particle positions described by coordinates on the plane does not resolve itself to the permutation only, as it was in 3D, but must be performed according to an appropriate element of the braid group, being in 2D not the same as the permutation group [25].

Figure 4. The cyclotron trajectories are repeated in the relative trajectory c, d with twice the radius of the individual particle trajectories a, b. In quantum language, with regards to classical multi-looped cyclotron trajecto-.

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From this observation it follows a simple rule: for q odd , each additional loop. This rule follows immediately from the definition of the cyclotron trajectory, which must be a closed individual particle trajectory related to a double interchange of the particle pair cf. Figure 5. In this way, the cyclotron trajectories of both interchanging particles are closed, just like the closed relative trajectory for the double interchange the braid trajectory for the elementary exchange of the indistinguishable particles is open in the geometrical presentation, and therefore the double interchange is needed to close this trajectory in this presentation.

If the interchange is simple, i. Nevertheless, when the interchange of particles is multi-looped, as associated with the q-type cyclotron subgroup , the. In 2D additional loops cannot enhance the total surface of the system. In this regard, it is important to emphasize the basic difference between the circumvolutions of a 3D winding e. Cyclotron trajectories of individual particles must be closed, therefore they correspond to double exchange braids, for both, simple exchanges upper and exchanges with additional loops lower , in the right part, quantization of flux per particle, for and , is indicated.

In 3D case, each circumvolution of the winding adds a new portion of the flux, just as a new circumvolution adds a new surface, which is, however, impossible in 2D. Thus in 2D all loops must share the same total flux, which results in diminishing flux-portion per a single loop and, effectively, in longer cyclotron radius allowing again particle interchanges. The additional loops in 2D take away the flux-portions equal to flux quanta just at , q odd si-.

Thus, it is clear that the CFs are actually not compositions of particles with flux-tubes, but are rightful particles in 2D corresponding to 1DURs of the cyclotron subgroups instead of the full braid group, which is unavoidably forced by too short ordinary single-looped cyclotron trajectories. It is important to emphasize that braid group approach in terms of trajectories does not describe detailed classical trajectories of particles in the system but only determines classes of trajectories which are available upon topological constraints.

Thus, if one considers physical factors which restrict availability of particular classes of trajectories e. In the case of N particles system on 2D plane in the presence of a perpendicular strong magnetic field such a condition can be based on the cyclotron radius. Although cyclotron radius is properly defined for free particles one can still consider some kind of focusing of charged particle motion on the cyclotron radius scale even in the presence of interaction, especially in homogeneous planar system with isotropic interaction.

One can also observe that for noninteracting fermions we have in the case of completely filled LLL exactly one external field. Here must be emphasized that in the case of the degenerated LLL all particles have the same cyclotron radius. Even though the velocity is not well determined in the LLL their coordinates do not commute as the operators , all particles have, however, the same kinetic energy, thus all particles have the same averaged velocity and the same cyclotron radius.

In order to determine whether a trajectory class is available for particles in the system can be brought to comparison of cyclotron orbit size with distance between particles, which for homogeneous system is defined from density and blocked by Coulomb repulsion preventing approaching one particle onto another one.

The distance between particles in homogeneous system is protected by the short-range part of the Coulomb interaction. Thus potentially available trajectory must ensure reaching neighboring particles, i. In 2D charged system in the presence of the perpendicular magnetic field only cyclotron trajectories are available for particles. Thus if one increases the magnetic field magnitude then the cyclotron radius will decrease causing the trajectory of particles exchange impossible. The simplest exchange was the implementation.

Figure 6.

If one excludes unavailable trajectories too short for exchanges, the rest of the full braid group occurs a subgroup generated by new generators describing in 2D multi-looped braids and corresponding to multi-looped cyclotron trajectories. These multi-looped cyclotron trajectories have the larger effective size allowing to match neighboring particles at strong magnetic field presence.

This subgroup we call cyclotron braid group or subgroup. Nevertheless, the cyclotron trajectories are closed trajectories despite the enumeration of particles only for closed trajectories one can define the piercing flux of the magnetic field. Thus, one must consider closed cyclotron trajectories. The smallest closed trajectory is a double exchange two semicircles create closed circle in the simplest case of single-looped exchanges.

Closed loops can be added only by one, therefore the simplest exchange with one additional loop results in three-looped cyclotron trajectory of individual particles. This explains why FQHE manifests in simplest case for. Summarizing this argumentation, we emphasize that additional loops can be added to single exchange trajectory braid group generator one by one in order to keep the exchange character of the trajectory.

Then to the closed trajectories double exchange must be added the double number of the additional loops, two in the case of. That is why braid trajectories are odd-looped trajectories 1, 3, 5, 7, etc. For LLL fillings particles traverse the closed individual cyclotron trajectories with additional loops and simultaneously open exchange trajectories with additional loops.

Those multi-looped closed trajectories in case of 2D have enhanced effective cyclotron radius which allows particles to exchange and to define the statistics. Each additional loop cannot add any new surface in 2D space. For e. For each particle at corresponding magnetic field we have 3 flux quanta. But the cyclotron trajectory is defined by a single flux quantum. However, if one considers 3-looped cyclotron trajectory in 2D with the size as trajectory corresponding to then the total flux of external field does not change.

The surface of the multi-looped trajectory also fits to the particle separation distance, but through every loop passes only single flux quantum because in 2D the total flux must be shared between all loops, which means that in the multi-looped case the effective cyclotron radius is greater than for single-looped trajectory. It is illustrated in Figure 6 for case and the arrow represents the single flux quantum defining the cyclotron radius. The Coulomb interaction plays a central role in the collective state with Laughlin correlations [17] -[19] protecting the uniform equidistant distribution of particles.

Nevertheless, in 2D systems upon the quantized magnetic field, the interaction of charges cannot be accounted for in a manner of the standard dressing of particles with the interaction as it was typical for quasiparticles in solids, because in 2D Hall regime this interaction does not have a continuous spectrum with respect to particle separation expressed by relative angular momentum projection [17] [18].

This non-continuous character of the interaction contribution in 2D charged systems upon sufficiently strong magnetic fields precludes the continuity of the mass operator which prevents the quasiparticle definition as a pole of the retarded single-particle Green function. Especially deeply developed with this regard is the present understanding of IQHE treated in single-particle and topological terms [7]. This is based on the observation that the IQHE states protected by Landau quantization gaps are not connected with 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 matter organization [6] [27].

These invariants are better and better recognized currently in terms of homotopy groups related to specially defined multidimensional transformations of physically conditioned objects like Green functions and their derivatives [28] [29] , previously developed for description of topology of textures in multicomponent condensed matter states with rich matrix order parameter, including superfluid He 3 or liquid crystals [4]. The role of various factors protecting gaps separating flat bands almost degenerated, as LLs at the interaction presence, and massively degenerated in the absence of the interaction are of particular interest in view of the role of the magnetic field breaking time reversion or other effects like spin-orbit interaction or special type time-reversion breaking traversing around the closed loop inside an elementary cell with complex hopping constants.

### References

Generalization of the familiar in mathematics Chern invariants [5] is developed in order to grasp the essential topology of various multiparticle structures [5] -[7]. 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 in analogy to their prototype in the form of IQHE.

The distinctive character of 2D space is linked in the latter case with the magnetic field flux quantization. Topological notions allow for definition of a new state of crystal called topological insulator. Despite of the 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 case of topological insulator one deals with insulating state inside the sample, whereas with conducting non-dissipative state on the sample edge, protected topologically, what is, however, no case for ordinary insulating state. 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 point of view of band organization must be characterized by flat bands which meet in summits of locally cone shaped valleys resembling Dirac points in graphene.

These Dirac points changes 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 high significance with regard to topological arrangement and related spin-type topological insulators are referred as to spin IQHE.

Commonly accepted definition of the topological insulator emphasizes the robust metallic character of the edge or surface states and extended bulk insulating states that are also robust against disorder. The edge states 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. This is the development of the interpretation of IQHE revealed a spectacular emergence of non-dissipative charge currents flowing around the edges of any finite IQHE sample.

The IQHE was observed only in the presence of an externally applied magnetic field. In Haldane presented a model of a condensed matter phase that exhibits IQHE without the need of a macroscopic magnetic field [30]. 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 per site,.

The difference between a quantum Hall state and an ordinary insulator is a matter of topology [8]. A 2D band structure consists of a mapping from the momentum k defined on a torus of the Brillouin zone to the Bloch Hamiltonian. Gapped band structures can be classified topologically by considering the equivalence classes of this mapping that can not be continuously deformed into one another without closing the energy gap.

The Chern numbers were introduced in the theory of fiber bundles [10] , and they can be understood physically in terms of the Berry phase [9] associated with the Bloch wave functions. When k traverses a closed loop, the Bloch function acquires a Berry phase given by the line integral of , or by a surface integral of the Berry field.

The Chern invariant is the total Berry field flux for the Brillouin zone,. Figure 7. The honeycomb structure similar as in graphene for the Haldane model [30] ; e 1 , e 2 are Bravais lattice vectors, nonequivalent site positions in the unit cell are indicated by. C is integer for reasons analogous to the quantization of the Dirac magnetic monopole. The Chern number, C, is a topological invariant in the sense that it cannot change when the Hamiltonian varies smoothly and this explains the quantization of conductivity in IQHE [8].