научная статья по теме THE COMMENSURABILITY CONDITION AND FRACTIONAL QUANTUM HALL EFFECT HIERARCHY IN HIGHER LANDAU LEVELS Физика

Текст научной статьи на тему «THE COMMENSURABILITY CONDITION AND FRACTIONAL QUANTUM HALL EFFECT HIERARCHY IN HIGHER LANDAU LEVELS»

Pis'ma v ZhETF, vol. 102, iss. 1, pp. 23-29

© 2015 July 10

The commensurability condition and fractional quantum Hall effect

hierarchy in higher Landau levels

J. Jacakl\ L. Jacak

Institute of Physics, Wroclaw University of Technology, 50-370 Wroclaw, Poland

Submitted 3 February 2015 Resubmitted 27 April 2015

The odd structure of the fractional filling hierarchy, which is referred to as the fractional quantum Hall effect, is studied in higher Landau levels using the commensurability condition. The hierarchy of fillings that are derived in this manner is consistent with the experimental observations in the first three Landau levels. The relative poverty of the fractional structure in higher Landau levels compared with the lowest Landau level is explained using commensurability topological arguments. The commensurability criterion for correlated states specific for higher Landau levels (with n > 1), including also the paired states at half fillings of the spin-subbands of these levels, is formulated.

DOI: 10.7868/S0370274X15130056

Introduction. The rich structure of the fractional quantum Hall effect (FQHE) hierarchy in the lowest Landau level (LLL) [1] is in contrast to the scarce manifestation of a similar effect in the higher Landau levels (LLs) (with n > 1) [2]. This observation has been widely illustrated in many experiments using continuously enhanced precision on increasingly higher quality 2DEG samples [2-6] but has not been explained, although it has been confirmed by various numerical exact diagonal-ization studies on finite model systems [2]. In the LLL, the main hierarchy for the FQHE is predicted by the composite fermion (CF) model by mapping the FQHE at the actual fillings onto the integer Hall effect (IQHE) in higher LLs in the resultant magnetic field, which is screened by the mean field of CF flux-tubes [7, 8]. However, a similar approach failed to explain the fractional hierarchy of correlated states in higher LLs (the CF model does not explain also some ratios outside the main hierarchy in the LLL, e.g., 4/11, 4/5, 5/13, 5/7, 3/8, 3/10, ...). To address these drawbacks, one can use the topology-based approach for the FQHE hierarchy [9, 10], which may be applied to the lowest and higher LLs. The good correspondence of the predicted hierarchy using the topological approach with experimental observations supports the applicability of this method. Using the topological commensurability condition, one can identify the reason for the experimentally observed difference between the FQHE hierarchies in the lowest and higher LLs. Moreover, analysis of the fillings of higher LLs using the commensurability criterion

-^e-mail: janusz.jacak@pwr.wroc.pl

provides information on the correlation type for particular ratios in the hierarchy in terms of the Halperin multicomponent generalization of the Laughlin function [ff, 12].

Commensurability condition in planar Hall systems. The specific and exceptional topology of a 2D plane can be expressed in terms of the first ho-motopy group of the multiparticle configuration space, which is called the braid group (infinite in 2D) [13]. In the case of a 2D manifold (plane, locally sphere or torus) and in the presence of a strong magnetic field, the special structure of the related braid group emerges referred to the cyclotron braid subgroup [9, 10]. The one-dimensional unitary representations (lDURs) of braid groups, in particular, of the cyclotron braid subgroup, weigh the contributions of nonhomotopic classes of trajectories to the path integral [14, 15]. The braids (elements of braid groups) describe the particle exchanges along classical trajectories and can be referred to exchanges of variables ..., zjy of the multiparticle wave function ^(zi,..., zjy). This wave function must gain the phase shift according to the fDUR of the braid, which describes the particular exchange of its variables [16, 17]. On the plane, the coordinate exchanges represented by braids are not permutations as in 3D. Distinct lDURs of the appropriate 2D braid groups, in particular of the cyclotron braid subgroups at the quantizing magnetic field presence, coincide with the Laughlin statistics correlations [9].

The construction of the cyclotron braids is based on the interaction of particles that fixes the interparti-cle separation in the uniform planar system. The par-

tide interaction, the flat band that assures the identical kinetic energy of all particles, and the 2D topology constitute the necessary prerequisites to organize a collective FQH state. The cyclotron braid groups enable the identification of LL fillings at which the correlated multiparticle state of FQHE can be arranged by verifying the so-called commensurability condition. With the magnetic field in the 2D interacting system, the classical cyclotron orbits may or may not be commensurate with the interparticle separation. This commensurability admits the definition of the braid exchanges of particles (because the exchange trajectories must be cyclotronic orbits at magnetic field presence) which subsequently determines the statistics of a collective state using the unitary representations of the braid group. This method is general and enables the identification of FQHE correlations in the system for both the lowest and higher LLs. Higher LLs have different commensurability of orbits (because of higher kinetic energy) in comparison to the LLL, which explains why the FQHE structure is distinct in higher LLs compared with that in the LLL, as illustrated below.

In more formal terms, one can invoke the proof [15] that in the case of a not simply connected configuration space (such as a multiparticle configuration space), the path integral formula for the propagator Ia~b (which expresses the probability of the system transition form point a to point b in its configuration space) has the following form: Ia~b = 2 e'a'1 J ciA,?ei's[A''(a'6)l/R, where

7Ti denotes the full braid group, 5'[A,?(o, b)] is the action for the trajectory Av(a, 6), which links a and b with the added loop from hi , d\ is the measure in the ho-motopy sector ij of trajectories. The sum over nonhomo-topic elements of hi reflects the impossibility of defining a uniform measure in the non-continuous path space divided into disjoint nonhomotopic fragments. The factors eiav form 1DTJR of the full braid group [15]. Different lDURs correspond to different types of quantum particles that are related to the same classical particles. For dimM > 3 (M is the manifold on which the particles are located), there exist only two lDURs of the full braid group because the full braid group in this case is the permutation group. These two lDURs correspond to bosons (av = 0) and fermions (av = ir). Nevertheless, for dimM = 2, particularly for M = R2, there exists an infinite number of full-braid group lDURs with av = a G [0, 2tt), which correspond to anyons.

However, anyons do not exhaust all topological resources of the 2D plane. Another topological effect occurs in the presence of a perpendicular magnetic field that is sufficiently strong to shorten the cyclotron trajec-

tories below the interparticle distances. The braid generators, a.,,, which are exchanges of neighboring particles along the cyclotron orbits, cannot be defined in this case because they are not sufficiently long for the exchanges. Nevertheless, it has been demonstrated [9, 18] that exclusively in the 2D case, the multi-loop exchanges can match the particles, and braid generators a,, must be substituted by (o-j)9, q — odd integer (note that (<7j)q = o-j(crj)9-1, and (a.,,makes additional loops). The resulting "cyclotron braid subgroup", which is spanned by generators (o-j)9, i = 1, 2,..., N, replaces the ordinary full braid group hi in the path integral. The lDURs specific to the cyclotron braid subgroup reproduce the phase shift the same as given by the Laughlin correlations for the LLL filling factor v = ^ [9, 10]. In this way the Laughlin statistics is obtained without a need to introduce CFs with auxiliary flux-tubes, which produces the required statistics by the Aharonov-Bohm effect [19].

Thus, the condition of commensurability of classical cyclotron orbits with the interparticle separation verifies whether the particle exchange trajectories along cyclotron orbits, which are required for the braid group definition in the presence of the magnetic field, can be implemented in the system or not. The details of this approach are described in Refs. [9] and [10]. For shorthand, let us refer here to the illustration in Fig. 1. In

AB = hc/e

o

A3B = 3hc/e

A/3 3B = hc/e

Fig. 1. Schematic illustration of the commensurability (left panel) of the cyclotron orbit with the interparticle separation: ideal fitting (a); overly short cyclotron radius, i.e., particles cannot be matched (b); overly large cyclotron radius, i.e., interparticle distances cannot be conserved (c). Schematic illustration of effective cyclotron orbit enhancement in 2D because of the multi-loop trajectory structure (right panel) (the third dimension is added for visual clarity)

the left panel, the picture displays the ideal fitting of the cyclotron orbit with a particle separation that allows for their interchange (a), overly short cyclotron orbits (b), and overly large cyclotron orbits (c). When the cyclotron orbits are overly short b) (e.g., for LLL filling factor v = 1/3), in 2D system another possibility of exchanges occurs along multi-loop cyclotron orbits (cf. right panel of Fig. 1). The cyclotron orbit defines the surface A (Fig. 1, right panel, left), which is accommodated to the particle velocity (being in average identi-

cal for all particles within the LL) and expressed by the magnetic field flux, i.e., BA = in the LLL. This orbit A fits to the sample surface per particle where S is the sample area and N is the number of particles, in the case of the completely filled LLL. In a p-fold-larger field than B for a completely f

Для дальнейшего прочтения статьи необходимо приобрести полный текст. Статьи высылаются в формате PDF на указанную при оплате почту. Время доставки составляет менее 10 минут. Стоимость одной статьи — 150 рублей.

Показать целиком