Bouketir, Ahmed

Hassan, Zainul Abidin

Talib, Zainal Abidin

Zainuddin, Hishamuddin

Universiti Putra Malaysia Institutional Repository

Physical SciencesGeometric Structures in Quantum Hall Systems*Hishamuddin Zainuddin, Ahmed 6ouketir, Zainul Al1idin Hassan, Zainal Al1idin Talil1Facultyof ScienceandEnvironmentalStudiesUniversitiPutraMalaysia43400 UPM.Serdang,SelangorMalaysiaE-mail ofCorrespondingAuthor:hisham@fsas.upm.edu.myKey words: quantization, geomrtrical & topological effects quantum hall ff dh. to-, elect, puncture surfaces, many bodyp YSICS.IntroductionQuantization being an ad-hoc proce-dure would require tailoring to suit theintrinsic properties of the physical sys-tem studied particularly the geometri-cal and topological properties. Oneparticular quantization technique thatdoes so is group-theoretic quantizationinitiated by Isham et al.(1984) forwhich the geometrical and topologicalproperties are intertwined and encodedwith some underlying group structure.This technique was later extended toinclude systems with external fields byZainuddin et al. (1989). In this project.this technique is specifically applied toHall systems of particle moving on anonlinear configuration space in anexternal homogeneous magnetic field.Materials and MethodsThe first phase of the project is to studyinteger quantum Hall systems modelledby a particle moving on the two-torusand the two-sphere. The canonicalgroup describing the kinematical sym-metries of the systems are sought forand eventually the classification of thegroups' representations are studied.The second phase is to study fractionalquantum Hall systems. which aretougher. Punctures are introduced intothe configuration space of the two-sphere to simulate the many-body ef-fect required for the fractional Halleffect. The effect of these punctures onthe (local) parental symmetry is studiedtogether with their exchange symme-tries in the hope of retrieving the ca-nonical group structure.Results and DiscussionThe canonical group for the integerquantum Hall systems on the two-torusand two-sphere are simply found to bethe universal cover and extension ofthe canonical group of the respectivesystems without the external magneticfield. The classification of their repre-sentations seem to reproduce the intgerquantum Hall states labeled by integral'magnetic monopole charge' (modulogeometrical factors) (Zainuddin et al.1998). The case of the two-torus seemto display a richer set of possibilitiesthan the case of the two-sphere whichis studied in a related research(Zainuddin et al., 2(00).The punctured case of the two-sphereproves to be a very difficult problem. Itwas shown that the SU(2) parentalsymmetry still exists locally by exam-ining the related Lie algebras. and ontop of this there is the braid groupBn(S2)giving the exchange symmetriesof the punctures with even the possi-bility of additional generators throughlifting Bn(S2)over a line bundle (Bou-ketir et al. & Zainuddin et al., 2(00).There is the difficulty of incorporatingthe braid group stucture into a whole-some canonical group and was left forfurther research. A simpler manageableapproach is to use uniformization theo-rem and pass to the universal cover ofthe punctured surface and to performthe quantization on this covering spacenamely the upper half plane. The ca-nonical (sub) group of SL(2. R) can befound but the classification of thegroup's representation does not repro-duce the wanted fractional feature.There is a slight hint of how fractionalfilling fractions can arise from thestudy of triply punctured 2-spherewhich involves the modular group asthe homotopy group but this is left forfuture research.ConclusionsGroup-theoretic quantization of Hallsystems reproduces quantum stateslabelled by integral magnetic chargeneeded for the usual integral quantumHall states. The case of the fractionalquantum Hall system can be studied byintroducing punctures for which non-trivial modification of the kinematicalsymmetries. A more tractable problemcan be obtained by using the coveringspace of the punctured surface and it ishoped that the fractional filling featurescan arise in a future study of the role ofthe homotopy group of punctured sur-faces in the quantization of such sys-tems.Benefits from the studyA coherent explanation of the topologi-cal character of the integer Hall effectis given. The study also opens up arelatively new area of research onpunctured surfaces.Literature cited in the textBouketir, A. Zainuddin, H. 2000.KinematicalSymmetrieson PuncturedSphere.InProceedings of theInternational Meeting on Frontiers ofPhysics 1998. Chia SP. BradleyDA (eds).621-627.Singapore:WorldScientific.Bouketir, A.• Zainuddin, H. 1999.Quantizationand HallEffect:NecessitiesandDifficulties.InMalaysian Science &Technology Congress '99 - Research &Development Priorities in the NewMillenium. pp 356-371. KualaLumpur:COSTAM.Isham.C.J. 1984. InRelativity. Groups andTopology II. DeWittBS, StoraR, (eds.).pp. 1061-1290.Amsterdam:North-Holland.Zainuddin,H. 1989. Group-TheoreticQuantizationof a ParticleonTorus in aConstantMagneticField.Physical ReviewD40: 636-641.Zainuddin,H. 2000.Groenewold-vanHoveProblemandGroup-TheoreticQuantizationof Hall Systemon Torus. InPresiding Fizik Kebangsaan 2000. Abd-ShukorR. MuniandySV (eds).pp 32-39.KualaLumpur:InstitutFizikMalaysia329UPM Research Report 1997-2000. Section 2-Extended AbstractsProject Publications inRefereed JournalsBouketir, A .• Zainuddin. H. 1998. AngularMomentum of Electromagnetic Field byGroup-Theoretic Quantization of IntegerHall System on a Sphere. Solid State Sci.'" Technol. 6: 74-83.Zainuddin, H .. Hassan. Z.A.•Talib, Z.A.1998a. Group- Theoretic Quantization.Topology and Integer Quantum HallEffect on Two-Torus. J. Solid State Sci.Techno!. Lett. 5: 53-60.330Project Publications InConference ProceedingsNone.Graduate ResearchAhmed Bouketir, 2000. MathematicalPhysics [Ph.DJ. Universiti PutraMalaysia.

Geometric structures in quantum Hall systems

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Geometric structures in quantum Hall systems

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