Strong Fermion Interactions in Fractional Quantum Hall States: Correlation Functions

1. (Chapter 1 Title: Introduction to Quantum Hall States.)Historical background of many-body interactions in the context of integral and fractional quantumHall effect. Review of experimentally observed families of IQL states and their interpretations2. (Chapter 2 Title: The composite Fermion hierarchy and justification of the CF approach.)Jain's mean field composite Fermion picture, Laughlin-Jastrow type correlations. The workof the University of Tennessee group in condensed matter physics justifying MFCF undercertain conditions on the interaction energy. The concept of "Effective CF angular momentum"associated to the lowest CF Landau level. Partially filled shells and Quasi-electons. Haldane'sheirarchy of Laughlin correlated daughter-states and Jain's sequence of IQL states3. (Chapter 3 Title: Correlation diagrams and the Algebra of correlation functions.)Justification of the CF approach. Establishing (with rigorous proofs) conditions under whichJain's elegant CF approach correctly predicts the angular momentum multiplets. Correlationdiagram approach to Moore-Read states provides a new insight and a more general view.Fully symmetric correlation functions for N Fermions in an IQL state for any filing factor lessthan ½ , obtained from balanced correlation diagrams. Detailed treatment of the algebraiccorrelation functions and diagrams for small values of N; overlap with the numerical workon diagonalization.4. (Chapter 4 Title: Invariant-theoretic essentials.)A self-contained brief introduction to the mathematical theory of invariants of binary formsin context of correlation diagrams and their associated correlation functions. Semi-invariantsof binary forms and dimension counting theorems. The relationship between the semi-invariantsand angular momentum multiplets in presence of quasi-electrons.5. (Chapter 5 Title: Constructions of correlation diagrams and their correlation functions.)Theorems proving existence of nontrivial fully symmetric correlation functions fora class of correlation diagrams. Symmetry groups of the balanced correlation diagramsand their utility from the computational point of view.6. (Chapter 6 Title: Trial wave functions for systems of Fermions in IQL state with generalfilling factor $n / (2 p n + - 1) < 1/2$.) Concrete constructions of fully symmetric correlationfunctions and detailed examination of their special algebraic features.7. (Chapter 7 Title: Correlation functions for some Configurations with quasi-electrons andrelated open questions. ) Concrete constructions of fully symmetric correlation functions forconfigurations of N interacting Fermions some of which are quasi-electrons, for a familytotal angular momenta. Various open mathematical questions directly related to suchconstructions.