Hilbert Space Structure of the Low Energy Sector of U(N) Quantum Hall Ferromagnets and Their Classical Limit
Abstract
1. Introduction
2. U(N) Ferromagnetism and Lieb–Mattis Ordering of Electronic Energy Levels
3. Low Energy Sector of U(N) Quantum Hall Ferromagnets at Filling Factor M
3.1. Boson Realization of U(N)-Spin Operators, Fock Space, Highest-Weight State and Ladder Operators
3.2. Young Tableaux, Gelfand and Fock Basis States
- The top row is read off the shape of the tableau, and it coincides with the highest weight. In terms of the occupancy numbers , we have
- The second row is read off the shape of the tableau that remains after all boxes containing the component/flavor are removed, that is, .
- ⋯
- is read off the shape of the tableau that remains after all boxes containing the flavors are removed, that is, .
- ⋯
- is read off the shape of the tableau that remains after all remaining boxes containing are removed.
- Finally, is read off the shape of the tableau that remains after all remaining boxes containing are removed.
3.2.1. U(2) Quantum Hall Ferromagnet at Filling Factor
3.2.2. U(4) Quantum Hall Ferromagnet at Filling Factor
3.2.3. U(6) Quantum Hall Ferromagnet at Filling Factor
3.3. General Dimension Formulas
4. Matrix Elements of U(N)-Spin Collective Operators
5. Grassmannian Coherent States and Nonlinear Sigma Models
5.1. Grassmannian Coherent States
5.2. Grassmannian Nonlinear Sigma Models
6. Conclusions and Outlook
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Quantum Hall Ferromagnets from Exchange Interactions
Appendix B. Proof of Proposition 1
Appendix C. Proof of Proposition 2
Appendix D. Relation between Gelfand–Tsetlin and Fock States
Appendix E. Single Landau Site Hilbert Space Basis for a Bilayer U(4) QHF at M=2
Appendix F. Explicit Particular Expressions of U(N)-Spin Matrix Elements
Appendix F.1. U(2)-Spin Matrices for M = 1 and L = 1
Appendix F.2. U(2)-Spin Matrices for M = 1 and L = 2
Appendix F.3. U(4)-Spin Matrices for M = 2 and L = 1
Appendix G. The Case of Non-Rectangular Young Tableaux
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Calixto, M.; Mayorgas, A.; Guerrero, J. Hilbert Space Structure of the Low Energy Sector of U(N) Quantum Hall Ferromagnets and Their Classical Limit. Symmetry 2022, 14, 872. https://doi.org/10.3390/sym14050872
Calixto M, Mayorgas A, Guerrero J. Hilbert Space Structure of the Low Energy Sector of U(N) Quantum Hall Ferromagnets and Their Classical Limit. Symmetry. 2022; 14(5):872. https://doi.org/10.3390/sym14050872
Chicago/Turabian StyleCalixto, Manuel, Alberto Mayorgas, and Julio Guerrero. 2022. "Hilbert Space Structure of the Low Energy Sector of U(N) Quantum Hall Ferromagnets and Their Classical Limit" Symmetry 14, no. 5: 872. https://doi.org/10.3390/sym14050872
APA StyleCalixto, M., Mayorgas, A., & Guerrero, J. (2022). Hilbert Space Structure of the Low Energy Sector of U(N) Quantum Hall Ferromagnets and Their Classical Limit. Symmetry, 14(5), 872. https://doi.org/10.3390/sym14050872