Homotopy Approach to Fractional Quantum Hall Effect
ABSTRACT
The topology-based explanation of the origin of the fractional quantum Hall effect is summarized. The cyclotron braid subgroups crucial for this approach are introduced in order to identify the origin of Laughlin correlations in 2D Hall systems. The so-called composite fermions are explained in terms of the homotopy cyclotron braids. Some new concept for fractional Chern insulator states is formulated in terms of the homotopy condition applied to the Berry field flux quantization.
The topology-based explanation of the origin of the fractional quantum Hall effect is summarized. The cyclotron braid subgroups crucial for this approach are introduced in order to identify the origin of Laughlin correlations in 2D Hall systems. The so-called composite fermions are explained in terms of the homotopy cyclotron braids. Some new concept for fractional Chern insulator states is formulated in terms of the homotopy condition applied to the Berry field flux quantization.
Cite this paper
Jacak, J. , Łydżba, P. and Jacak, L. (2015) Homotopy Approach to Fractional Quantum Hall Effect. Applied Mathematics, 6, 345-358. doi: 10.4236/am.2015.62033.
Jacak, J. , Łydżba, P. and Jacak, L. (2015) Homotopy Approach to Fractional Quantum Hall Effect. Applied Mathematics, 6, 345-358. doi: 10.4236/am.2015.62033.
References
[1] Ryder, L.H. (1996) Quantum Field Theory. 2nd Edition, Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9780511813900 [2] Spanier, E. (1966) Algebraic Topology. Springer-Verlag, Berlin. [3] Hatcher, A. (2002) Algebraic Topology. Cambridge University Press, Cambridge. [4] Mermin, N. (1979) The Topological Theory of Defects in Ordered Media. Reviews of Modern Physics, 51, 591.
http://dx.doi.org/10.1103/RevModPhys.51.591 [5] Prodan, E. (2011) Disordered Topological Insulators: A Non-Commutative Geometry Perspective. Journal of Physics A: Mathematical and Theoretical, 44, Article ID: 113001.
http://dx.doi.org/10.1088/1751-8113/44/11/113001 [6] Hasan, M.Z. and Kane, C.L. (2010) Colloquium: Topological Insulators. Reviews of Modern Physics, 82, 3045-3067. [7] Qi, X.L. and Zhang, S.C. (2011) Topological Insulators and Superconductors. Reviews of Modern Physics, 83, 1057.
http://dx.doi.org/10.1103/RevModPhys.83.1057 [8] Thouless, D.J., Kohmoto, M., Nightingale, M.P. and den Nijs, M. (1982) Quantized Hall Conductance in a Two- Dimensional Periodic Potential. Physical Review Letters, 49, 405.
http://dx.doi.org/10.1103/PhysRevLett.49.405 [9] Berry, M.V. (1984) Quantal Phase Factors Accompanying Adiabatic Changes. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 392, 45-57. [10] Nakahara, M. (1990) Geometry, Topology and Physics. Adam Hilger, Bristol. [11] Birman, J.S. (1974) Braids, Links and Mapping Class Groups. Princeton University Press, Princeton. [12] Laidlaw, M.G. and DeWitt, C.M. (1971) Feynman Functional Integrals for Systems of Indistinguishable Particles. Physical Review D, 3, 1375-1378.
http://dx.doi.org/10.1103/PhysRevD.3.1375 [13] Jacak, J., Józwiak, I. and Jacak, L. (2009) New Implementation of Composite Fermions in Terms of Subgroups of a Braid Group. Physics Letters A, 374, 346-350.
http://dx.doi.org/10.1016/j.physleta.2009.10.075 [14] Jacak, J., Józwiak, I., Jacak, L. and Wieczorek, K. (2010) Cyclotron Braid Group Structure for Composite Fermions. Journal of Physics: Condensed Matter, 22, Article ID: 355602.
http://dx.doi.org/10.1088/0953-8984/22/35/355602 [15] Pan, W., Störmer, H.L., Tsui, D.C., Pfeiffer, L.N., Baldwin, K.W. and West, K.W. (2003) Fractional Quantum Hall Effect of Composite Fermions. Physical Review Letters, 90, Article ID: 016801.
http://dx.doi.org/10.1103/PhysRevLett.90.016801 [16] Laughlin, R.B. (1983) Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations. Physical Review Letters, 50, 1395-1398.
http://dx.doi.org/10.1103/PhysRevLett.50.1395 [17] Haldane, F.D.M. (1983) Fractional Quantization of the Hall Effect: A Hierarchy of Incompressible Quantum Fluid States. Physical Review Letters, 51, 605-608.
http://dx.doi.org/10.1103/PhysRevLett.51.605 [18] Prange, R.E. and Girvin, S.M. (1990) The Quantum Hall Effect. Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4612-3350-3 [19] Laughlin, R.B. (1983) Quantized Motion of Three Two-Dimensional Electrons in a Strong Magnetic Field. Physical Review B, 27, 3383-3389.
http://dx.doi.org/10.1103/PhysRevB.27.3383 [20] Landau, L.D. and Lifshitz, E.M. (1972) Quantum Mechanics: Non-Relativistic Theory. Nauka, Moscow. [21] Abrikosov, A.A., Gorkov, L.P. and Dzialoshinskii, I.E. (1975) Methods of Quantum Field Theory in Statistical Physics. Dover Publications Inc., Dover. [22] Jain, J.K. (1989) Composite-Fermion Approach for the Fractional Quantum Hall Effect. Physical Review Letters, 63, 199-202.
http://dx.doi.org/10.1103/PhysRevLett.63.199 [23] Wilczek, F. (1990) Fractional Statistics and Anyon Superconductivity. World Scientific, Singapore City.
http://dx.doi.org/10.1142/0961 [24] Wu, Y.S. (1984) General Theory for Quantum Statistics in Two Dimensions. Physical Review Letters, 52, 2103-2106.
http://dx.doi.org/10.1103/PhysRevLett.52.2103 [25] Sudarshan, E.C.G., Imbo, T.D. and Govindarajan, T.R. (1988) Configuration Space Topology and Quantum Internal Symmetries. Physics Letters B, 213, 471-476.
http://dx.doi.org/10.1016/0370-2693(88)91294-4 [26] Avron, J.E., Osadchy, D. and Seiler, R. (2003) A Topological Look at the Quantum Hall Effect. Physics Today, 56, 38- 42. [27] Qi, X.L. and Zhang, S.C. (2010) The Quantum Spin Hall Effect and Topological Insulators. arXiv:1001.1602v1 [cond-mat.mtrl-sci] [28] Wang, Z., Qi, X.L. and Zhang, S.C. (2010) Topological Order Parameters for Interacting Topological Insulators. Physical Review Letters, 105, Article ID: 256803.
http://dx.doi.org/10.1103/PhysRevLett.105.256803 [29] Qi, X.L. (2011) Generic Wave-Function Description of Fractional Quantum Anomalous Hall States and Fractional Topological Insulators. Physical Review Letters, 107, Article ID: 126803.
http://dx.doi.org/10.1103/PhysRevLett.107.126803 [30] Haldane, F.D.M. (1988) Model of Quantum Hall Effect without Landau Levels: Condensed Matter Realization of the “Parity Anomaly”. Physical Review Letters, 61, 2015-2018.
http://dx.doi.org/10.1103/PhysRevLett.61.2015 [31] Kourtis, S., Venderbos, J.W.F. and Daghofer, M. (2012) Fractional Chern Insulator on a Triangular Lattice of Strongly Correlated t2g Electrons. Physical Review B, 86, Article ID: 235118.
http://dx.doi.org/10.1103/PhysRevB.86.235118 [32] Parameswaran, S.A., Roy, R. and Sondhi, S.L. (2013) Fractional Quantum Hall Physics in Topological Flat Bands. Comptes Rendus Physique, 14, 816-839.
http://dx.doi.org/10.1016/j.crhy.2013.04.003 [33] Sun, K., Gu, Z., Katsura, H. and Das Sarma, S. (2011) Nearly Flatbands with Nontrivial Topology. Physical Review Letters, 106, Article ID: 236803.
http://dx.doi.org/10.1103/PhysRevLett.106.236803 [34] Sheng, D.N., Gu, Z.C., Sun, K. and Sheng, L. (2011) Fractional Quantum Hall Effect in the Absence of Landau Levels. arXiv:1102.2658v1 [cond-mat.str-el]
[1] Ryder, L.H. (1996) Quantum Field Theory. 2nd Edition, Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9780511813900 [2] Spanier, E. (1966) Algebraic Topology. Springer-Verlag, Berlin. [3] Hatcher, A. (2002) Algebraic Topology. Cambridge University Press, Cambridge. [4] Mermin, N. (1979) The Topological Theory of Defects in Ordered Media. Reviews of Modern Physics, 51, 591.
http://dx.doi.org/10.1103/RevModPhys.51.591 [5] Prodan, E. (2011) Disordered Topological Insulators: A Non-Commutative Geometry Perspective. Journal of Physics A: Mathematical and Theoretical, 44, Article ID: 113001.
http://dx.doi.org/10.1088/1751-8113/44/11/113001 [6] Hasan, M.Z. and Kane, C.L. (2010) Colloquium: Topological Insulators. Reviews of Modern Physics, 82, 3045-3067. [7] Qi, X.L. and Zhang, S.C. (2011) Topological Insulators and Superconductors. Reviews of Modern Physics, 83, 1057.
http://dx.doi.org/10.1103/RevModPhys.83.1057 [8] Thouless, D.J., Kohmoto, M., Nightingale, M.P. and den Nijs, M. (1982) Quantized Hall Conductance in a Two- Dimensional Periodic Potential. Physical Review Letters, 49, 405.
http://dx.doi.org/10.1103/PhysRevLett.49.405 [9] Berry, M.V. (1984) Quantal Phase Factors Accompanying Adiabatic Changes. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 392, 45-57. [10] Nakahara, M. (1990) Geometry, Topology and Physics. Adam Hilger, Bristol. [11] Birman, J.S. (1974) Braids, Links and Mapping Class Groups. Princeton University Press, Princeton. [12] Laidlaw, M.G. and DeWitt, C.M. (1971) Feynman Functional Integrals for Systems of Indistinguishable Particles. Physical Review D, 3, 1375-1378.
http://dx.doi.org/10.1103/PhysRevD.3.1375 [13] Jacak, J., Józwiak, I. and Jacak, L. (2009) New Implementation of Composite Fermions in Terms of Subgroups of a Braid Group. Physics Letters A, 374, 346-350.
http://dx.doi.org/10.1016/j.physleta.2009.10.075 [14] Jacak, J., Józwiak, I., Jacak, L. and Wieczorek, K. (2010) Cyclotron Braid Group Structure for Composite Fermions. Journal of Physics: Condensed Matter, 22, Article ID: 355602.
http://dx.doi.org/10.1088/0953-8984/22/35/355602 [15] Pan, W., Störmer, H.L., Tsui, D.C., Pfeiffer, L.N., Baldwin, K.W. and West, K.W. (2003) Fractional Quantum Hall Effect of Composite Fermions. Physical Review Letters, 90, Article ID: 016801.
http://dx.doi.org/10.1103/PhysRevLett.90.016801 [16] Laughlin, R.B. (1983) Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations. Physical Review Letters, 50, 1395-1398.
http://dx.doi.org/10.1103/PhysRevLett.50.1395 [17] Haldane, F.D.M. (1983) Fractional Quantization of the Hall Effect: A Hierarchy of Incompressible Quantum Fluid States. Physical Review Letters, 51, 605-608.
http://dx.doi.org/10.1103/PhysRevLett.51.605 [18] Prange, R.E. and Girvin, S.M. (1990) The Quantum Hall Effect. Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4612-3350-3 [19] Laughlin, R.B. (1983) Quantized Motion of Three Two-Dimensional Electrons in a Strong Magnetic Field. Physical Review B, 27, 3383-3389.
http://dx.doi.org/10.1103/PhysRevB.27.3383 [20] Landau, L.D. and Lifshitz, E.M. (1972) Quantum Mechanics: Non-Relativistic Theory. Nauka, Moscow. [21] Abrikosov, A.A., Gorkov, L.P. and Dzialoshinskii, I.E. (1975) Methods of Quantum Field Theory in Statistical Physics. Dover Publications Inc., Dover. [22] Jain, J.K. (1989) Composite-Fermion Approach for the Fractional Quantum Hall Effect. Physical Review Letters, 63, 199-202.
http://dx.doi.org/10.1103/PhysRevLett.63.199 [23] Wilczek, F. (1990) Fractional Statistics and Anyon Superconductivity. World Scientific, Singapore City.
http://dx.doi.org/10.1142/0961 [24] Wu, Y.S. (1984) General Theory for Quantum Statistics in Two Dimensions. Physical Review Letters, 52, 2103-2106.
http://dx.doi.org/10.1103/PhysRevLett.52.2103 [25] Sudarshan, E.C.G., Imbo, T.D. and Govindarajan, T.R. (1988) Configuration Space Topology and Quantum Internal Symmetries. Physics Letters B, 213, 471-476.
http://dx.doi.org/10.1016/0370-2693(88)91294-4 [26] Avron, J.E., Osadchy, D. and Seiler, R. (2003) A Topological Look at the Quantum Hall Effect. Physics Today, 56, 38- 42. [27] Qi, X.L. and Zhang, S.C. (2010) The Quantum Spin Hall Effect and Topological Insulators. arXiv:1001.1602v1 [cond-mat.mtrl-sci] [28] Wang, Z., Qi, X.L. and Zhang, S.C. (2010) Topological Order Parameters for Interacting Topological Insulators. Physical Review Letters, 105, Article ID: 256803.
http://dx.doi.org/10.1103/PhysRevLett.105.256803 [29] Qi, X.L. (2011) Generic Wave-Function Description of Fractional Quantum Anomalous Hall States and Fractional Topological Insulators. Physical Review Letters, 107, Article ID: 126803.
http://dx.doi.org/10.1103/PhysRevLett.107.126803 [30] Haldane, F.D.M. (1988) Model of Quantum Hall Effect without Landau Levels: Condensed Matter Realization of the “Parity Anomaly”. Physical Review Letters, 61, 2015-2018.
http://dx.doi.org/10.1103/PhysRevLett.61.2015 [31] Kourtis, S., Venderbos, J.W.F. and Daghofer, M. (2012) Fractional Chern Insulator on a Triangular Lattice of Strongly Correlated t2g Electrons. Physical Review B, 86, Article ID: 235118.
http://dx.doi.org/10.1103/PhysRevB.86.235118 [32] Parameswaran, S.A., Roy, R. and Sondhi, S.L. (2013) Fractional Quantum Hall Physics in Topological Flat Bands. Comptes Rendus Physique, 14, 816-839.
http://dx.doi.org/10.1016/j.crhy.2013.04.003 [33] Sun, K., Gu, Z., Katsura, H. and Das Sarma, S. (2011) Nearly Flatbands with Nontrivial Topology. Physical Review Letters, 106, Article ID: 236803.
http://dx.doi.org/10.1103/PhysRevLett.106.236803 [34] Sheng, D.N., Gu, Z.C., Sun, K. and Sheng, L. (2011) Fractional Quantum Hall Effect in the Absence of Landau Levels. arXiv:1102.2658v1 [cond-mat.str-el]