Abelian and Non-Abelian Statistics in the Coherent State Representation

Physical Review X

Volumn 1, Issue 2, 2011, Pages 1-37

  Flavin, John ^a   Seidel, Alexander ^a  

^a WASHINGTON UNIVERSITY   (United States)

Author keywords

Condensed Matter Physics; Mesoscopics; Strongly Correlated Materials

Indexed keywords

ADIABATIC TRANSPORT; COHERENT STATE; COHERENT-STATE REPRESENTATIONS; CONFORMAL BLOCKS; FILLING FACTOR; FRACTIONAL QUANTUM HALL STATE; MESOSCOPICS; NON-ABELIAN STATISTICS; QUANTUM HALL STATE; QUASIPARTICLES; SELF-CONSISTENT DERIVATION; STRONGLY CORRELATED MATERIALS;

CONDENSED MATTER PHYSICS; MACHINERY; MATHEMATICAL TRANSFORMATIONS;

SEMICONDUCTING GALLIUM ARSENIDE;

EID: 84865131040     PISSN: None     EISSN: 21603308     Source Type: Journal    
DOI: 10.1103/PhysRevX.1.021015     Document Type: Article

References (77)

1. D. C. Tsui, H. L. Stormer, and A. C. Gossard, Two- Dimensional Magnetotransport in the Extreme Quantum Limit, Phys. Rev. Lett. 48, 1982, 1559.
2. X.-G. Wen and Q. Niu, Ground-State Degeneracy of the Fractional Quantum Hall States in the Presence of a Random Potential and on High-Genus Riemann Surfaces, Phys. Rev. B 41, 1990, 9377.
3. R. B. Laughlin, Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations, Phys. Rev. Lett. 50, 1983, 1395.
4. G. Moore and N. Read, Non-Abelions in the Fractional Quantum Hall Effect, Nucl. Phys. B360, 1991, 362.
5. B. Blok and X.-G. Wen, Many-Body Systems with Non- Abelian Statistics, Nucl. Phys. B374, 1992, 615.
6. A.Y. Kitaev, Fault-Tolerant Quantum Computation by Anyons, Ann. Phys. (N.Y.) 303, 2003, 2.
7. S. Das Sarma, M. Freedman, and C. Nayak, Topologically Protected Qubits from a Possible Non-Abelian Fractional Quantum Hall State, Phys. Rev. Lett. 94, 2005, 166802.
8. C. Nayak and F. Wilczek, 2n-Quasihole States Realize 2n-1-Dimensional Spinor Braiding Statistics in Paired Quantum Hall States, Nucl. Phys. B479, 1996, 529.
9. D. Arovas J. R. Schrieffer, and F. Wilczek, Fractional Statistics and theQuantum Hall Effect, Phys. Rev. Lett. 53, 1984, 722.
10. P. Bonderson, V. Gurarie, and C. Nayak, Plasma Analogy and Non-Abelian Statistics for Ising-Type Quantum Hall States, Phys. Rev. B 83, 2011, 075303.
11. V. Gurarie and C. Nayak, A Plasma Analogy and Berry Matrices for Non-Abelian Quantum Hall States, Nucl. Phys. B506, 1997, 685.
12. N. Read, Non-Abelian Adiabatic Statistics and Hall Viscosity in Quantum Hall States and pxipy Paired Superfluids, Phys. Rev. B 79, 2009, 045308.
13. N. Read, Quasiparticle Spin from Adiabatic Transport in Quantum Hall Trial Wavefunctions, arXiv:0807.3107.
14. R. Willett, J. P. Eisenstein, H. L. Stormer, D.C. Tsui, A. C. Gossard, and J. H. English, Observation of an Even-Denominator Quantum Number in the Fractional Quantum Hall effect, Phys. Rev. Lett. 59, 1987, 1776.
15. N. Read and D. Green, Paired States of Fermions in Two Dimensions with Breaking of Parity and Time-Reversal Symmetries and the Fractional Quantum Hall Effect, Phys. Rev. B 61, 2000, 10267.
16. D. A. Ivanov, Non-Abelian Statistics of Half-Quantum Vortices in p-Wave Superconductors, Phys. Rev. Lett. 86, 2001, 268.
17. A. Stern, F. von Oppen, and E. Mariani, Geometric Phases and Quantum Entanglement as Building Blocks for Non- Abelian Quasiparticle Statistics, Phys. Rev. B 70, 2004, 205338.
18. M. Stone and S.-B. Chung, Fusion Rules and Vortices in pxipy Superconductors, Phys. Rev. B 73, 2006, 014505.
19. M. Oshikawa, Y. B. Kim, K. Shtengel, C. Nayak, and S. Tewari, Topological Degeneracy of Non-Abelian States for Dummies, Ann. Phys. (N.Y.) 322, 2007, 1477.
20. J. K. Jain, Composite-Fermion Approach for the Fractional Quantum Hall Effect, Phys. Rev. Lett. 63, 1989, 199.
21. A. Seidel, H. Fu, D.-H. Lee, J. M. Leinaas, and J. E. Moore, Incompressible Quantum Liquids and New Conservation Laws, Phys. Rev. Lett. 95, 2005, 266405.
22. A. Seidel and D.-H. Lee, Abelian and Non-Abelian Hall Liquids and Charge-Density Wave: Quantum Number Fractionalization in One and Two Dimensions, Phys. Rev. Lett. 97, 2006, 56804.
23. A. Seidel and D.-H. Lee, Domain-Wall-Type Defects as Anyons in Phase Space, Phys. Rev. B 76, 2007, 155101.
24. A. Seidel and K. Yang, Halperin m;m0; n Bilayer Quantum Hall States on Thin Cylinders, Phys. Rev. Lett. 101, 2008, 036804.
25. A. Seidel, Pfaffian Statistics Through Adiabatic Transport in the 1D Coherent State Representation, Phys. Rev. Lett. 101, 2008, 196802.
26. A. Seidel, S-Duality Constraints on 1D Patterns Associated with Fractional Quantum Hall States, Phys. Rev. Lett. 105, 2010, 026802.
27. A. Seidel and K. Yang, Gapless Excitations in the Haldane-Rezayi State: The Thin-Torus Limit, Phys. Rev. B 84, 2011, 085122.
28. E. J. Bergholtz and A. Karlhede, Half-Filled Lowest Landau Level on a Thin Torus, Phys. Rev. Lett. 94, 2005, 26802.
29. E. J. Bergholtz and A. Karlhede, 'One-Dimensional' Theory of the Quantum Hall System, J. Stat. Mech. (2006) L04001.
30. E. J. Bergholtz, J. Kailasvuori, E. Wikberg, T. H. Hansson, and A. Karlhede, Pfaffian Quantum Hall State Made Simple: Multiple Vacua and Domain Walls on a Thin Torus, Phys. Rev. B 74, 2006, 081308.
31. E. J. Bergholtz, T. H. Hansson, M. Hermanns, and A. Karlhede, Microscopic Theory of the Quantum Hall Hierarchy, Phys. Rev. Lett. 99, 2007, 256803.
32. E. J. Bergholtz and A. Karlhede, Quantum Hall System in Tao-Thouless Limit, Phys. Rev. B 77, 2008, 155308.
33. E. Wikberg, E. J. Bergholtz, and A. Karlhede, Spin Chain Description of Rotating Bosons at 1, J. Stat. Mech. (2009) P07038.
34. F. D. M. Haldane and E. H. Rezayi, Finite-Size Studies of the Incompressible State of the Fractionally Quantized Hall Effect and its Excitations, Phys. Rev. Lett. 54, 1985, 237.
35. B. A. Bernevig and F. D. M. Haldane, Model Fractional Quantum Hall States and Jack Polynomials, Phys. Rev. Lett. 100, 2008, 246802.
36. B. A. Bernevig and F. D. M. Haldane, Generalized Clustering Conditions of Jack Polynomials at Negative Jack Parameter, Phys. Rev. B 77, 2008, 184502.
37. B. A. Bernevig and F. D. M. Haldane, Properties of Non- Abelian Fractional Quantum Hall States at Filling k=r, Phys. Rev. Lett. 101, 2008, 246806.
38. X.-G. Wen and Z. Wang, Classication of Symmetric Polynomials of Infinite Variables: Construction of Abelian and Non-Abelian Quantum Hall States, Phys. Rev. B 77, 2008, 235108.
39. X.-G. Wen and Z. Wang, Topological properties of Abelian and non-Abelian Quantum Hall States from the Pattern of Zeros, Phys. Rev. B 78, 2008, 155109.
40. M. Barkeshli and X.-G. Wen, Non-Abelian Two- Component Fractional Quantum Hall States, Phys. Rev. B 82, 2010, 233301.
41. M. Barkeshli and X.-G. Wen, Classification of Abelian and non-Abelian Multilayer Fractional Quantum Hall States through the Pattern of Zeros, Phys. Rev. B 82, 2010, 245301.
42. P. Bonderson, Non-Abelian Anyons and Interferometry, Ph.D. thesis, California Institute of Technology, 2007
43. A.Y. Kitaev, Anyons in an Exactly Solved Model and Beyond, Ann. Phys. (N.Y.) 321, 2006, 2.
44. G. Moore and N. Seiberg, Polynomial Equations for Rational Conformal Field Theories, Phys. Lett. B 212, 1988, 451.
45. E. Ardonne, E. J. Bergholtz, J. Kailasvuori, and E. Wikberg, Degeneracy of Non-Abelian Quantum Hall States on the Torus: Domain Walls and Conformal Field Theory, J. Stat. Mech. (2008) P04016.
46. E. Ardonne, Domain Walls, Fusion Rules, and Conformal Field Theory in the Quantum Hall Regime, Phys. Rev. Lett. 102, 2009, 180401.
47. N. Read and E. H. Rezayi, Beyond Paired Quantum Hall States: Parafermions and Incompressible States in the First Excited Landau Level, Phys. Rev. B 59, 1999, 8084.
48. Y.-M. Lu, X.-G. Wen, Z. Wang, and Z. Wang, Non- Abelian Quantum Hall States and their Quasiparticles: From the Pattern of Zeros to Vertex Algebra, Phys. Rev. B 81, 2010, 115124.
49. F. D. M. Haldane, Many-Particle Translational Symmetries of Two-Dimensional Electrons at Rational Landau-Level Filling, Phys. Rev. Lett. 55, 1985, 2095.
50. E. H. Rezayi and F. D. M. Haldane, Laughlin State on Stretched and Squeezed Cylinders and Edge Excitations in the Quantum Hall Effect, Phys. Rev. B 50, 1994, 17199.
51. N. Read and E. H. Rezayi, Quasiholes and Fermionic Zero Modes of Paired Fractional Quantum Hall States: The Mechanism for Non-Abelian Statistics, Phys. Rev. B 54, 1996, 16864.
52. F. D. M. Haldane, Fractional Quantization of the Hall Effect: A Hierarchy of Incompressible Quantum Fluid States, Phys. Rev. Lett. 51, 1983, 605.
53. T. Einarsson Fractional Statistics on a Torus, Phys. Rev. Lett. 64, 1990, 1995.
54. S. A. Trugman and S. Kivelson, Exact Results for the Fractional Quantum Hall Effect with General Interactions, Phys. Rev. B 31, 1985, 5280.
55. R. Tao and D. J. Thouless, Fractional Quantization of Hall Conductance, Phys. Rev. B 28, 1983, 1142.
56. S. Jansen, E. H. Lieb, and R. Seiler, Symmetry Breaking in Laughlin's State on a Cylinder, Commun. Math. Phys. 285, 2008, 503.
57. W. P. Su, J. R. Schrieffer, and A. J. Heeger, Soliton Excitations in Polyacetylene, Phys. Rev. B 22, 1980, 2099.
58. The other sector can be reached by multiplication with i i.
59. An hy-dependent phase has been dropped for simplicity. As a result, note that the original Laughlin state Eq. (24) is single valued in hy, whereas Eq. (25) is not.
60. M.V. Berry, Quantal Phase Factors Accompanying Adiabatic Changes, Proc. R. Soc. A 392, 1984, 45.
61. B. Simon, Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase, Phys. Rev. Lett. 51, 1983, 2167.
62. F. Wilczek and A. Zee, Appearance of Gauge Structure in Simple Dynamical Systems, Phys. Rev. Lett. 52, 1984, 2111.
63. Some care must be given to fermion negative signs at odd denominator filling factors, in equations such as (39), (40), and (17). See Ref. [23].
64. M. Greiter, X.-G.Wen, and F.Wilczek, Paired Hall States, Nucl. Phys. B374, 1992, 567.
65. Note that a coordinate shift in particular, changes both the magnetic vector potential and thequasiperiodic boundary condition in x on wave functions. The constant in A 0 determines the locations of the LLL orbitals 'n. An additional phase twist in the magnetic boundary condition in x does the same for the orbitals 'n. In this sense, fixing and the magnetic boundary conditions leads to a preferred set of coordinate systems on the torus, which, up to scaling (!), is symmetric with respect to the LLL basesn and n. Here, the index n is always defined via properties under magnetic translations, Eq. (6).
66. One may consider a generalized version of the coherent state (66), with replaced by1 in the first factor, and by2 in the second. Consistent behavior of this expression under requires12 mod= . Consistent behavior under I requires12 mod= . This indicates that either1;2 0 mod=
67. or1;2=2mod=2 . We can then take12 without loss of generality, since shiftingi by=
68. only results in an overall change of phase; cf. Sec. III B, where furthermore
69. 1=2 is derived.
70. Here, an additional phase factor eithat was present in Eq. (79), which would arise in the off-diagonal matrix element with the conventions of the preceding sections, has been absorbed into a sign convention for the adiabatically continued domain-wall state basis.
71. Other choices for the configurations of the quasiholes will result in the same braiding matrices, as it should be.
72. We leave it understood that this relation holds modulo 2.
73. J. K. Slingerland and F. A. Bais, Quantum Groups and Non-Abelian Braiding in Quantum Hall Systems, Nucl. Phys. B612, 2001, 229.
74. J. Preskill Lecture Notes for Physics 219: Quantum Computation (2004),http://www.theory.caltech.edu/ ~preskill/ph219/topological.pdf.
75. P. Bonderson (private communication).
76. S. H. Simon, E. H. Rezayi, N. R. Cooper, and I. Berdnikov, Construction of a Paired Wave Function for Spinless Electrons at Filling Fraction 2=5, Phys. Rev. B 75, 2007, 75317.
77. In this case, Eqs. (B5d) and (B8b) imply 3 p , and 44j from Eq. (B5e). From Eq. (B12), we must then have 1 1 0, hence, 2p3. The absolute values in Eq. (B14) therefore work out, and Eq. (B14) must thus hold for some phase2, which is then defined through this equation.