Three Models and a Ground State

Chapter
Part of the Springer Tracts in Modern Physics book series (STMP, volume 244)

Abstract

Laughlin′s theory [1, 2, 3, 4, 5, 6] for a series of fractionally quantized Hall states is first and foremost the key to an explanation for the experimentally observed, fractionally quantized plateaus in the Hall resistivity of a spin-polarized, two-dimensional electron gas realized in semiconductor inversion layers [7, 8, 9, 10, 5]. For our purposes here, however, we will view it primarily as an exact model, that is, a ground state which supports fractionally quantized excitations, and a model Hamiltonian for which this ground state is exact.

Keywords

Landau Level Braid Group Young Tableau Majorana Fermion Lower Landau Level 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    R.B. Laughlin, Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett. 50, 1395 (1983)ADSCrossRefGoogle Scholar
  2. 2.
    B.I. Halperin, Theory of the quantized Hall conductance. Helv. Phys. Acta 56, 75 (1983)Google Scholar
  3. 3.
    F.D.M. Haldane, Fractional quantization of the Hall effect: a hierarchy of incompressible quantum fluid states. Phys. Rev. Lett. 51, 605 (1983)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    R. Laughlin, Primitive and composite ground states in the fractional quantum Hall effect. Surf. Sci. 142, 163 (1984)ADSCrossRefGoogle Scholar
  5. 5.
    R. Prange, S. Girvin (eds.), The Quantum Hall Effect, 2nd edn. (Springer, New York, 1990)Google Scholar
  6. 6.
    T. Chakraborty, P. Pietiläinen, The Fractional Quantum Hall Effect, 2nd edn. (Springer, New York, 1995)CrossRefGoogle Scholar
  7. 7.
    D.C. Tsui, H.L. Stormer, A.C. Gossard, Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett. 48, 1559 (1982)ADSCrossRefGoogle Scholar
  8. 8.
    A.M. Chang, P. Berglund, D.C. Tsui, H.L. Stormer, J.C.M. Hwang, Higherorder states in the multiple-series, fractional, quantum Hall effect. Phys. Rev. Lett. 53, 997 (1984)ADSCrossRefGoogle Scholar
  9. 9.
    R. Clark, R. Nicholas, A. Usher, C. Foxon, J. Harris, Odd and even fractionally quantized states in GaAs–GaAlAs heterojunctions. Surf. Sci. 170, 141 (1986)ADSCrossRefGoogle Scholar
  10. 10.
    R. Willett, J.P. Eisenstein, H.L. Störmer, D.C. Tsui, A.C. Gossard, J.H. English, Observation of an even-denominator quantum number in the fractional quantum Hall effect. Phys. Rev. Lett. 59, 1776 (1987)ADSCrossRefGoogle Scholar
  11. 11.
    L. Landau, Diamagnetismus der Metalle. Z. Phys. 64, 629 (1930)ADSCrossRefGoogle Scholar
  12. 12.
    D.P. Arovas, Fear and loathing in the lowest Landau level, Ph.D. thesis, University of California, Santa Barbara, 1986Google Scholar
  13. 13.
    A.H. MacDonald, Laughlin states in higher Landau levels. Phys. Rev. B 30, 3550 (1984)ADSCrossRefGoogle Scholar
  14. 14.
    S.M. Girvin, T. Jach, Interacting electrons in two-dimensional Landau levels: results for small clusters. Phys. Rev. B 28, 4506 (1983)ADSCrossRefGoogle Scholar
  15. 15.
    F.D.M. Haldane, E.H. Rezayi, Periodic Laughlin–Jastrow wave functions for the fractional quantized Hall effect. Phys. Rev. B 31, 2529 (1985)ADSCrossRefGoogle Scholar
  16. 16.
    D. Yoshioka, B.I. Halperin, P.A. Lee, Ground state of two-dimensional electrons in strong magnetic fields and \({\frac{1}{3}}\) quantized Hall effect. Phys. Rev. Lett. 50, 1219 (1983)Google Scholar
  17. 17.
    M. Greiter, Quantum Hall quarks. Phys. E 1, 1 (1997)CrossRefGoogle Scholar
  18. 18.
    M. Greiter, F. Wilczek, Heuristic principle for quantized Hall states. Mod. Phys. Lett. B 4, 1063 (1990)ADSCrossRefGoogle Scholar
  19. 19.
    S.A. Trugman, S. Kivelson, Exact results for the fractional quantum Hall effect with general interactions. Phys. Rev. B 31, 5280 (1985)ADSCrossRefGoogle Scholar
  20. 20.
    B.I. Halperin, Statistics of quasiparticles and the hierarchy of fractional quantized Hall states. Phys. Rev. Lett. 52, 1583 (1984)ADSCrossRefGoogle Scholar
  21. 21.
    B.I. Halperin, Statistics of quasiparticles and the hierarchy of fractional quantized Hall states. Phys. Rev. Lett. 52, E2390 (1984)ADSCrossRefGoogle Scholar
  22. 22.
    D. Arovas, J.R. Schrieffer, F. Wilczek, Fractional statistics and the quantum Hall effect. Phys. Rev. Lett. 53, 722 (1984)ADSCrossRefGoogle Scholar
  23. 23.
    J.M. Leinaas, J. Myrheim, On the theory of identical particles. Nuovo Cimento B 37, 1 (1977)ADSCrossRefGoogle Scholar
  24. 24.
    F. Wilczek, Magnetic flux, angular momentum, and statistics. Phys. Rev. Lett. 48, 1144 (1982)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    F. Wilczek, Quantum mechanics of fractional-spin particles. Phys. Rev. Lett. 49, 957 (1982)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Y.-S. Wu, General theory for quantum statistics in two dimensions. Phys. Rev. Lett. 52, 2103 (1984)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    D.P. Arovas, R. Schrieffer, F. Wilczek, A. Zee, Statistical mechanics of anyons. Nucl. Phys. B 251, 117 (1985)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    J. Fröhlich, P.-A. Marchetti, Quantum field theory of anyons. Lett. Math. Phys. 16, 347 (1988)MATHADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    A.S. Goldhaber, R. MacKenzie, F. Wilczek, Field corrections to induced statistics. Mod. Phys. Lett. A 4, 21 (1989)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    F. Wilczek, Fractional Statistics and Anyon Superconductivity (World Scientific, Singapore, 1990)Google Scholar
  31. 31.
    A. Khare, Fractional Statistics and Quantum Theory (World Scientific, New Jersey, 2005)MATHCrossRefGoogle Scholar
  32. 32.
    M.V. Berry, Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 4557 (1984)ADSGoogle Scholar
  33. 33.
    B. Simon, Holonomy, the quantum adiabatic theorem, and Berry’s phase. Phys. Rev. Lett. 51, 2167 (1983)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    F. Wilczek, A. Zee, Appearance of gauge structure in simple dynamical systems. Phys. Rev. Lett. 52, 2111 (1984)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    F. Wilczek, A. Shapere, Geometric Phases in Physics (World Scientific, Singapore, 1989)MATHGoogle Scholar
  36. 36.
    G. Fano, F. Ortolani, E. Colombo, Configuration-interaction calculations on the fractional quantum Hall effect. Phys. Rev. B 34, 2670 (1986)ADSCrossRefGoogle Scholar
  37. 37.
    M. Greiter, Landau level quantization on the sphere. Phys. Rev. B 83, 115129 (2011)ADSCrossRefGoogle Scholar
  38. 38.
    P.A.M. Dirac, Quantised singularities in the electromagnetic field. Proc. Roy. Soc. Lon. Ser. A 133, 60 (1931)ADSCrossRefGoogle Scholar
  39. 39.
    G. Baym, Lectures on Quantum Mechanics (Benjamin/Addison Wesley, New York, 1969)MATHGoogle Scholar
  40. 40.
    M. Greiter, Microscopic formulation of the hierarchy of quantized Hall states. Phys. Lett. B 336, 48 (1994)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    Z.N.C. Ha, F.D.M. Haldane, Exact Jastrow-Gutzwiller resonant-valence-bond ground state of the spin- 12 antiferromagnetic Heisenberg chain with \(1/r^{2}\) exchange. Phys. Rev. Lett. 60, 635 (1988)Google Scholar
  42. 42.
    B.S. Shastry, Exact solution of an \(S={\frac{1}{2}}\) Heisenberg antiferromagnetic chain with long-ranged interactions. Phys. Rev. Lett. 60, 639 (1988)Google Scholar
  43. 43.
    V. I. Inozemtsev, On the connection between the one-dimensional \(S={\frac{1}{2}}\) Heisenberg chain and Haldane–Shastry model. J. Stat. Phys. 59, 1143 (1990)Google Scholar
  44. 44.
    Z.N.C. Ha, F.D.M. Haldane, "Spinon gas" description of the \(S = {\frac{1}{2}}\) Heisenberg chain with inverse-square exchange: exact spectrum and thermodynamics. Phys. Rev. Lett. 66 1529 (1991)Google Scholar
  45. 45.
    B.S. Shastry, Taking the square root of the discrete \(1/r^{2}\) model. Phys. Rev. Lett. 69, 164 (1992)Google Scholar
  46. 46.
    F.D.M. Haldane, Z.N.C. Ha, J.C. Talstra, D. Bernard, V. Pasquier, Yangian symmetry of integrable quantum chains with long-range interactions and a new description of states in conformal field theory. Phys. Rev. Lett. 69, 2021 (1992)MATHADSMathSciNetCrossRefGoogle Scholar
  47. 47.
    N. Kawakami, Asymptotic Bethe-ansatz solution of multicomponent quantum systems with \(1/r^{2}\) long-range interaction. Phys. Rev. B 46, 1005 (1992)Google Scholar
  48. 48.
    N. Kawakami, SU(N) generalization of the Gutzwiller–Jastrow wave function and its critical properties in one dimension. Phys. Rev. B 46, 3191 (1992)ADSCrossRefGoogle Scholar
  49. 49.
    J.C. Talstra, Integrability and applications of the exactly-solvable Haldane–Shastry one-dimensional quantum spin chain, Ph.D. thesis, Department of Physics, Princeton University, 1995Google Scholar
  50. 50.
    R.B. Laughlin, D. Giuliano, R. Caracciolo, O.L. White, Quantum number fractionalization in antiferromagnets. In: G. Morandi, P. Sodano, A. Tagliacozzo, V. Tognetti (eds) Field Theories for Low-Dimensional Condensed Matter Systems (Springer, Berlin, 2000)Google Scholar
  51. 51.
    B.A. Bernevig, D. Giuliano, R.B. Laughlin, Spinon attraction in spin-1/2 antiferromagnetic chains. Phys. Rev. Lett. 86, 3392 (2001)ADSCrossRefGoogle Scholar
  52. 52.
    B.A. Bernevig, D. Giuliano, R.B. Laughlin, Coordinate representation of the two-spinon wave function and spinon interaction in the Haldane–Shastry model. Phys. Rev. B 64, 024425 (2001)ADSCrossRefGoogle Scholar
  53. 53.
    M. Greiter, D. Schuricht, Many-spinon states and the secret significance of Young tableaux. Phys. Rev. Lett. 98, 237202 (2007)ADSCrossRefGoogle Scholar
  54. 54.
    Z.N.C. Ha, F.D.M. Haldane, Squeezed strings and Yangian symmetry of the Heisenberg chain with long-range interaction. Phys. Rev. B 47, 12459 (1993)ADSCrossRefGoogle Scholar
  55. 55.
    V.G. Drinfel’d, Hopf algebras and the quantum Yang–Baxter equation. Sov. Math. Dokl. 31, 254 (1985)Google Scholar
  56. 56.
    V. Chari, A. Pressley, A Guide to Quantum Groups (Cambridge University Press, Cambridge, 1998)Google Scholar
  57. 57.
    V.E. Korepin, N.M. Bogoliubov, A.G. Izergin, Quantum Inverse Scattering Method and Correlation Functions (Cambridge University Press, Cambridge, 1997)Google Scholar
  58. 58.
    J.C. Talstra, F.D.M. Haldane, Integrals of motion of the Haldane–Shastry model. J. Phys. A Math. Gen. 28, 2369 (1995)MATHADSMathSciNetCrossRefGoogle Scholar
  59. 59.
    B. Sutherland, Quantum many-body problem in one dimension: ground state. J. Math. Phys. 12, 246 (1971)ADSCrossRefGoogle Scholar
  60. 60.
    B. Sutherland, Quantum many-body problem in one dimension: thermodynamics. J. Math. Phys. 12, 251 (1971)ADSCrossRefGoogle Scholar
  61. 61.
    B. Sutherland, Exact result for a quantum many-body problem in one dimension. Phys. Rev. A 4, 2019 (1971)ADSCrossRefGoogle Scholar
  62. 62.
    B. Sutherland, Exact result for a quantum many-body problem in one dimension. II, Phys. Rev. A 5, 1372 (1972)ADSCrossRefGoogle Scholar
  63. 63.
    Z.N.C. Ha, F.D.M. Haldane, Elementary excitations of one-dimensional t–J model with inverse-square exchange. Phys. Rev. Lett. 73, 2887 (1994)ADSCrossRefGoogle Scholar
  64. 64.
    Z.N.C. Ha, F.D.M. Haldane, Elementary excitations of one-dimensional t–J model with inverse-square exchange. Phys. Rev. Lett. 74, E3501 (1995)ADSCrossRefGoogle Scholar
  65. 65.
    M.C. Gutzwiller, Effect of correlation on the ferromagnetism of transition metals. Phys. Rev. Lett. 10, 159 (1963)ADSCrossRefGoogle Scholar
  66. 66.
    M. Gaudin, Gaz coulombien discretà une dimension. J. Phys. (Paris) 34, 511 (1973)MathSciNetCrossRefGoogle Scholar
  67. 67.
    M.L. Mehta, G.C. Mehta, Discrete Coulomb gas in one dimension: correlation functions. J. Math. Phys. 16, 1256 (1975)ADSCrossRefGoogle Scholar
  68. 68.
    T.A. Kaplan, P. Horsch, P. Fulde, Close relation between localized-electron magnetism and the paramagnetic wave function of completely itinerant electrons. Phys. Rev. Lett. 49, 889 (1982)ADSCrossRefGoogle Scholar
  69. 69.
    C. Gros, R. Joynt, T.M. Rice, Antiferromagnetic correlations in almostlocalized fermi liquids. Phys. Rev. B 36, 381 (1987)ADSCrossRefGoogle Scholar
  70. 70.
    W. Metzner, D. Vollhardt, Ground-state properties of correlated fermions: exact analytic results for the Gutzwiller wave function. Phys. Rev. Lett. 59, 121 (1987)ADSCrossRefGoogle Scholar
  71. 71.
    F. Gebhard, D. Vollhardt, Correlation functions for Hubbard-type models: the exact results for the Gutzwiller wave function in one dimension. Phys. Rev. Lett. 59, 1472 (1987)ADSCrossRefGoogle Scholar
  72. 72.
    W. Marshall, Antiferromagnetism. Proc. R. Soc. (London), Ser. A 232, 48 (1955)MATHADSCrossRefGoogle Scholar
  73. 73.
    K. Gottfried, Quantum mechanics, vol. I, Fundamentals (Benjamin/Addison Wesley, New York, 1966)Google Scholar
  74. 74.
    K.G. Wilson, Proof of a conjecture by Dyson. J. Math. Phys. 3, 1040 (1962)MATHADSCrossRefGoogle Scholar
  75. 75.
    V. Kalmeyer, R.B. Laughlin, Equivalence of the resonating-valence-bond and fractional quantum Hall states. Phys. Rev. Lett. 59, 2095 (1987)ADSCrossRefGoogle Scholar
  76. 76.
    S.A. Kivelson, D.S. Rokhsar, Quasiparticle statistics in time-reversal invariant states. Phys. Rev. Lett. 61, 2630 (1988)ADSCrossRefGoogle Scholar
  77. 77.
    Z. Zou, B. Doucot, B.S. Shastry, Equivalence of fractional Hall and resonatingvalence-bond states on a square lattice. Phys. Rev. B 39, 11424 (1989)ADSCrossRefGoogle Scholar
  78. 78.
    X.G. Wen, F. Wilczek, A. Zee, Chiral spin states and superconductivity. Phys. Rev. B 39, 11413 (1989)ADSCrossRefGoogle Scholar
  79. 79.
    V. Kalmeyer, R.B. Laughlin, Theory of the spin liquid state of the heisenberg antiferromagnet. Phys. Rev. B 39, 11879 (1989)ADSCrossRefGoogle Scholar
  80. 80.
    R.B. Laughlin, Z. Zou, Properties of the chiral-spin-liquid state. Phys. Rev. B 41, 664 (1990)ADSCrossRefGoogle Scholar
  81. 81.
    D.F. Schroeter, E. Kapit, R. Thomale, M. Greiter, Spin Hamiltonian for which the chiral spin liquid is the exact ground state. Phys. Rev. Lett. 99, 097202 (2007)ADSCrossRefGoogle Scholar
  82. 82.
    R. Thomale, E. Kapit, D.F. Schroeter, M. Greiter, Parent Hamiltonian for the chiral spin liquid. Phys. Rev. B 80, 104406 (2009)ADSCrossRefGoogle Scholar
  83. 83.
    M. Greiter, D. Schuricht, No attraction between spinons in the Haldane–Shastry model. Phys. Rev. B 71, 224424 (2005)ADSCrossRefGoogle Scholar
  84. 84.
    M. Greiter, D. Schuricht, Comment on “Spinon Attraction in Spin-1/2 Antiferromagnetic Chains”. Phys. Rev. Lett. 96, 059701 (2006)ADSCrossRefGoogle Scholar
  85. 85.
    M. Greiter, Statistical phases and momentum spacings for one-dimesional anyons. Phys. Rev. B 79, 064409 (2009)ADSCrossRefGoogle Scholar
  86. 86.
    F.H.L. Eßler, A note on dressed S-matrices in models with long-range interactions. Phys. Rev. B 51, 13357 (1995)ADSCrossRefGoogle Scholar
  87. 87.
    Z.N.C. Ha, F.D.M. Haldane, "Fractional statistics" in arbitrary dimensions: a generalization of the Pauli principle. Phys. Rev. Lett. 67, 937 (1991)ADSMathSciNetCrossRefGoogle Scholar
  88. 88.
    M. Hamermesh, Group Theory and its Application to Physical Problems (Addison-Wesley, Reading, 1962)MATHGoogle Scholar
  89. 89.
    T. Inui, Y. Tanabe, Y. Onodera, Group Theory and Its Applications in Physics (Springer, Berlin, 1996)MATHGoogle Scholar
  90. 90.
    G. Moore, N. Read, Nonabelions in the fractional quantum Hall effect. Nucl. Phys. B 360, 362 (1991)ADSMathSciNetCrossRefGoogle Scholar
  91. 91.
    M. Greiter, X.G. Wen, F. Wilczek, Paired Hall state at half filling. Phys. Rev. Lett. 66, 3205 (1991)ADSCrossRefGoogle Scholar
  92. 92.
    M. Greiter, X.G. Wen, F. Wilczek, Paired Hall states. Nucl. Phys. B 374, 567 (1992)MATHADSMathSciNetCrossRefGoogle Scholar
  93. 93.
    W. Pan, J.-S. Xia, V. Shvarts, D.E. Adams, H.L. Stormer, D.C. Tsui, L.N. Pfeiffer, K.W. Baldwin, K.W. West, Exact quantization of the even-denominator fractional quantum Hall state at \(\nu = 5/2\) Landau level filling factor. Phys. Rev. Lett. 83, 3530 (1999)Google Scholar
  94. 94.
    J.S. Xia, W. Pan, C.L. Vicente, E.D. Adams, N.S. Sullivan, H.L. Stormer, D.C. Tsui, L.N. Pfeiffer, K.W. Baldwin, K.W. West, Electron correlation in the second Landau level: a competition between many nearly degenerate quantum phases. Phys. Rev. Lett. 93, 176809 (2004)ADSCrossRefGoogle Scholar
  95. 95.
    W. Pan, J.S. Xia, H.L. Stormer, D.C. Tsui, C. Vicente, E.D. Adams, N.S. Sullivan, L.N. Pfeiffer, K.W. Baldwin, K.W. West, Experimental studies of the fractional quantum Hall effect in the first excited Landau level. Phys. Rev. B 77, 075307 (2008)ADSCrossRefGoogle Scholar
  96. 96.
    C. Zhang, T. Knuuttila, Y. Dai, R.R. Du, L.N. Pfeiffer, K.W. West, \(\nu = 5/2\) fractional quantum Hall effect at 10 T: implications for the Pfaffian state. Phys. Rev. Lett. 104, 166801 (2010)Google Scholar
  97. 97.
    R.H. Morf, Transition from quantum Hall to compressible states in the second Landau level: new light on the \(\nu = 5/2\) enigma. Phys. Rev. Lett. 80, 1505 (1998)Google Scholar
  98. 98.
    G. Möller, S.H. Simon, Paired composite-fermion wave functions. Phys. Rev. B 77, 075319 (2008)ADSCrossRefGoogle Scholar
  99. 99.
    M. Storni, R.H. Morf, S.D. Sarma Fractional quantum Hall state at \(\nu = 5/2\) and the Moore-Read Pfaffian. Phys. Rev. Lett. 104, 076803 (2010)Google Scholar
  100. 100.
    R. Thomale, A. Sterdyniak, N. Regnault, B.A. Bernevig, Entanglement gap and a new principle of adiabatic continuity. Phys. Rev. Lett. 104, 180502 (2010)ADSCrossRefGoogle Scholar
  101. 101.
    M. Dolev, M. Heiblum, V. Umansky, A. Stern, D. Mahalu, Observation of a quarter of an electron charge at the \(\nu = 5/2\) quantum Hall state. Nature 452, 829 (2008)Google Scholar
  102. 102.
    I.P. Radu, J.B. Miller, C.M. Marcus, M.A. Kastner, L.N. Pfeiffer, K.W. West, Quasi-particle properties from tunneling in the \(\nu = 5/2\) fractional quantum Hall state. Science 320, 899 (2008)Google Scholar
  103. 103.
    J. Bardeen, L.N. Cooper, J.R. Schrieffer, Microscopic theory of superconductivity. Phys. Rev. 106, 162 (1957)ADSMathSciNetCrossRefGoogle Scholar
  104. 104.
    J.R. Schrieffer, Theory of Superconductivity (Benjamin/Addison Wesley, New York, 1964)MATHGoogle Scholar
  105. 105.
    de P.G. Gennes, Superconductivity of Metals and Alloys (Benjamin/Addison Wesley, New York, 1966)MATHGoogle Scholar
  106. 106.
    M. Tinkham, Introduction to Superconductivity (McGraw Hill, New York, 1996)Google Scholar
  107. 107.
    F. Dyson, quoted in [104], page 42.Google Scholar
  108. 108.
    M. Greiter, Is electromagnetic gauge invariance spontaneously violated in superconductors?. Ann. Phys. 319, 217 (2005)MATHADSCrossRefGoogle Scholar
  109. 109.
    P. Anderson, Considerations on the flow of superfluid helium. Rev. Mod. Phys. 38, 298 (1966)ADSCrossRefGoogle Scholar
  110. 110.
    G. Frobenius, Über die elliptischen Funktionen zweiter Art. J. Reine Angew. Math. 93, 53 (1882)CrossRefGoogle Scholar
  111. 111.
    M. Greiter, F. Wilczek, Exact solutions and the adiabatic heuristic for quantum Hall states. Nucl. Phys. B 370, 577 (1992)ADSMathSciNetCrossRefGoogle Scholar
  112. 112.
    C. Nayak, F. Wilczek, 2n-quasihole states realize \(2n-1\)-dimensional spinor braiding statistics in paired quantum Hall states. Nucl. Phys. B 479, 529 (1996)Google Scholar
  113. 113.
    N. Read, 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, 10267 (2000)ADSCrossRefGoogle Scholar
  114. 114.
    M.H. Freedman, A. Kitaev, Z. Wang, Simulation of topological field theories by quantum computer. Comm. Math. Phys. 227, 587 (2002)MATHADSMathSciNetCrossRefGoogle Scholar
  115. 115.
    S.D. Sarma, M. Freedman, C. Nayak, Topologically-protected qubits from a possible non-Abelian fractional quantum Hall state. Phys. Rev. Lett. 94, 166802 (2005)ADSCrossRefGoogle Scholar
  116. 116.
    C. Nayak, S.H. Simon, A. Stern, M. Freedman, S.D. Sarma, Rev. Mod. Phys. 80, 1083 (2008)MATHADSCrossRefGoogle Scholar
  117. 117.
    W. Bishara, P. Bonderson, C. Nayak, K. Shtengel, J.K. Slingerland, Interferometric signature of non-Abelian anyons. Phys. Rev. B 80, 155303 (2009)ADSCrossRefGoogle Scholar
  118. 118.
    J.E. Moore, Quasiparticles do the twist. Physics 2, 82 (2009)CrossRefGoogle Scholar
  119. 119.
    A. Stern, Non-Abelian states of matter. Nature 464, 187 (2010)ADSCrossRefGoogle Scholar
  120. 120.
    N.B. Kopnin, M. M. Salomaa, Mutual friction in superfluid \(^3{\rm he}{:}\) effects of bound states in the vortex core. Phys. Rev. B 44, 9667 (1991)Google Scholar
  121. 121.
    D.A. Ivanov, Non-Abelian statistics of half-quantum vortices in p-wave superconductors. Phys. Rev. Lett. 86, 268 (2001)ADSCrossRefGoogle Scholar
  122. 122.
    A. Stern, von F. Oppen, E. Mariani, Geometric phases and quantum entanglement as building blocks for non-Abelian quasiparticle statistics. Phys. Rev. B 70, 205338 (2004)ADSCrossRefGoogle Scholar
  123. 123.
    L.H. Kauffman, Knots and Physics (World Scientific, Singapore, 1993)Google Scholar
  124. 124.
    E. Fradkin, C. Nayak, A. Tsvelik, F. Wilczek, A chern-simons effective field theory for the Pfaffian quantum Hall state. Nucl. Phys. B 516, 704 (1998)MATHADSMathSciNetCrossRefGoogle Scholar
  125. 125.
    M. Levin, B.I. Halperin, B. Rosenow, Particle-hole symmetry and the Pfaffian state. Phys. Rev. Lett. 99, 236806 (2007)ADSCrossRefGoogle Scholar
  126. 126.
    S.-S. Lee, S. Ryu, C. Nayak, M.P.A. Fisher, Particle-hole symmetry and the \(\nu = 5/2\) quantum Hall state. Phys. Rev. Lett. 99, 236807 (2007)Google Scholar
  127. 127.
    S.H. Simon, E.H. Rezayi, N.R. Cooper, Pseudopotentials for multiparticle interactions in the quantum Hall regime. Phys. Rev. B 75, 195306 (2007)ADSCrossRefGoogle Scholar
  128. 128.
    M. Greiter, S=1 spin liquids: broken discrete symmetries restored. J. Low Temp. Phys. 126, 1029 (2002)CrossRefGoogle Scholar
  129. 129.
    M. Greiter, R. Thomale, Non-Abelian statistics in a quantum antiferromagnet. Phys. Rev. Lett. 102, 207203 (2009)ADSCrossRefGoogle Scholar
  130. 130.
    F.D.M. Haldane, Contiuum dynamics of the 1-D Heisenberg antiferromagnet: identification with the O(3) nonlinear sigma model. Phys. Lett. 93 A, 464 (1983)ADSMathSciNetGoogle Scholar
  131. 131.
    F.D.M. Haldane, Nonlinear field theory of large-spin Heisenberg antiferromagnets: semiclassically quantized solitons of the one-dimensional easy-axis Néel state. Phys. Rev. Lett. 50, 1153 (1983)ADSMathSciNetCrossRefGoogle Scholar
  132. 132.
    I. Affleck, Field theory methods and quantum critical phenomena, in Fields Strings and Critical Phenomena, vol. XLIX, Les Houches Lectures, ed. by E. Brézin, J. Zinn-Justin (Elsevier, Amsterdam, 1990)Google Scholar
  133. 133.
    E. Fradkin, Field Theories of Condensed Matter Systems number 82 in Frontiers in Physics (Addison Wesley, Redwood City, 1991)Google Scholar
  134. 134.
    I. Affleck, T. Kennedy, E.H. Lieb, H. Tasaki, Rigorous results on valence-bond ground states in antiferromagnets. Phys. Rev. Lett. 59, 799 (1987)ADSCrossRefGoogle Scholar
  135. 135.
    I. Affleck, T. Kennedy, E.H. Lieb, H. Tasaki, Valence bond ground states in isotropic quantum antiferromagnets. Commun. Math. Phys. 115, 477 (1988)ADSMathSciNetCrossRefGoogle Scholar
  136. 136.
    M. Greiter, S. Rachel, Valence bond solids for SU(n) spin chains: exact models, spinon confinement, and the Haldane gap. Phys. Rev. B 75, 184441 (2007)ADSCrossRefGoogle Scholar
  137. 137.
    M. Greiter, Quantum many-body physics: confinement in a quantum magnet. Nat. Phys. 6, 5 (2010)CrossRefGoogle Scholar
  138. 138.
    J. Schwinger, in Quantum Theory of Angular Momentum, ed. by L. Biedenharn, H. van Dam (Academic Press, New York, 1965)Google Scholar
  139. 139.
    A. Auerbach, Interacting Electrons and Quantum Magnetism (Springer, New York, 1994)CrossRefGoogle Scholar
  140. 140.
    D.P. Arovas, A. Auerbach, F.D.M. Haldane, Extended Heisenberg models of antiferromagnetism: analogies to the fractional quantum Hall effect. Phys. Rev. Lett. 60, 531 (1988)ADSMathSciNetCrossRefGoogle Scholar
  141. 141.
    B. Scharfenberger, M. Greiter, manuscript in preparationGoogle Scholar
  142. 142.
    N. Read, E. Rezayi, Beyond paired quantum Hall states: parafermions and incompressible states in the first excited Landau level. Phys. Rev. B 59, 8084 (1999)ADSCrossRefGoogle Scholar
  143. 143.
    P. Bouwknegt, A.W.W. Ludwig, K. Schoutens, Spinon basis for higher level SU(2) WZW models. Phys. Lett. B 359, 304 (1995)ADSMathSciNetCrossRefGoogle Scholar
  144. 144.
    J. Wess, B. Zumino, Consequences of anomalous ward identities. Phys. Lett. 37, 95 (1971)MathSciNetGoogle Scholar
  145. 145.
    E. Witten, Non-Abelian bosonization in two dimensions. Commun. Math. Phys. 92, 455 (1984)MATHADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institut für Theorie der Kondensierten Materie (TKM)Karlsruhe Institut für Technologie (KIT)KarlsruheGermany

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