The Spin pp 71-103 | Cite as

Anyons and Lowest Landau Level Anyons

  • Stéphane Ouvry
Part of the Progress in Mathematical Physics book series (PMP, volume 55)


Intermediate statistics interpolating from Bose statistics to Fermi statistics are allowed in two dimensions. This is due to the particular topology of the two dimensional configuration space of identical particles, leading to non trivial braiding of particles around each other. One arrives at quantum many-body states with a multivalued phase factor, which encodes the any- onic nature of particle windings. Bosons and fermions appear as two limiting cases. Gauging away the phase leads to the so-called anyon model, where the charge of each particle interacts “à la Aharonov-Bohm” with the fluxes carried by the other particles. The multivaluedness of the wave function has been traded off for topological interactions between ordinary particles. An alternative Lagrangian formulation uses a topological Chern-Simons term in 2+1 dimensions. Taking into account the short distance repulsion between particles leads to an Hamiltonian well defined in perturbation theory, where all perturbative divergences have disappeared. Together with numerical and semi-classical studies, perturbation theory is a basic analytical tool at disposal to study the model, since finding the exact N-body spectrum seems out of reach (except in the 2-body case which is solvable, or for particular classes of N-body eigenstates which generalize some 2-body eigenstates). However, a simplification arises when the anyons are coupled to an external homogeneous magnetic field. In the case of a strong field, by projecting the system on its lowest Landau level (LLL, thus the LLL-anyon model), the anyon model becomes solvable, i.e., the classes of exact eigenstates alluded to above provide for a complete interpolation from the LLL-Bose spectrum to the LLL-Fermi spectrum. Being a solvable model allows for an explicit knowledge of the equation of state and of the mean occupation number in the LLL, which do interpolate from the Bose to the Fermi cases. It also provides for a combinatorial interpretation of LLL-anyon braiding statistics in terms of occupation of single particle states. The LLL-anyon model might also be relevant experimentally: a gas of electrons in a strong magnetic field is known to exhibit a quantized Hall conductance, leading to the integer and fractional quantum Hall effects. Haldane/exclusion statistics, introduced to describe FQHE edge excitations, is a priori different from anyon statistics, since it is not defined by braiding considerations, but rather by counting arguments in the space of available states. However, it has been shown to lead to the same kind of thermodynamics as the LLL-anyon thermodynamics (or, in other words, the LLL-anyon model is a microscopic quantum mechanical realization of Hal- dane’s statistics). The one dimensional Calogero model is also shown to have the same kind of thermodynamics as the LLL-anyons thermodynamics. This is not a coincidence: the LLL-anyon model and the Calogero model are intimately related, the latter being a particular limit of the former. Finally, on the purely combinatorial side, the minimal difference partition problem — partition of integers with minimal difference constraints on their parts — can also be mapped on an abstract exclusion statistics model with a constant one-body density of states, which is neither the LLL-anyon model nor the Calogero model.


Partition Function Thermodynamic Limit Landau Level Identical Particle Partition Problem 
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  1. [1]
    J.M. Leinaas, J. Marnheim, On the theory of identical particles, Nuovo Cimento B37 (1977), 1–23. For an earlier work on the subject see: M.G.G. Laidlaw, C.M. de Witt, Feynman Functional Integrals for Systems of Indistinguishable Particles, Phys. Rev D3 (1971), 1375.Google Scholar
  2. [3]
    G. Moore, N. Seiberg, Polynomial equations for rational conformal field theories, Phys. Lett. B212 (1988), 451. Classical and quantum conformal field theory, Commun. Math. Phys. 123 (1989), 177. G. Moore, N. Read, Nonabelions in the fractional quantum Hall effect, Nucl. Phys. B 360(2-3) (1991), 362.MathSciNetGoogle Scholar
  3. [4]
    A.Yu. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303 (2003), 2–30.MATHCrossRefADSMathSciNetGoogle Scholar
  4. [5]
    F. Wilczek, Magnetic flux, angular momentum, and statistics, Phys. Rev. Lett. 48 (1982), 1144–1146. Quantum mechanics of fractional-spin particles, Phys. Rev. Lett. 49 (1982), 957-959.CrossRefADSMathSciNetGoogle Scholar
  5. [6]
    Y. Aharonov, D. Bohm, Significance of electromagnetic potentials in quantum theory, Phys. Rev. 115 (1959), 485–491. For an earlier work on the subject see: W. Ehrenberg, R.W. Siday, The Refractive index in electron optics and the principles of dynamics, Proc. Phys. Soc. London B62 (1949), 8-21.MATHCrossRefADSMathSciNetGoogle Scholar
  6. [7]
    R.G. Chambers, Shift of an electron interference pattern by enclosed magnetic Flux, Phys. Rev. Lett. 5 (1960), 3.CrossRefADSGoogle Scholar
  7. [8]
    Tsuyoshi Matsuda, Shuji Hasegawa, Masukazu Igarashi, Toshio Kobayashi, Masayoshi Naito, Hiroshi Kajiyama, Junji Endo, Nobuyuki Osakabe, Akira Tonomura, Ryozo Aoki, Magnetic field observation of a single flux quantum by electronholographic interferometry, Phys. Rev. Lett. 62 (1989), 2519.CrossRefADSGoogle Scholar
  8. [9]
    W. Siegel, Unextended superfields in extended super symmetry, Nucl. Phys. B156 (1979), 135–143. J.F. Schonfeld, A mass term for three-dimensional gauge fields, Nucl. Phys. B185 (1981) 157-171. R. Jackiw, S. Templeton, How superrenormalizable interactions cure their infrared divergences, Phys. Rev. D23 (1981), 2291-2304. S. Deser, R. Jackiw, S. Templeton, Three-dimensional massive gauge theories, Phys. Rev. Lett. 48 (1982), 975-978. Topologically massive gauge theories, Ann. Phys. (N.Y.) 140 (1982), 372-411.CrossRefADSGoogle Scholar
  9. [10]
    J. McCabe, S. Ouvry, Perturbative three-body spectrum and the third virial coefficient in the anyon model, Phys. Lett. B260 (1991), 113–119.MathSciNetGoogle Scholar
  10. [11]
    S. Ouvry, δ perturbative interactions in the Aharonov-Bohm and anyon models, Phys. Rev. D50 (1994), 5296–5299. A. Comtet, S.V. Mashkevich, S. Ouvry, Magnetic moment and perturbation theory with singular magnetic fields, Phys. Rev. D52 (1995), 2594-2597.ADSGoogle Scholar
  11. [12]
    C. Manuel, R. Tarrach, Contact interactions of anyons, Phys. Lett. B268 (1991), 222–226.MathSciNetGoogle Scholar
  12. [13]
    For a review on the anyon model, see (among others): J. Myrheim, Anyons, Les Houches LXIX Summer School “Topological aspects of low dimensional systems” (1998) 265–414.Google Scholar
  13. [14]
    Y.-S. Wu, Multiparticle quantum mechanics obeying fractional statistics, Phys. Rev. Lett. 53 (1984), 111–114.CrossRefADSMathSciNetGoogle Scholar
  14. [15]
    A.P. Polychronakos, Exact anyonic states for a general quadratic hamiltonian, Phys. Lett. B264 (1991), 362–366. C. Chou, Multianyon spectra and wave functions, Phys. Rev. D44 (1991), 2533-2547. S.V. Mashkevich, Exact solutions of the many-anyon problem, Int. J. Mod. Phys. A7 (1992), 7931-7942. G. Dunne, A. Lerda, S. Sciuto, C.A. Trugenberger, Exact multi-anyon wave functions in a magnetic field, Nucl. Phys. B370 (1992), 601-635. A. Karlhede, E. Westerberg, Anyons in a magnetic field, Int. J. Mod. Phys. B6 (1992), 1595-1621. S.V. Mashkevich, Towards the exact spectrum of the three-anyon problem, Phys. Lett. B295 (1992), 233-236.MathSciNetGoogle Scholar
  15. [16]
    D. Arovas, R. Schrieffer, F. Wilczek, A. Zee, Statistical mechanics of anyons, Nucl. Phys. B251 (1985), 117–126.CrossRefADSMathSciNetGoogle Scholar
  16. [17]
    A. Comtet, Y. Georgelin, S. Ouvry, Statistical aspects of the anyon model, J. Phys. A: Math. Gen. 22 (1989), 3917–3926.CrossRefADSMathSciNetGoogle Scholar
  17. [18]
    D. Sen, Spectrum of three anyons in a harmonic potential and the third virial coefficient, Phys. Rev. Lett. 68 (1992), 2977–2980. M. Sporre, J.J.M. Verbaarschot, I. Zahed, Anyon spectra and the third virial coefficient, Nucl. Phys. B389 (1993), 645-665.MATHCrossRefADSMathSciNetGoogle Scholar
  18. [19]
    M. Sporre, J.J.M. Verbaarschot, I. Zahed, Numerical solution of the three-anyon problem, Phys. Rev. Lett. 67 (1991), 1813–1816. M.V.N. Murthy, J. Law, M. Brack, R.K. Bhaduri, Quantum spectrum of three anyons in an oscillator potential, Phys. Rev. Lett. 67 (1991), 1817-1820. M. Sporre, J.J.M. Verbaarschot, I. Zahed, Four anyons in a harmonic well, Phys. Rev. B46 (1992), 5738-5741.CrossRefADSGoogle Scholar
  19. [20]
    R.K. Bhaduri, R.S. Bhalerao, A. Khare, J. Law, M.V.N. Murthy, Semiclassical two-and three-anyon partition functions, Phys. Rev. Lett. 66 (1991), 523–526. F. Illuminati, F. Ravndal, J.Aa. Ruud, A semi-classical approximation to the three-anyon spectrum, Phys. Lett. A161 (1992), 323-325. J.Aa. Ruud, F. Ravndal, Systematics of the N-anyon spectrum, Phys. Lett. B291 (1992), 137-141.CrossRefADSGoogle Scholar
  20. [21]
    A. Comtet, J. McCabe, S. Ouvry, Perturbative equation of state for a gas of anyons, Phys. Lett. B260 (1991), 372–376.MathSciNetGoogle Scholar
  21. [22]
    A. Dasnières deVeigy, S. Ouvry, Perturbative equation of state for a gas of anyons: Second order, Phys. Lett. B291 (1992), 130–136. Perturbative anyon gas, Nucl. Phys. B388 (1992), 715-755.Google Scholar
  22. [23]
    J. Marnheim, K. Olaussen, The third virial coefficient of free anyons, Phys. Lett. B299 (1993), 267–272. S.V. Mashkevich, J. Marnheim, K. Olaussen, The third virial coefficient of anyons revisited, Phys. Lett. B382 (1996), 124-130. A. Kristoffersen, S.V. Mashkevich, J. Marnheim, K. Olaussen, The fourth virial coefficient of anyons, Int. J. Mod. Phys. A11 (1998), 3723-3747. S.V. Mashkevich, J. Marnheim, K. Olaussen, R. Rietman, The nature of the three-anyon wave functions, Phys. Lett. B348 (1995), 473-480.Google Scholar
  23. [24]
    R.B. Laughlin, Quantized Hall conductivity in two dimensions, Phys. Rev. B23(1981), 5632–5633. Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations, Phys. Rev. Lett. 50 (1983), 1395-1398. Quantized motion of three two-dimensional electrons in a strong magnetic field, Phys. Rev. B27 (1983), 3383-3389. See also: F.D.M. Haldane, Fractional quantization of the Hall effect: A hierarchy of incompressible quantum fluid States, Phys. Rev. Lett. 51 (1983), 605-608.ADSCrossRefGoogle Scholar
  24. [25]
    K. von Klitzing, G. Dorda, M. Pepper, New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance, Phys. Rev. Lett. 45 (1980), 494–497. D.C. Tsui, H.L. Störmer, A.C. Gossard, Zero-resistance state of two-dimensional electrons in a quantizing magnetic field, Phys. Rev. B25 (1982), 1405-1407. M.A. Paalanen, D.C. Tsui, A.C. Gossard, Quantized Hall effect at low temperatures, Phys. Rev. B25 (1982), 5566-5569. H.L. Störmer, A. Chang, D.C. Tsui, J.C.M. Hwang, A.C. Gossard, W. Wiegmann, Fractional quantization of the Hall effect, Phys. Rev. Lett. 50 (1983), 1953-1956.CrossRefADSGoogle Scholar
  25. [26]
    D.P. Arovas, R. Schrieffer, F. Wilczek, Fractional statistics and the quantum Hall effect, Phys. Rev. Lett. 53 (1994), 722–725. B.I. Halperin, Statistics of quasiparticles and the hierarchy of fractional quantized Hall states, Phys. Rev. Lett. 52 (1984), 1583-1586.CrossRefADSGoogle Scholar
  26. [27]
    H. Kjønsberg, J. Marnheim, Numerical study of charge and statistics of Laughlin quasiparticles, Int. J. Mod. Phys. A14 (1999), 537–557. D. Banerjee, Topological aspects of phases in fractional quantum Hall effect, Phys. Lett. A269 (2000), 138-143.CrossRefADSGoogle Scholar
  27. [29]
    A. Dasnières de Veigy, S. Ouvry, Equation of state of an any on gas in a strong magnetic field, Phys. Rev. Lett. 72 (1994), 600–603.CrossRefADSGoogle Scholar
  28. [30]
    E. Fermi was the first to introduce an harmonic well confinement to compute thermodynamical quantities: E. Fermi, Sulla quantizzazione del gas perfetto monoatomico, Rend. Lincei 3 (1926), 145. In the anyon context, the harmonic well confinement was first used in [17]. See also: K. Olaussen, On the harmonic oscillator regularization of partition functions, Trondheim preprint No. 13 (1992).Google Scholar
  29. [31]
    S.V. Mashkevich, J. Marnheim, K. Olaussen, R. Rietman, Anyon trajectories and the systematics of the three-anyon spectrum, Int. J. Mod. Phys. A11 (1996), 1299–1313.MATHCrossRefADSGoogle Scholar
  30. [32]
    S. Ouvry, On the relation between the anyon and the Calogero Models, Phys. Lett. B510 (2001), 335.MATHMathSciNetGoogle Scholar
  31. [33]
    S. Isakov, G. Lozano, S. Ouvry, Non abelian Chern-Simons particles in an external magnetic field, Nucl. Phys. B552 [FS] (1999), 677.MATHCrossRefADSMathSciNetGoogle Scholar
  32. [34]
    F.D.M. Haldane, “Fractional statistics” in arbitrary dimensions: A generalization of the Pauli principle, Phys. Rev. Lett. 67 (1991), 937–940.MATHCrossRefADSMathSciNetGoogle Scholar
  33. [35]
    S.V. Mashkevich, S. Ouvry, The lowest Landau level anyon equation of state in the anti-screening regime, Phys. Lett. A310 (2003), 85–94.CrossRefGoogle Scholar
  34. [36]
    N.K. Wilkin, J.M. Gunn, R.A. Smith, Do attractive bosons condense?, Phys. Rev. Lett. 80 (1998), 2265.CrossRefADSGoogle Scholar
  35. [37]
    A. Dasnières de Veigy, S. Ouvry, One-dimensional statistical mechanics for identical particles: the Calogero and anyon cases, Mod. Phys. Lett. B9 (1995), 271.CrossRefADSGoogle Scholar
  36. [38]
    A.P. Polychronakos, Probabilities and path-integral realization of exclusion statistics. arXiv: hep-th/9503077. See also: Generalized statistics in one dimension, Les Houches LXIX Summer School “Topological aspects of low dimensional systems” (1998) 415–472.Google Scholar
  37. [39]
    A.G. Bytsko, Haldane-Wu statistics and Rogers dilogarithm, Zap. Nauchn. Semin. POMI 291 (2002), 64–77, J. Math. Sciences 125 (2005), 136-143.Google Scholar
  38. [40]
    Y.S. Wu, Statistical distribution for generalized ideal gas of fractional-statistics particles, Phys. Rev. Lett. 73 (1994), 922–925.CrossRefADSGoogle Scholar
  39. [41]
    C. Nayak, F. Wilczek, Exclusion statistics: Low-temperature properties, fluctuations, duality, and applications, Phys. Rev. Lett. 73 (1994), 2740. S. Chaturvedi, V. Srinivasan, Microscopic interpretation of Haldane’s semion statistics, Phys. Rev. Lett. 78 (1997), 4316. M.V.N. Murthy, R. Shankar, Exclusion statistics: A resolution of the problem of negative weights, Phys. Rev. B60 (1999), 6517.CrossRefADSGoogle Scholar
  40. [42]
    G. Gomila, L. Reggiani, Fractional exclusion statistics and shot noise in ballistic conductors, Phys. Rev. B63 (2001), 165404.CrossRefGoogle Scholar
  41. [43]
    F. Calogero, Solution of a three-body problem in one dimension, J. Math. Phys. 10 (1969), 2191–2196. Ground state of a one-dimensional N-body system, J. Math. Phys. 10 (1969), 2197-2200. Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971), 419-436.CrossRefADSMathSciNetGoogle Scholar
  42. [45]
    A.P. Polychronakos, Non-relativistic bosonization and fractional statistics, Nucl. Phys. B324 (1989), 597. Exchange operator formalism for integrable systems of particles, Phys. Rev. Lett. 69 (1992), 703. S. B. Isakov, Fractional statistics in one dimension: Modeling by means of 1/x 2 interaction and statistical mechanics, Int. J. Mod. Phys. A9 (1994), 2563. Generalization of statistics for several species of identical particles, Mod. Phys. Lett. B8 (1994), 319. Bosonic and fermionic single-particle states in the Haldane approach to statistics for identical particles, Phys. Rev. B53 (1996), 6585-6590. D. Bernard, Y.-S. Wu, A Note on statistical interactions and the thermodynamic Bethe ansatz, (1994). arXiv: cond-mat/9404025.CrossRefADSMathSciNetGoogle Scholar
  43. [46]
    B. Sutherland, Quantum many-body problem in one dimension: Ground state, J. Math. Phys. 12 (1971), 246–250. Quantum many-body problem in one dimension: Thermodynamics, J. Math. Phys. 12 (1971), 251-256.CrossRefADSGoogle Scholar
  44. [47]
    L. Brink, T.H. Hansson, S. Konstein, M.A. Vasiliev, The Calogero model anyonic representation, fermionic extension and super symmetry, Nucl. Phys. B401, issue 3 (1993), 591–612.MATHCrossRefADSMathSciNetGoogle Scholar
  45. [48]
    K. Schoutens, Exclusion statistics in conformal field theory spectra, Phys. Rev. Lett. 79 (1997) 2608–2611. P. Fendley, K. Schoutens, Cooper pairs and exclusion statistics from coupled free-fermion chains, J. Stat. Mech. 0207 (2007), 17.MATHCrossRefADSMathSciNetGoogle Scholar
  46. [49]
    S. Isakov, T. Martin, S. Ouvry, Conductance and shot noise for particles with exclusion statistics, Phys. Rev. Lett. 83 (1999), 580.CrossRefADSGoogle Scholar
  47. [50]
    G.E. Andrews, The Theory of partitions, Cambridge University Press, Cambridge (1998). G.E. Andrews, R. Askey, R. Roy, Special functions, Encyclopedia of Mathematics and its applications 71, Cambridge University Press, Cambridge (1999).MATHGoogle Scholar
  48. [51]
    A. Comtet, S.N. Majumdar, S. Ouvry, Integer partitions and exclusion statistics, J. Phys. A: Math. Theor. 40 (2007), 11255.MATHCrossRefADSMathSciNetGoogle Scholar
  49. [52]
    G.H. Hardy, S. Ramanujan, Proc. London. Math. Soc. 17 (1918), 75.CrossRefGoogle Scholar
  50. [53]
    A. Comtet, S.N. Majumdar, S. Ouvry, S. Sabhapandit, Integer partitions and exclusion statistics: Limit shapes and the largest part of Young diagrams, J. Stat. Mech. (2007) P10001.Google Scholar
  51. [54]
    A.M. Vershik, Statistical mechanics of combinatorial partitions and their limit shapes, Functional Analysis and Its Applications 30 (1996), 90.MATHCrossRefMathSciNetGoogle Scholar
  52. [56]
    L. Saminadayar, D.C. Glattli, Y. Jin, B. Etienne, Observation of the e/3 fractionally charged Laughlin quasiparticle, Phys. Rev. Lett 79 (1997), 162. R. de-Picciotto et al, Nature 389 (1997), 162.CrossRefGoogle Scholar
  53. [57]
    D.E. Feldman, Y. Gefen, A.Yu. Kitaev, K.T. Law, A. Stern, Shot noise in anyonic Mach-Zehnder interferometer, Phys. Rev. B76 (2007), 085333.CrossRefGoogle Scholar

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© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • Stéphane Ouvry
    • 1
  1. 1.Laboratoire de Physique Théorique et Modèles StatistiquesUniversité Paris-SudOrsay CedexFrance

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