Skip to main content

Part of the book series: Les Houches - Ecole d’Ete de Physique Theorique ((LHSUMMER,volume 69))

  • 1700 Accesses

Abstract

These lecture notes were prepared rather soon after I completed my book on Topological quantum numbers in nonrelativistic physics, which was published by World Scientific Publishing Co. Pte. Ltd., Singapore, in early 1998. I have not attempted to make a completely fresh presentation, but have cannibalized the text of my book to produce something shorter, with a different ordering of topics. I wish to thank the publishers for allowing me to do this self-plagiarization.

This work was supported in part by the U.S. National Science Foundation, grant number DMR-9528345.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Onsager L., Nuovo Cimento 6 (1949) 249–250.

    MathSciNet  Google Scholar 

  2. Vinen W.F., The detection of single quanta of circulation in liquid helium II, Proc. Roy. Soc. (London) A 260 (1961) 218–236.

    ADS  Google Scholar 

  3. Rayfield G.W. and Reif F., Evidence for the creation and motion of quantized vortex rings in superfluid helium, Phys. Rev. Lett. 11 (1963) 305–308.

    ADS  Google Scholar 

  4. Dirac PAM, Quantized singularities in the electromagnetic field, Proc. Roy. Soc. (London) 133 (1931) 60–72.

    ADS  Google Scholar 

  5. Dirac PAM, The theory of magnetic poles, Phys. Rev. 74 (1948) 817–830.

    ADS  MATH  MathSciNet  Google Scholar 

  6. Piccard A. and Kessler E., Determination of the ratio between the electrostatic charges of the proton and of the electron, Arch. Sci. Phys. Nat. 7 (1925) 340–342.

    Google Scholar 

  7. Petley B.W., The fundamental physical constants and the frontier of measurement (A. Hilger, Bristol, 1985) pp. 282–287.

    Google Scholar 

  8. Donnelly R.J., in The collected works of Lars Onsager: with commentary (World Scientific, Singapore, 1996) pp. 693–696.

    Google Scholar 

  9. London F., Superfluids, Vol. II, (1954) pp. 151–155.

    Google Scholar 

  10. Thuneberg E.V., Introduction to the vortex sheet of superfluid 3He, Physica B 210 (1995) 287–299.

    ADS  Google Scholar 

  11. Parts U., Ruutu V.M.H., Koivuniemi J.H., Krusius M., Thuneberg E.V. and Volovik G.E., Measurements on the vortex sheet in rotating superfluid 3He-A, Physica B 210 (1995) 311–333.

    ADS  Google Scholar 

  12. Pitaevskii L.P., Vortex lines in an imperfect Bose gas, Zhur. Eksp. Teor. Fiz. 40 (1961) 454–477 [Translation in Soviet Physics JETP 13 (1961) 451].

    Google Scholar 

  13. Gross E.P., Structure of a quantized vortex in boson systems, Nuovo Cimento 20 (1961) 454–477.

    MATH  Google Scholar 

  14. Hall H.E. and Vinen W.F., The rotation of liquid helium II. I: Experiments on the propagation of second sound in uniformly rotating helium II, Proc. Roy. Soc. (London) A 238 (1956) 204.

    ADS  Google Scholar 

  15. Hall H.E. and Vinen W.F., The rotation of liquid helium II. II: The theory of mutual friction in uniformly rotating helium II, Proc. Roy. Soc. (London) A 238 (1956) 215.

    ADS  MATH  Google Scholar 

  16. Vinen W.F., Critical velocities in liquid helium II, in Proceedings of the International School of Physics Enrico Fermi Course XXI. Liquid Helium, edited by G. Careri (Academic Press, New York 1963) pp. 336–355.

    Google Scholar 

  17. Langer J.S. and Fisher M.E., Intrinsic critical velocity of a superfluid, Phys. Rev. Lett. 19 (1967) 560–563.

    ADS  Google Scholar 

  18. Muirhead C.M., Vinen W.F. and Donnelly R.J., The nucleation of vorticity by ions in rotating superfluid 4He, Phil. Trans. R. Soc. A 311 (1984) 433–467.

    ADS  Google Scholar 

  19. Donnelly R.J., Quantized vortices in helium II (Cambridge University Press, 1991).

    Google Scholar 

  20. Feynman R.P., Application of quantum mechanics to liquid helium, in Progress in Low Temperature Physics 1, edited by C.J. Gorter (North-Holland, Amsterdam, 1955), pp. 17–53.

    Google Scholar 

  21. Bardeen J., Cooper L.N. and Schrieffer J.R., Theory of superconductivity, Phys. Rev. 108 (1957) 1175–1204.

    ADS  MATH  MathSciNet  Google Scholar 

  22. London F., On the problem of the molecular theory of superconductivity, Phys. Rev. 74 (1948) 562–573.

    ADS  MATH  Google Scholar 

  23. Putterman S.J., Superfluid Hydrodynamics (North-Holland, Amesterdam, 1974) pp. 404–407.

    Google Scholar 

  24. Sonin E.B., Magnus force in superfluids and superconductors, Phys. Rev. B 55 (1997) 485–501.

    ADS  Google Scholar 

  25. Whitmore S.C. and Zimmermann W., Observation of quantized circulation in superfluid helium, Phys. Rev. 166 (1968) 181–196.

    ADS  Google Scholar 

  26. Zieve R.J., Close J.D., Davis J.C. and Packard R.E., New experiments on quantized circulation in superfluid 4He, J. Low Temp. Phys. 90 (1993) 243–268.

    ADS  Google Scholar 

  27. Zieve R.J., Mukharsky Y.M., Close J.D., Davis J.C. and Packard R.E., Investigation of quantized circulation in superfluid 3He-B, J. Low Temp. Phys. 91 (1993) 315–339.

    ADS  Google Scholar 

  28. Rayfield G.W. and Reif F., Quantized vortex rings in superfluid helium, Phys. Rev. 136 (1964) A1194–1208.

    ADS  Google Scholar 

  29. Volovik G.E., Quantum-mechanical formation of vortices in a superfluid liquid, Zhur. Eksp. Teor. Fiz. Pizma 15 (1972) 116–120 [translation in JETP Lett. 15 (1972) 81-83].

    Google Scholar 

  30. Deaver B.S. and Fairbank W.M., Experimental evidence for quantized flux in superconducting cylinders, Phys. Rev. Lett. 7 (1961) 43–46.

    ADS  Google Scholar 

  31. Doll R. and Näbauer M., Experimental proof of magnetic flux quantization in a superconducting ring, Phys. Rev. Lett. 7 (1961) 51–52.

    ADS  Google Scholar 

  32. Parks R.D. and Little W.A., Fluxoid quantization in a multiply-connected superconductor, Phys. Rev. 133 (1964) A97–102.

    ADS  Google Scholar 

  33. Gough C.E., Colcough M.S., Forgan E.M., Jordan R.G., Keene M., Muirhead C.M., Rae A.I.M., Thomas N., Abell J.S. and Sutton S., Flux quantization in a high-T c superconductor, Nature (London) 326 (1987) 855.

    ADS  Google Scholar 

  34. Abrikosov A.A., On the magnetic properties of superconductors of the second type, Zhur. Eksp. Teor. Fiz. 32 (1957) 1442–1452.; Sov. Phys. JETP 5 (1957) 1174.

    Google Scholar 

  35. Cribier D., Jacrot B., Madhov Rao L. and Farnoux B., Evidence from neutron diffraction for a periodic structure of the magnetic field in a niobium superconductor, Phys. Lett. 9 (1964) 106–107.

    ADS  Google Scholar 

  36. Essmann U. and Träuble H. The direct observation of individual flux lines in type II superconductors, Phys. Lett. 24 (1967) 526–527.

    Google Scholar 

  37. Cubitt R., Forgan E.M., Yang G., Lee S.L., Paul D.Mc.K., Mook H.A., Yethiraj M., Kes P.H., Li T.W., Menovsky A.A., Tarnawski Z. and Mortensen K., Direct observation of magnetic flux lattice melting and decomposition in the high-T c superconductor Bi2.15Sr1.95CaCu2O8+x, Nature 365 (1993) 407–411.

    ADS  Google Scholar 

  38. Bishop D.J. Gammel P.L., Huse D.A. and Murray C.A., Magnetic flux-line lattices and vortices in the copper oxide superconductors, Science 255 (1992) 165–172.

    ADS  Google Scholar 

  39. Yarmchuk E. J., Gordon M.J.V. and Packard R.E., Observation of stationary vortex arrays in rotating superfluid helium, Phys. Rev. Lett. 43 (1979) 214–217.

    ADS  Google Scholar 

  40. Yarmchuk E.J. and Packard R.E., Photographic studies of quantized vortex lines, J. Low Temp. Phys. 46 (1982) 479–515.

    ADS  Google Scholar 

  41. Josephson B.D., Possible new effects in superconductive tunnelling, Phys. Lett. 1 (1962) 251–253.

    ADS  MATH  Google Scholar 

  42. Josephson B.D., Supercurrents through barriers, Adv. Phys. 14 (1965) 419–451.

    ADS  Google Scholar 

  43. Anderson P.W., The Josephson effect and quantum coherence measurements in superconductors and superfluids, in Progress in Low Temperature Physics 5, edited by C.J. Gorter (North-Holland, Amsterdam, 1967) pp. 1–43.

    Google Scholar 

  44. Parker W.H., Taylor B.N. and Langenberg D.N., Measurement of 2e/h using the ac Josephson effect and its implications for quantum electrodynamics, Phys. Rev. Lett. 18 (1967) 287–291.

    ADS  Google Scholar 

  45. Cohen E.R. and Taylor B.N., The 1986 adjustment of the fundamental physical constants, Rev. Mod. Phys. 59 (1987) 1121.

    ADS  Google Scholar 

  46. Clarke J., Experimental comparison on the Josephson voltage-frequency relation in different superconductors, Phys. Rev. Lett. 21 (1968) 1566–1569.

    ADS  Google Scholar 

  47. Tsai J.S., Jain A.K. and Lukens E., High-precision test of the universality of the Josephson voltage-frequency relation, Phys. Rev. Lett. 51 (1983) 316–319.

    ADS  Google Scholar 

  48. Kautz R.L. and Lloyd F.L., Precision of series-array Josephson voltage standards, Appl. Phys. Lett. 51 (1987) 2043–2045.

    ADS  Google Scholar 

  49. Duan J.M., Mass of a vortex line in superfluid 4He: effects of gauge-symmetry breaking, Phys. Rev. B 49 (1994) 12381–12383.

    ADS  Google Scholar 

  50. Demircan E., Ao P. and Niu Q., Vortex dynamics in superfluids: cyclotron-type motion, Phys. Rev. B 54 (1996) 10027–10034.

    ADS  Google Scholar 

  51. Thouless D.J., Ao P., Niu Q., Geller M.R. and Wexler C., Quantized vortices in superfluids and superconductors. In the Proceedings of the 9th International Conference on Recent Progress in Many-Body Theories, edited by D. Neilson and R.F. Bishop (World Scientific, Singapore, 1998) 387–398.

    Google Scholar 

  52. Berry M.V., Quantal phase factors accompanying adiabatic changes, Proc. Roy. Soc. (London) A 392 (1984) 45–57.

    ADS  MATH  Google Scholar 

  53. Haldane F.D.M. and Wu Y.S., Quantum dynamics and statistics of vortices in two-dimensional superfluids, Phys. Rev. Lett. 55 (1985) 2887–2890.

    ADS  Google Scholar 

  54. Thouless D.J., Ao P. and Niu Q., Vortex dynamics in superfluids and the Berry phase, Physica A 200 (1993) 42–49.

    ADS  Google Scholar 

  55. Thouless D.J., Ao P. and Niu Q., Transverse force on a quantized vortex in a superfluid, Phys. Rev. Lett. 76 (1996) 3758–3761.

    ADS  Google Scholar 

  56. Tang J.M. and Thouless D.J., Longitudinal force on moving potential, Phys. Rev. B 58 (1998) 14179–2.

    ADS  Google Scholar 

  57. Wexler C., Magnus and Iordanskii forces in superfluids, Phys. Rev. Lett. 79 (1997) 1321–1324.

    ADS  Google Scholar 

  58. Laughlin R.B., Quantized Hall conductivity in two dimensions, Phys. Rev. B 23 (1981) 5632–5633.

    ADS  Google Scholar 

  59. Volovik G.E., Comment on “Transverse force on a quantized vortex in a superfluid”, Phys. Rev. Lett. 77 (1997) 4687.

    ADS  Google Scholar 

  60. Iordanskii S.V., On the mutual friction between the normal and superfluid components in a rotating Bose gas, Ann. Phys. (NY) 29 (1964) 335–349; Iordanskii S.V., Zhur. Eksp. Teor. Fiz. 49 (1965) 225-236 [translation in Soviet Phys. JETP 22 (1965) 160-167].

    ADS  Google Scholar 

  61. Geller M.R., Wexler C. and Thouless D.J., Transverse force on a quantized vortex in a superconductor, Phys. Rev. B 57 (1998) R8119–8122.

    ADS  Google Scholar 

  62. Dirac P.A.M., Peierls R.E. and Pryce M.H.L., On Lorentz invariance in the quantum theory, Proc. Cambridge Philos. Soc. 44 (1942) 143–157.

    Google Scholar 

  63. Nozières P. and Vinen W.F., Phil. Mag. 14 (1966) 667.

    ADS  Google Scholar 

  64. Klitzing Kv., Dorda G. and Pepper M., New method for high-accuracy determination of fine-structure constant based on quantized Hall resistance, Phys. Rev. Lett. 45 (1980) 494–497.

    ADS  Google Scholar 

  65. Yoshihiro K., Kinoshita J., Inayaki K., Yamanouchi C., Endo T., Murayama Y., Koyanagi M., Yagi A., Wakabayashi J. and Kawaji S., Quantum Hall effect in silicon metal-oxide-semiconductor inversion layers: Experimental conditions for determination of h/e 2, Phys. Rev. B 33 (1986) 6874–6896.

    ADS  Google Scholar 

  66. Hartland A., Jones K., Williams J.M., Gallagher B.L. and Galloway T., Direct comparison of the quantized Hall resistance in gallium arsenide and silicon, Phys. Rev. Lett. 66 (1991) 969–973.

    ADS  Google Scholar 

  67. Taylor B.N., New measurement standards for 1990, Physics Today 42 (1989) 23–26.

    Google Scholar 

  68. Aoki H. and Ando T., Effect of localization on the Hall conductivity in the two-dimensional system in strong magnetic fields, Solid St. Commun. 38 (1981) 1079–1082.

    ADS  Google Scholar 

  69. Prange R.E., Quantized Hall resistance and the measurement of the fine-structure constant, Phys. Rev. B 23 (1981) 4802–4805.

    ADS  Google Scholar 

  70. Thouless D.J., Localization and the two-dimensional Hall effect, J. Phys. C 14 (1981) 3475–3480.

    ADS  Google Scholar 

  71. Thouless D.J., Kohmoto M., Nightingale M.P. and den Nijs M., Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett. 49 (1982) 405–408.

    ADS  Google Scholar 

  72. Streda P., Theory of quantised Hall conductivity in two dimensions, J. Phys. C 15 (1982) L717–721.

    ADS  Google Scholar 

  73. Avron J.E. and Seiler R., Quantization of the Hall conductance of general multi-particle Schrödinger Hamiltonians, Phys. Rev. Lett. 54 (1985) 259–262.

    ADS  MathSciNet  Google Scholar 

  74. Niu Q., Thouless D.J. and Wu Y.S., Quantized Hall conductance as a topological invariant, Phys. Rev. B 31 (1985) 3372–3377.

    ADS  MathSciNet  Google Scholar 

  75. Büttiker M., Absence of backscattering in the quantum Hall effect in multiprobe conductors, Phys. Rev. B 38 (1988) 9375–9389.

    ADS  Google Scholar 

  76. Tsui D.C., Stormer H.L. and Gossard A.C., Two-dimensional magneto-transport in the extreme quantum limit, Phys. Rev. Lett. 48 (1982) 1559–1562.

    ADS  Google Scholar 

  77. Kubo R., Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems, J. Phys. Soc. Jpn. 12 (1957) 570–586.

    ADS  MathSciNet  Google Scholar 

  78. Kubo R. The fluctuation-dissipation theorem, Rep. Progr. Phys. 29 (1966) 255–284.

    ADS  Google Scholar 

  79. Lee P.A. and Ramakrishnan T.V., Dirordered electronic systems, Rev. Mod. Phys. 57 (1985) 287–337.

    ADS  Google Scholar 

  80. Bohm D., Note on a theorem of Bloch concerning possible causes of superconductivity, Phys. Rev. 75 (1949) 502–504.

    ADS  MATH  Google Scholar 

  81. Halperin B.I., Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential, Phys. Rev. B 25 (1982) 2185–2190.

    ADS  MathSciNet  Google Scholar 

  82. Thouless D.J., Edge voltages and distributed currents in the quantum Hall effect, Phys. Rev. Lett. 71 (1993) 1879–1882.

    ADS  Google Scholar 

  83. Azbel M.Ya., Energy spectrum of conduction electrons in a magnetic field, Zh. Eksp. Teor. Fiz. 46 (1964) 929–947 [translation in Soviet Phys. JETP 19 (1964)].

    Google Scholar 

  84. Hofstadter D., Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B 14 (1976) 2239–2249.

    ADS  Google Scholar 

  85. Niu Q. and Thouless D.J., Quantum Hall effect with realistic boundary conditions, Phys. Rev. B 35 (1987) 2188–2197.

    ADS  Google Scholar 

  86. Choquet-Bruhat Y., DeWitt-Morette C. and Dillard-Bleick M., Analysis, Manifolds and Physics (North-Holland Publishing Co., Amsterdam 1982), pp. 393–396.

    MATH  Google Scholar 

  87. Thouless D.J., Wannier functions for magnetic sub-bands, J. Phys. C 17 (1984) L325–327.

    ADS  MathSciNet  Google Scholar 

  88. Kohmoto M., Topological invariant and the quantization of the Hall conductance, Ann. Phys. (NY) 160 (1985) 343–354.

    ADS  MathSciNet  Google Scholar 

  89. Arovas D.P., Bhatt R.N., Haldane F.D.M., Littlewood P.B. and Rammal R., Localization, wave-function topology, and the integer quantized Hall effect, Phys. Rev. Lett. 60 (1988) 619–622.

    ADS  Google Scholar 

  90. Laughlin R.B., Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations, Phys. Rev. Lett. 50 (1983) 1395–1398.

    ADS  Google Scholar 

  91. Haldane F.D.M. and Rezayi E.H., Finite-size studies of the incompressible state of the fractionally quantized Hall effect and its excitations, Phys. Rev. Lett. 54 (1985) 237–240.

    ADS  Google Scholar 

  92. Jain J.K., Composite fermion approach for the fractional quantum Hall effect, Phys. Rev. Lett. 63 (1989) 199–202.

    ADS  Google Scholar 

  93. Jain J.K., Microscopic theory of the fractional quantum Hall effect, Adv. Phys. 41 (1992) 105–146.

    ADS  Google Scholar 

  94. Jain J.K. and Kamilla R.K., Composite fermions in the Hilbert space of the lowest electronic Landau level, Int. J. Mod. Phys. B 11 (1997) 2621–2660.

    ADS  Google Scholar 

  95. Anderson P. W., Remarks on the Laughlin theory of the fractionally quantized Hall effect, Phys. Rev. B 28 (1983) 2264–2265.

    ADS  Google Scholar 

  96. Tao R. and Wu Y.S., Gauge invariance and the fractional quantum Hall effect, Phys. Rev. B 30 (1984) 1097–1098.

    ADS  Google Scholar 

  97. Thouless D.J., Level crossing and the fractional quantum Hall effect, Phys. Rev. B 40 (1989) 12034–12036.

    ADS  Google Scholar 

  98. Thouless D.J. and Gefen Y., Fractional quantum Hall effect and multiple Aharonov-Bohm periods, Phys. Rev. Lett. 66 (1991) 806–909.

    ADS  Google Scholar 

  99. Gefen Y. and Thouless D.J., Detection of fractional charge and quenching of the quantum Hall effect, Phys. Rev. B 47 (1993) 10423–104236.

    ADS  Google Scholar 

  100. Halperin B.I., Theory of the quantized Hall conductance, Helv. Phys. Acta 56 (1983) 75–102.

    Google Scholar 

  101. Sondhi S.L., Karlhede A., Kivelson S.A. and Rezayi E.H., Skyrmions and the crossover from the integer to fractional quantum Hall effect at small Zeeman energies, Phys. Rev. B 47 (1993) 16419–16426.

    ADS  Google Scholar 

  102. Fertig H.A., Brey L., Cote R. and MacDonald A.H., Charged spin-texture excitations and the Hartree-Fock approximation in the quantum Hall effect, Phys. Rev. B 50 (1994) 11018–11021.

    ADS  Google Scholar 

  103. Xie X.C. and He S., Skyrmion excitations in quantum Hall systems, Phys. Rev. B 53 (1996) 1046–1049.

    ADS  Google Scholar 

  104. MacDonald A.H., Fertig H.A. and Brey L., Skyrmions without sigma models in quantum Hall ferromagnets, Phys. Rev. Lett. 76 (1996) 2153–2156.

    ADS  Google Scholar 

  105. Fertig H.A., Brey L., Côté R., Macdonald A.H., Karlhede A. and Sondhi S.L., Hartree-Fock theory of skyrmions in quantum Hall ferromagnets, Phys. Rev. 55 (1997) 10671–10680.

    Google Scholar 

  106. Barrett S.E., Dabbagh G., Pfeiffer L.N. and West K.W., Optically pumped NMR evidence for finite-size Skyrmions in GaAs quantum wells near Landau level filling υ = 1, Phys. Rev. Lett. 74 (1995) 5112–5115.

    ADS  Google Scholar 

  107. Schmeller A., Eisenstein J.P., Pfeiffer L.N. and West K.W., Evidence for Skyrmions and single spin flips in the integer quantized Hall effect, Phys. Rev. Lett. 75 (1995) 4290–4293.

    ADS  Google Scholar 

  108. Aifer E.H., Goldberg B.B. and Broido D.A., Evidence of Skyrmion excitations about υ = 1 in n-modulation-doped single quantum wells by interband optical transmission, Phys. Rev. Lett. 76 (1996) 680–683.

    ADS  Google Scholar 

  109. Bayot V., Grivei E., Melinte S., Santos M.B. and Shayegan M., Giant low temperature heat capacity of GaAs quantum wells near Landau level filling υ = 1, Phys. Rev. Lett. 76 (1996) 4584–4587.

    ADS  Google Scholar 

  110. Maude D.K., Potemski M., Portal J.C., Henini M., Eaves L., Hill G. and Pate M.A., Spin excitations of a two-dimensional electron gas in the limit of vanishing Lande g factor, Phys. Rev. Lett. 77 (1996) 4604–4607.

    ADS  Google Scholar 

  111. Peierls R.E., Remarks on transition temperatures, Helv. Phys. Acta 7 (1934) 81–83. Peierls R.E., Some properties of solids, Ann. Inst. Henri Poincaré 5 (1935) 177-222.

    Google Scholar 

  112. Mermin N.D. and Wagner H., Absence of ferromagnetism in one-and two-dimensional isotropic Heisenberg models, Phys. Rev. Lett. 17 (1966) 1133.

    ADS  Google Scholar 

  113. Hohenberg P.C., Existence of long-range order in one and two dimensions, Phys. Rev. 158 (1967) 383.

    ADS  Google Scholar 

  114. Imry Y. and Gunther L., Fluctuations and physical properties of the two-dimensional crystal lattice, Phys. Rev. B 3 (1971) 3939–3945.

    ADS  Google Scholar 

  115. Berezinskii V.L., Destruction of long-range order in one-dimensional and two-dimensional systems with a continuous symmetry group. I. Classical systems, Zhur. Eksp. Teor. Fiz. 59 (1970) 907 [translation in Sov. Phys. JETP 32 (1970) 493].

    Google Scholar 

  116. Berezinskii V.L., Destruction of long-range order in one-dimensional and two-dimensional systems with a continuous symmetry group. II. Quantum systems, Zhur. Eksp. Teor. Fiz. 61 (1971) 1144.

    Google Scholar 

  117. Kosterlitz J.M. and Thouless D.J., Long range order and metastability in two dimensional solids and superfluids, J. Phys. C 5 (1972) L124–126.

    ADS  Google Scholar 

  118. Kosterlitz J.M. and Thouless D.J., Ordering, metastability and phase transitions in two-dimensional systems, J. Phys. C 6 (1973) 1181–1203.

    ADS  Google Scholar 

  119. Wegner F.J., Spin-ordering in a planar classical Heisenberg model, Z. Phys. 206 (1967) 465–470.

    ADS  Google Scholar 

  120. Jancovici B., Infinite susceptibility without long-range order: the two-dimensional harmonic “solid”, Phys. Rev. Lett. 19 (1967) 20–22.

    ADS  Google Scholar 

  121. Nelson D.R. and Kosterlitz J.M., Universal jump in the superfluid density of two-dimensional superfluids, Phys. Rev. Lett. 39 (1977) 1201–1205.

    ADS  Google Scholar 

  122. Anderson P.W. and Yuval G., Some numerical results on the Kondo problem and the inverse square one-dimensional Ising model, J. Phys. C 4 (1971) 607–620.

    ADS  Google Scholar 

  123. Kosterlitz J.M., The critical properties of the two-dimensional xy model, J. Phys. C 7 (1974) 1046–1060.

    ADS  Google Scholar 

  124. José J., Kadanoff L.P., Kirkpatrick S. and Nelson D.R., Renormalization, vortices and symmetry breaking perturbations in the two-dimensional planar model, Phys. Rev. B 16 (1977) 1217–1241.

    ADS  Google Scholar 

  125. Nelson D.R., Defect-mediated phase transitions, in Phase transitions and critical phenomena Vol. 7, pp. 1–99, edited by C. Domb and J.L. Lebowitz (Academic Press Ltd., London and New York, 1983).

    Google Scholar 

  126. Rudnick I., Critical surface density of the superfluid component in 4He films, Phys. Rev. Lett. 40 (1978) 1454–1455.

    ADS  Google Scholar 

  127. Bishop D.J. and Reppy J.D., Study of the superfluid transition in two-dimensional 4He films, Phys. Rev. Lett. 40 (1978) 1727–1730.

    ADS  Google Scholar 

  128. Bishop D.J. and Reppy J.D., Study of the superfluid transition in two-dimensional 4He films, Phys. Rev. B 22 (1979) 5171–5185.

    ADS  Google Scholar 

  129. Ambegaokar V., Halperin B.I., Nelson D.R. and Siggia E.D., Dynamics of superfluid films, Phys. Rev. B 21 (1980) 1806–1826.

    ADS  Google Scholar 

  130. McQueeney D., Agnolet G.T. and Reppy J.D., Surface superfluidity in dilute 4He-3He mixtures, Phys. Rev. Lett. 52 (1984) 1325–1328.

    ADS  Google Scholar 

  131. Polyakov A.M., Interaction of Goldstone particles in two dimensions. Applications to ferromagnets and massive Yang-Mills fields, Phys. Lett. B 59 (1975) 79–81.

    ADS  MathSciNet  Google Scholar 

  132. Kléman M., Relationship between Burgers circuit, Volterra process and homotopy groups, J. Phys. Lett. (Paris) 38 (1977) L199–202.

    Google Scholar 

  133. Nabarro F.R.N., Theory of crystal dislocations (Clarendon Press, Oxford, 1967).

    Google Scholar 

  134. Young A.P., Melting and the vector Coulomb gas in two dimensions, Phys. Rev. B 19 (1979) 1855–1866.

    ADS  Google Scholar 

  135. Halperin B.I. and Nelson D.R., Theory of two-dimensional melting, Phys. Rev. Lett. 41 (1978) 121–124 and 1519.

    ADS  MathSciNet  Google Scholar 

  136. Nelson D.R. and Halperin B.I., Dislocation-mediated melting in two dimensions, Phys. Rev. B 19 (1979) 2457–2484.

    ADS  Google Scholar 

  137. Grimes C.C. and Adams G., Evidence for a liquid-to-crystal phase transition in a classical, two-dimensional sheet of electrons, Phys. Rev. Lett. 42 (1979) 795–798.

    ADS  Google Scholar 

  138. Gallet F., Deville G., Valdes A. and Williams F.I.B., Fluctuations and shear modulus of a classical two-dimensional electron solid: experiment, Phys. Rev. Lett. 49 (1982) 212–215.

    ADS  Google Scholar 

  139. Morf R.H., Temperature dependence of the shear modulus and melting of the two-dimensional electron solid, Phys. Rev. Lett. 43 (1979) 931–935.

    ADS  Google Scholar 

  140. Strandburg K.J., Two-dimensional melting, Rev. Mod. Phys. 60 (1988) 161–207.

    ADS  Google Scholar 

  141. Beasley M.R., Mooij J.E. and Orlando T.P., Possibility of vortex—antivortex pair dissociation in two-dimensional superconductors, Phys. Rev. Lett. 42 (1979) 1165–1168.

    ADS  Google Scholar 

  142. Doniach S. and Huberman B.A., Topological excitations in two-dimensional superconductors, Phys. Rev. Lett. 42 (1979) 1169–1172.

    ADS  Google Scholar 

  143. Hebard A.F. and Fiory A.T., Evidence for the Kosterlitz-Thouless transition in thin superconducting aluminum films, Phys. Rev. Lett. 44 (1981) 291–294.

    ADS  Google Scholar 

  144. Fiory A.T., Hebard A.F. and Glaberson W.I., Superconducting phase transitions in indium/indium-oxide thin-film composites, Phys. Rev. B 28 (1983) 5075–5087.

    ADS  Google Scholar 

  145. Hebard A.F. and Fiory A.T., Critical-exponent measurements of a two-dimensional superconductor, Phys. Rev. Lett. 50 (1983) 1603–1606.

    ADS  Google Scholar 

  146. Minnhagen P., The two-dimensional Coulomb gas, vortex unbinding, and superfluid—superconducting films, Rev. Mod. Phys. 59 (1987) 1001–1066.

    ADS  Google Scholar 

  147. Huberman B.A. and Doniach S., Melting of two-dimensional vortex lattices, Phys. Rev. Lett. 43 (1979) 950–952.

    ADS  Google Scholar 

  148. Fisher D.S., Flux-lattice melting in thin-film superconductors, Phys. Rev. B 22 (1980) 1190–1199.

    ADS  Google Scholar 

  149. Abraham D.A., Lobb C.J., Tinkham M. and Klapwijk T.M., Resistive transition in two-dimensional arrays of superconducting weak links, Phys. Rev. B 26 (1982) 5268–5271.

    ADS  Google Scholar 

  150. Toulouse G. and Kléman M., Principles of a classification of defects in disordered media, J. Phys. Lett. France 37 (1976) L149–151.

    Google Scholar 

  151. Volovik G.E. and Mineev V.P., Investigation of singularities in superfluid He3 and liquid crystals by homotopic topology methods, Zhur. Eksp. Teor. Fiz. 72 (1977) 2256–2274 [translation in Soviet Phys. JETP 45 (1977) 1186-1196].

    MathSciNet  Google Scholar 

  152. Volovik G.E., Exotic Properties of Superfluid 3He (World Scientific, Singapore, 1992).

    Google Scholar 

  153. Anderson P.W. and Toulouse G., Phase slippage without vortex cores: vortex textures in superfluid 3He, Phys. Rev. Lett. 38 (1977) 508–511.

    ADS  Google Scholar 

  154. Parts Ü., Avilov V.V., Koivuniemi J.H., Krusius M., Ruohio J.J. and Ruutu V.M.H., Vortex arrays of coexisting singly and doubly quantized vortex lines in 3He-A, Czechoslovak J. Phys. 46 (1996) 13–14.

    ADS  Google Scholar 

  155. Krusius M., The vortices of superfluid 3He, J. Low Temp. Phys. 91 (1993) 233–273.

    ADS  Google Scholar 

  156. Thuneberg E.V., Identification of vortices in superfluid 3He-B, Phys. Rev. Lett. 56 (1986) 359–362.

    ADS  Google Scholar 

  157. Salomaa M.M. and Volovik G.E., Topological transition of v-vortex core matter in 3He-B, Europhys. Lett. 2 (1986) 781–787.

    ADS  Google Scholar 

  158. Kondo Y., Korhonen J.S., Krusius M., Dmitriev V.V., Mukharsky Y.M., Sonin E.B. and Volovik G.E., Direct observation of the nonaxisymmetric vortex in superfluid 3He-B, Phys. Rev. Lett. 67 (1991) 81–84.

    ADS  Google Scholar 

  159. Mermin N.D., Surface singularities and superflow in 3He-A, in Quantum Fluids and Solids, edited by S.B Trickey, E.D. Adams and J.W. Dufty (Plenum Press, New York, 1977), pp. 3–22.

    Google Scholar 

  160. de Gennes P.G. and Prost J., The Physics of Liquid Crystals (Clarendon Press, Oxford, 1993).

    Google Scholar 

  161. Kléman M. and Michel L., Spontaneous breaking of Euclidean invariance and classification of topologically stable defects and configurations of crystals and liquid crystals, Phys. Rev. Lett. 40 (1978) 1387–1390.

    ADS  Google Scholar 

  162. Volovik G.E. and Mineev V.P., Line and point singularities in superfluid He3, Pis’ma Zh. Eksp. Teor. Fiz. 24 (1976) 605–608 [translation in JETP Lett. 24 (1976) 561-563].

    Google Scholar 

  163. Kléman M., Points, Lines and Walls (John Wiley & Sons, Chichester, 1983).

    Google Scholar 

  164. Kurik M.V. and Lavrentovich O.D., Defects in liquid crystals: homotopy theory and experimental studies, Usp. Fiz. Nauk 154 (1988) 381–431 [translation in Sov. Phys. Usp. 31 (1988) 196-224].

    MathSciNet  Google Scholar 

  165. Poénaru V. and Toulouse G., The crossing of defects in ordered media and the topology of 3-manifolds, J. Phys. (Paris) 38 (1977) 887–895.

    Google Scholar 

  166. Toulouse G., On biaxial nematics, J. Phys. Lett. (Paris) 38 (1977) L67–68.

    MathSciNet  Google Scholar 

  167. Kléman M. and Friedel J., J. Phys. France Colloq 30 (1969) 43.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

A. Comtet T. Jolicœur S. Ouvry F. David

Rights and permissions

Reprints and permissions

Copyright information

© 1999 EDP Sciences, Springer-Verlag

About this paper

Cite this paper

Thouless, D.J. (1999). Introduction to Topological Quantum Numbers. In: Comtet, A., Jolicœur, T., Ouvry, S., David, F. (eds) Aspects topologiques de la physique en basse dimension. Topological aspects of low dimensional systems. Les Houches - Ecole d’Ete de Physique Theorique, vol 69. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46637-1_11

Download citation

  • DOI: https://doi.org/10.1007/3-540-46637-1_11

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66909-8

  • Online ISBN: 978-3-540-46637-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics