Introduction to Topological Quantum Numbers

  • D. J. Thouless
Conference paper
Part of the Les Houches - Ecole d’Ete de Physique Theorique book series (LHSUMMER, volume 69)


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.


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  1. [1]
    Onsager L., Nuovo Cimento 6 (1949) 249–250.MathSciNetGoogle Scholar
  2. [2]
    Vinen W.F., The detection of single quanta of circulation in liquid helium II, Proc. Roy. Soc. (London) A 260 (1961) 218–236.ADSGoogle Scholar
  3. [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.ADSGoogle Scholar
  4. [4]
    Dirac PAM, Quantized singularities in the electromagnetic field, Proc. Roy. Soc. (London) 133 (1931) 60–72.ADSGoogle Scholar
  5. [5]
    Dirac PAM, The theory of magnetic poles, Phys. Rev. 74 (1948) 817–830.ADSzbMATHMathSciNetGoogle Scholar
  6. [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. [7]
    Petley B.W., The fundamental physical constants and the frontier of measurement (A. Hilger, Bristol, 1985) pp. 282–287.Google Scholar
  8. [8]
    Donnelly R.J., in The collected works of Lars Onsager: with commentary (World Scientific, Singapore, 1996) pp. 693–696.Google Scholar
  9. [9]
    London F., Superfluids, Vol. II, (1954) pp. 151–155.Google Scholar
  10. [10]
    Thuneberg E.V., Introduction to the vortex sheet of superfluid 3He, Physica B 210 (1995) 287–299.ADSGoogle Scholar
  11. [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.ADSGoogle Scholar
  12. [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. [13]
    Gross E.P., Structure of a quantized vortex in boson systems, Nuovo Cimento 20 (1961) 454–477.zbMATHGoogle Scholar
  14. [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.ADSGoogle Scholar
  15. [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.ADSzbMATHGoogle Scholar
  16. [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. [17]
    Langer J.S. and Fisher M.E., Intrinsic critical velocity of a superfluid, Phys. Rev. Lett. 19 (1967) 560–563.ADSGoogle Scholar
  18. [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.ADSGoogle Scholar
  19. [19]
    Donnelly R.J., Quantized vortices in helium II (Cambridge University Press, 1991).Google Scholar
  20. [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. [21]
    Bardeen J., Cooper L.N. and Schrieffer J.R., Theory of superconductivity, Phys. Rev. 108 (1957) 1175–1204.ADSzbMATHMathSciNetGoogle Scholar
  22. [22]
    London F., On the problem of the molecular theory of superconductivity, Phys. Rev. 74 (1948) 562–573.ADSzbMATHGoogle Scholar
  23. [23]
    Putterman S.J., Superfluid Hydrodynamics (North-Holland, Amesterdam, 1974) pp. 404–407.Google Scholar
  24. [24]
    Sonin E.B., Magnus force in superfluids and superconductors, Phys. Rev. B 55 (1997) 485–501.ADSGoogle Scholar
  25. [25]
    Whitmore S.C. and Zimmermann W., Observation of quantized circulation in superfluid helium, Phys. Rev. 166 (1968) 181–196.ADSGoogle Scholar
  26. [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.ADSGoogle Scholar
  27. [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.ADSGoogle Scholar
  28. [28]
    Rayfield G.W. and Reif F., Quantized vortex rings in superfluid helium, Phys. Rev. 136 (1964) A1194–1208.ADSGoogle Scholar
  29. [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. [30]
    Deaver B.S. and Fairbank W.M., Experimental evidence for quantized flux in superconducting cylinders, Phys. Rev. Lett. 7 (1961) 43–46.ADSGoogle Scholar
  31. [31]
    Doll R. and Näbauer M., Experimental proof of magnetic flux quantization in a superconducting ring, Phys. Rev. Lett. 7 (1961) 51–52.ADSGoogle Scholar
  32. [32]
    Parks R.D. and Little W.A., Fluxoid quantization in a multiply-connected superconductor, Phys. Rev. 133 (1964) A97–102.ADSGoogle Scholar
  33. [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.ADSGoogle Scholar
  34. [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. [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.ADSGoogle Scholar
  36. [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. [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.ADSGoogle Scholar
  38. [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.ADSGoogle Scholar
  39. [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.ADSGoogle Scholar
  40. [40]
    Yarmchuk E.J. and Packard R.E., Photographic studies of quantized vortex lines, J. Low Temp. Phys. 46 (1982) 479–515.ADSGoogle Scholar
  41. [41]
    Josephson B.D., Possible new effects in superconductive tunnelling, Phys. Lett. 1 (1962) 251–253.ADSzbMATHGoogle Scholar
  42. [42]
    Josephson B.D., Supercurrents through barriers, Adv. Phys. 14 (1965) 419–451.ADSGoogle Scholar
  43. [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. [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.ADSGoogle Scholar
  45. [45]
    Cohen E.R. and Taylor B.N., The 1986 adjustment of the fundamental physical constants, Rev. Mod. Phys. 59 (1987) 1121.ADSGoogle Scholar
  46. [46]
    Clarke J., Experimental comparison on the Josephson voltage-frequency relation in different superconductors, Phys. Rev. Lett. 21 (1968) 1566–1569.ADSGoogle Scholar
  47. [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.ADSGoogle Scholar
  48. [48]
    Kautz R.L. and Lloyd F.L., Precision of series-array Josephson voltage standards, Appl. Phys. Lett. 51 (1987) 2043–2045.ADSGoogle Scholar
  49. [49]
    Duan J.M., Mass of a vortex line in superfluid 4He: effects of gauge-symmetry breaking, Phys. Rev. B 49 (1994) 12381–12383.ADSGoogle Scholar
  50. [50]
    Demircan E., Ao P. and Niu Q., Vortex dynamics in superfluids: cyclotron-type motion, Phys. Rev. B 54 (1996) 10027–10034.ADSGoogle Scholar
  51. [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. [52]
    Berry M.V., Quantal phase factors accompanying adiabatic changes, Proc. Roy. Soc. (London) A 392 (1984) 45–57.ADSzbMATHGoogle Scholar
  53. [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.ADSGoogle Scholar
  54. [54]
    Thouless D.J., Ao P. and Niu Q., Vortex dynamics in superfluids and the Berry phase, Physica A 200 (1993) 42–49.ADSGoogle Scholar
  55. [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.ADSGoogle Scholar
  56. [56]
    Tang J.M. and Thouless D.J., Longitudinal force on moving potential, Phys. Rev. B 58 (1998) 14179–2.ADSGoogle Scholar
  57. [57]
    Wexler C., Magnus and Iordanskii forces in superfluids, Phys. Rev. Lett. 79 (1997) 1321–1324.ADSGoogle Scholar
  58. [58]
    Laughlin R.B., Quantized Hall conductivity in two dimensions, Phys. Rev. B 23 (1981) 5632–5633.ADSGoogle Scholar
  59. [59]
    Volovik G.E., Comment on “Transverse force on a quantized vortex in a superfluid”, Phys. Rev. Lett. 77 (1997) 4687.ADSGoogle Scholar
  60. [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].ADSGoogle Scholar
  61. [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.ADSGoogle Scholar
  62. [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. [63]
    Nozières P. and Vinen W.F., Phil. Mag. 14 (1966) 667.ADSGoogle Scholar
  64. [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.ADSGoogle Scholar
  65. [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.ADSGoogle Scholar
  66. [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.ADSGoogle Scholar
  67. [67]
    Taylor B.N., New measurement standards for 1990, Physics Today 42 (1989) 23–26.Google Scholar
  68. [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.ADSGoogle Scholar
  69. [69]
    Prange R.E., Quantized Hall resistance and the measurement of the fine-structure constant, Phys. Rev. B 23 (1981) 4802–4805.ADSGoogle Scholar
  70. [70]
    Thouless D.J., Localization and the two-dimensional Hall effect, J. Phys. C 14 (1981) 3475–3480.ADSGoogle Scholar
  71. [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.ADSGoogle Scholar
  72. [72]
    Streda P., Theory of quantised Hall conductivity in two dimensions, J. Phys. C 15 (1982) L717–721.ADSGoogle Scholar
  73. [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.ADSMathSciNetGoogle Scholar
  74. [74]
    Niu Q., Thouless D.J. and Wu Y.S., Quantized Hall conductance as a topological invariant, Phys. Rev. B 31 (1985) 3372–3377.ADSMathSciNetGoogle Scholar
  75. [75]
    Büttiker M., Absence of backscattering in the quantum Hall effect in multiprobe conductors, Phys. Rev. B 38 (1988) 9375–9389.ADSGoogle Scholar
  76. [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.ADSGoogle Scholar
  77. [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.ADSMathSciNetGoogle Scholar
  78. [78]
    Kubo R. The fluctuation-dissipation theorem, Rep. Progr. Phys. 29 (1966) 255–284.ADSGoogle Scholar
  79. [79]
    Lee P.A. and Ramakrishnan T.V., Dirordered electronic systems, Rev. Mod. Phys. 57 (1985) 287–337.ADSGoogle Scholar
  80. [80]
    Bohm D., Note on a theorem of Bloch concerning possible causes of superconductivity, Phys. Rev. 75 (1949) 502–504.ADSzbMATHGoogle Scholar
  81. [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.ADSMathSciNetGoogle Scholar
  82. [82]
    Thouless D.J., Edge voltages and distributed currents in the quantum Hall effect, Phys. Rev. Lett. 71 (1993) 1879–1882.ADSGoogle Scholar
  83. [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. [84]
    Hofstadter D., Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B 14 (1976) 2239–2249.ADSGoogle Scholar
  85. [85]
    Niu Q. and Thouless D.J., Quantum Hall effect with realistic boundary conditions, Phys. Rev. B 35 (1987) 2188–2197.ADSGoogle Scholar
  86. [86]
    Choquet-Bruhat Y., DeWitt-Morette C. and Dillard-Bleick M., Analysis, Manifolds and Physics (North-Holland Publishing Co., Amsterdam 1982), pp. 393–396.zbMATHGoogle Scholar
  87. [87]
    Thouless D.J., Wannier functions for magnetic sub-bands, J. Phys. C 17 (1984) L325–327.ADSMathSciNetGoogle Scholar
  88. [88]
    Kohmoto M., Topological invariant and the quantization of the Hall conductance, Ann. Phys. (NY) 160 (1985) 343–354.ADSMathSciNetGoogle Scholar
  89. [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.ADSGoogle Scholar
  90. [90]
    Laughlin R.B., Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations, Phys. Rev. Lett. 50 (1983) 1395–1398.ADSGoogle Scholar
  91. [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.ADSGoogle Scholar
  92. [92]
    Jain J.K., Composite fermion approach for the fractional quantum Hall effect, Phys. Rev. Lett. 63 (1989) 199–202.ADSGoogle Scholar
  93. [93]
    Jain J.K., Microscopic theory of the fractional quantum Hall effect, Adv. Phys. 41 (1992) 105–146.ADSGoogle Scholar
  94. [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.ADSGoogle Scholar
  95. [95]
    Anderson P. W., Remarks on the Laughlin theory of the fractionally quantized Hall effect, Phys. Rev. B 28 (1983) 2264–2265.ADSGoogle Scholar
  96. [96]
    Tao R. and Wu Y.S., Gauge invariance and the fractional quantum Hall effect, Phys. Rev. B 30 (1984) 1097–1098.ADSGoogle Scholar
  97. [97]
    Thouless D.J., Level crossing and the fractional quantum Hall effect, Phys. Rev. B 40 (1989) 12034–12036.ADSGoogle Scholar
  98. [98]
    Thouless D.J. and Gefen Y., Fractional quantum Hall effect and multiple Aharonov-Bohm periods, Phys. Rev. Lett. 66 (1991) 806–909.ADSGoogle Scholar
  99. [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.ADSGoogle Scholar
  100. [100]
    Halperin B.I., Theory of the quantized Hall conductance, Helv. Phys. Acta 56 (1983) 75–102.Google Scholar
  101. [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.ADSGoogle Scholar
  102. [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.ADSGoogle Scholar
  103. [103]
    Xie X.C. and He S., Skyrmion excitations in quantum Hall systems, Phys. Rev. B 53 (1996) 1046–1049.ADSGoogle Scholar
  104. [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.ADSGoogle Scholar
  105. [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. [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.ADSGoogle Scholar
  107. [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.ADSGoogle Scholar
  108. [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.ADSGoogle Scholar
  109. [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.ADSGoogle Scholar
  110. [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.ADSGoogle Scholar
  111. [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. [112]
    Mermin N.D. and Wagner H., Absence of ferromagnetism in one-and two-dimensional isotropic Heisenberg models, Phys. Rev. Lett. 17 (1966) 1133.ADSGoogle Scholar
  113. [113]
    Hohenberg P.C., Existence of long-range order in one and two dimensions, Phys. Rev. 158 (1967) 383.ADSGoogle Scholar
  114. [114]
    Imry Y. and Gunther L., Fluctuations and physical properties of the two-dimensional crystal lattice, Phys. Rev. B 3 (1971) 3939–3945.ADSGoogle Scholar
  115. [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. [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. [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.ADSGoogle Scholar
  118. [118]
    Kosterlitz J.M. and Thouless D.J., Ordering, metastability and phase transitions in two-dimensional systems, J. Phys. C 6 (1973) 1181–1203.ADSGoogle Scholar
  119. [119]
    Wegner F.J., Spin-ordering in a planar classical Heisenberg model, Z. Phys. 206 (1967) 465–470.ADSGoogle Scholar
  120. [120]
    Jancovici B., Infinite susceptibility without long-range order: the two-dimensional harmonic “solid”, Phys. Rev. Lett. 19 (1967) 20–22.ADSGoogle Scholar
  121. [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.ADSGoogle Scholar
  122. [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.ADSGoogle Scholar
  123. [123]
    Kosterlitz J.M., The critical properties of the two-dimensional xy model, J. Phys. C 7 (1974) 1046–1060.ADSGoogle Scholar
  124. [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.ADSGoogle Scholar
  125. [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. [126]
    Rudnick I., Critical surface density of the superfluid component in 4He films, Phys. Rev. Lett. 40 (1978) 1454–1455.ADSGoogle Scholar
  127. [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.ADSGoogle Scholar
  128. [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.ADSGoogle Scholar
  129. [129]
    Ambegaokar V., Halperin B.I., Nelson D.R. and Siggia E.D., Dynamics of superfluid films, Phys. Rev. B 21 (1980) 1806–1826.ADSGoogle Scholar
  130. [130]
    McQueeney D., Agnolet G.T. and Reppy J.D., Surface superfluidity in dilute 4He-3He mixtures, Phys. Rev. Lett. 52 (1984) 1325–1328.ADSGoogle Scholar
  131. [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.ADSMathSciNetGoogle Scholar
  132. [132]
    Kléman M., Relationship between Burgers circuit, Volterra process and homotopy groups, J. Phys. Lett. (Paris) 38 (1977) L199–202.Google Scholar
  133. [133]
    Nabarro F.R.N., Theory of crystal dislocations (Clarendon Press, Oxford, 1967).Google Scholar
  134. [134]
    Young A.P., Melting and the vector Coulomb gas in two dimensions, Phys. Rev. B 19 (1979) 1855–1866.ADSGoogle Scholar
  135. [135]
    Halperin B.I. and Nelson D.R., Theory of two-dimensional melting, Phys. Rev. Lett. 41 (1978) 121–124 and 1519.ADSMathSciNetGoogle Scholar
  136. [136]
    Nelson D.R. and Halperin B.I., Dislocation-mediated melting in two dimensions, Phys. Rev. B 19 (1979) 2457–2484.ADSGoogle Scholar
  137. [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.ADSGoogle Scholar
  138. [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.ADSGoogle Scholar
  139. [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.ADSGoogle Scholar
  140. [140]
    Strandburg K.J., Two-dimensional melting, Rev. Mod. Phys. 60 (1988) 161–207.ADSGoogle Scholar
  141. [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.ADSGoogle Scholar
  142. [142]
    Doniach S. and Huberman B.A., Topological excitations in two-dimensional superconductors, Phys. Rev. Lett. 42 (1979) 1169–1172.ADSGoogle Scholar
  143. [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.ADSGoogle Scholar
  144. [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.ADSGoogle Scholar
  145. [145]
    Hebard A.F. and Fiory A.T., Critical-exponent measurements of a two-dimensional superconductor, Phys. Rev. Lett. 50 (1983) 1603–1606.ADSGoogle Scholar
  146. [146]
    Minnhagen P., The two-dimensional Coulomb gas, vortex unbinding, and superfluid—superconducting films, Rev. Mod. Phys. 59 (1987) 1001–1066.ADSGoogle Scholar
  147. [147]
    Huberman B.A. and Doniach S., Melting of two-dimensional vortex lattices, Phys. Rev. Lett. 43 (1979) 950–952.ADSGoogle Scholar
  148. [148]
    Fisher D.S., Flux-lattice melting in thin-film superconductors, Phys. Rev. B 22 (1980) 1190–1199.ADSGoogle Scholar
  149. [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.ADSGoogle Scholar
  150. [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. [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].MathSciNetGoogle Scholar
  152. [152]
    Volovik G.E., Exotic Properties of Superfluid 3He (World Scientific, Singapore, 1992).Google Scholar
  153. [153]
    Anderson P.W. and Toulouse G., Phase slippage without vortex cores: vortex textures in superfluid 3He, Phys. Rev. Lett. 38 (1977) 508–511.ADSGoogle Scholar
  154. [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.ADSGoogle Scholar
  155. [155]
    Krusius M., The vortices of superfluid 3He, J. Low Temp. Phys. 91 (1993) 233–273.ADSGoogle Scholar
  156. [156]
    Thuneberg E.V., Identification of vortices in superfluid 3He-B, Phys. Rev. Lett. 56 (1986) 359–362.ADSGoogle Scholar
  157. [157]
    Salomaa M.M. and Volovik G.E., Topological transition of v-vortex core matter in 3He-B, Europhys. Lett. 2 (1986) 781–787.ADSGoogle Scholar
  158. [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.ADSGoogle Scholar
  159. [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. [160]
    de Gennes P.G. and Prost J., The Physics of Liquid Crystals (Clarendon Press, Oxford, 1993).Google Scholar
  161. [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.ADSGoogle Scholar
  162. [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. [163]
    Kléman M., Points, Lines and Walls (John Wiley & Sons, Chichester, 1983).Google Scholar
  164. [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].MathSciNetGoogle Scholar
  165. [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. [166]
    Toulouse G., On biaxial nematics, J. Phys. Lett. (Paris) 38 (1977) L67–68.MathSciNetGoogle Scholar
  167. [167]
    Kléman M. and Friedel J., J. Phys. France Colloq 30 (1969) 43.Google Scholar

Copyright information

© EDP Sciences, Springer-Verlag 1999

Authors and Affiliations

  • D. J. Thouless
    • 1
  1. 1.Dept. of PhysicsUniversity of WashingtonSeattleUSA

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