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Microscopic Theories of Quantum Lattice Systems

  • Raymond F. Bishop

Abstract

Quantum-mechanical systems described by a Hamiltonian defined on an extended regular spatial lattice have become the subjects of a great deal of theoretical work in various fields of physics. Examples of considerable topical interest include: (i) spin-lattice systems such as the solid phases of He,and models of interest in magnetism, exemplified by the spin1/2 anisotropic Heisenberg (or XXZ) model, the spin-1 Heisenberg–biquadratic model, and models with nearest and next-nearest neighbour interactions; (ii) models of strongly interacting electrons on lattices, such as the t-J and Hubbard models; and (iii) lattice gauge field theories, such as the Abelian U (1) and the non-Abelian SU (2) models.

Keywords

Wilson Loop Hubbard Model Heisenberg Model Lattice Gauge Theory Correlation Operator 
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References

  1. 1.
    J.G. Bednorz and K.A. Müller, Z. Phys. B64:189(1986); Rev. Mod. Phys. 60:585(1988).Google Scholar
  2. 2.
    P.W. Anderson, Science 235:1196(1987); in: “Frontiers and Borderlines in Many-Body Physics,” J.R. Schrieffer and R.A. Broglia, ed., North-Holland, Amsterdam (1987).Google Scholar
  3. 3.
    J. Hubbard, Proc. R. Soc. London Ser. A 276:238(1963); ibid. 281:401(1964).Google Scholar
  4. 4.
    H.G. Kümmel, in: “Recent Progress in Many-Body Theories,” Vol. 2, Y. Avishai, ed., Plenum, New York (1990), p. 353.CrossRefGoogle Scholar
  5. 5.
    L. Yu, in: “Recent Progress in Many-Body Theories,” Vol. 3, T.L. Ainsworth, C.E. Campbell, B.E. Clements, and E. Krotscheck, ed., Plenum, New York (1992), p. 157.Google Scholar
  6. 6.
    F. Coester, Nucl. Phys. 7:421(1958); F. Coester and H. Kümmel, ibid. 17:477(1960).Google Scholar
  7. 7.
    J. Cizek, J. Chem. Phys. 45:4256(1966); Advan. Chem. Phys. 14:35(1969).Google Scholar
  8. 8.
    H. Kümmel, K.H. Lührmann, and J.G. Zabolitzky, Phys. Rep. 36C:1(1978).ADSCrossRefGoogle Scholar
  9. 9.
    R.F. Bishop and K.H. Lührmann, Phys. Rev. B17:3757(1978); ibid. B26:5523(1982).Google Scholar
  10. 10.
    H.G. Kümmel, in: “Nucleon-Nucleon Interaction and Nuclear Many-Body Problems,” S.S. Wu and T.T.S. Kuo, ed., World Scientific, Singapore (1984), p. 46.Google Scholar
  11. 11.
    R.F. Bishop and H.G. Kümmel, Phys. Today 40(3):52(1987).CrossRefGoogle Scholar
  12. 12.
    . J.S. Arponen, R.F. Bishop, and E. Pajanne, Phys. Rev. A36:2519(1987); ibid.A36:2539(1987).Google Scholar
  13. 13.
    R.F. Bishop, Theor. Chim. Acta 80:95(1991).Google Scholar
  14. 14.
    R.F. Bishop, in: “Dirkfest ‘82-A Symposium in Honor of J. Dirk Walecka’s Sixtieth Birthday,” W.W. Buck, K.M. Maung, and B.D. Serot, ed., World Scientific, Singapore (1992), p. 21.Google Scholar
  15. 15.
    R.F. Bishop, in: “Many-Body Physics,” C. Fiolhais, M. Fiolhais, C. Sousa, and J.N. Urbano, ed., World Scientific, Singapore (1994), p. 3.Google Scholar
  16. 16.
    E. Feenberg, “Theory of Quantum Fluids,” Academic Press, New York (1969).Google Scholar
  17. 17.
    J.W. Clark and E. Feenberg, Phys. Rev. 113:388(1959); H.W. Jackson and E. Feenberg, Ann. Phys. (NY) 15:266(1961); E. Feenberg and C.W. Woo, Phys. Rev. 137:A391(1965); J.W. Clark and P. Westhaus, Phys. Rev. 141:833(1966); ibid. 149:990(1966).Google Scholar
  18. 18.
    J.W. Clark, L.R. Mead, E. Krotscheck, K.E. Kürten, and M.L. Ristig, Nucl. Phys. A328:45(1979); E. Manousakis and V.R. Pandharipande, Phys. Rev. B30:5062 (1984).Google Scholar
  19. 19.
    E. Krotscheck and J.W. Clark, Nucl. Phys. A328:73(1979).Google Scholar
  20. 20.
    J.W. Clark, Prog. Part. Nucl. Phys. 2:89(1979).Google Scholar
  21. 21.
    J.W. Clark, in: “The Many-Body Problem, Jastrow Correlations Versus Brueckner Theory,” R. Guardiola and J. Ros, ed., (Lect. Notes in Physics, Vol. 138), Springer, Berlin (1981), p. 184.CrossRefGoogle Scholar
  22. 22.
    E. Krotscheck, J.W. Clark, and A.D. Jackson, Phys. Rev. B28:5088(1983).Google Scholar
  23. 23.
    J.W. Clark, E. Krotscheck, and B. Schwesinger, Phys. Lett. 143B:287(1984).ADSCrossRefGoogle Scholar
  24. 24.
    W. Heisenberg, Z. Phys. 49:619(1928).Google Scholar
  25. 25.
    P.A.M. Dirac, Proc. R. Soc. London Ser. A 123:714(1929).Google Scholar
  26. 26.
    J.H. Van Vleck, “Theory of Electric and Magnetic Susceptibilities,” Clarendon Press, Oxford (1932).Google Scholar
  27. 27.
    E. Manousakis, Rev. Mod. Phys. 63:1(1991).Google Scholar
  28. 28.
    T. Barnes, Int. J. Mod. Phys. C2:659(1991).Google Scholar
  29. 29.
    F.D.M. Haldane, Phys. Lett. 93A:464(1983); Phys. Rev. Lett. 50:1153(1983).Google Scholar
  30. 30.
    I. Affleck, J. Phys.: Condens. Matter 1:3047(1989).Google Scholar
  31. 31.
    D.S. Greywall and P.A. Busch, Phys. Rev. Lett. 65:2788(1990); D.S. Greywall, Phys. Rev. B41:1842(1990).Google Scholar
  32. 32.
    H.A. Bethe, Z. Phys. 71:205(1931).Google Scholar
  33. 33.
    L. Hulthén, Ark. Mat. Astron. Fys. A26, No. 11 (1938).Google Scholar
  34. 34.
    R. Orbach, Phys. Rev. 112:309(1958); C.N. Yang and C.P. Yang, ibid. 150:321(1966); ibid. 150:327(1966).Google Scholar
  35. 35.
    M. Gaudin, “La Fonction d’Onde de Bethe,” Masson, Paris (1983).Google Scholar
  36. 36.
    J. des Cloiseaux and J.J. Pearson, Phys. Rev. 128:2131(1962); J.D. Johnson, S. Krinsky, and B.M. McCoy, Phys. Rev. A8:2526(1973); L.D. Faddeev and L.A. Takhtajan, Phys. Lett. 85A:375(1981).Google Scholar
  37. 37.
    R.J. Baxter, J. Stat. Phys. 9:145(1973).Google Scholar
  38. 38.
    N.M. Bogoliubov, A.G. Izergin, and V.E. Korepin, Nucl. Phys. B275:687(1986).MathSciNetGoogle Scholar
  39. 39.
    E.K. Sklyanin, L.A. Takhtajan, and L.D. Faddeev, Theor. Math. Phys. 40:688(1980); V.E. Korepin, N.M. Bogoliubov, and A.G. Izergin, “Quantum Inverse Scattering Method and Correlation Functions,” Cambridge Univ. Press (1993).Google Scholar
  40. 40.
    A. Luther and I. Peschel, Phys. Rev. B12:3908(1975).Google Scholar
  41. 41.
    F.J. Dyson, E.H. Lieb, and B.J. Simon, J. Stat. Phys. 18:335(1978).MathSciNetGoogle Scholar
  42. 42.
    E. Jordao Neves and J. Fernando Perez, Phys. Lett. 114A:331(1986); I. Affleck, T. Kennedy, E.H. Lieb, and H. Tasaki, Phys. Rev. Lett. 59:799(1987); Commun. Math. Phys. 115:447(1988); T. Kennedy, E.H. Lieb, and B.S.Shastry, J. Stat. Phys. 53:1019(1988).Google Scholar
  43. 43.
    B. Bernu, P. Lecheminant, C. Lhuillier, and L. Pierre, Phys. Scripta T49:192(1993); Phys. Rev. B50:10048(1994).Google Scholar
  44. 44.
    J. Carlson, Phys. Rev. B40:846(1989); N. Trivedi and D.M. Ceperley, ibid. B41:4552 (1990); K.J. Runge, ibid. B45:12292(1992).Google Scholar
  45. 45.
    P.W. Anderson, Mater. Res. Bull. 8:153(1973); P. Fazekas and P.W. Anderson, Phil. Mag. 30:423(1974).Google Scholar
  46. 46.
    K. Kubo and T. Kishi, Phys. Rev. Lett. 61:2585(1988).Google Scholar
  47. 47.
    C.K. Majumdar and D.K. Ghosh, J. Math. Phys. 10:1388(1969); ibid. 10:1399(1969).Google Scholar
  48. 48.
    F.D.M. Haldane, Phys. Rev. B25:4925(1982).Google Scholar
  49. 49.
    L.A. Takhtajan, Phys. Lett. 87A:479(1982); H.M. Babujian, ibid. 90A:479(1982).Google Scholar
  50. 50.
    C.K. Lai, J. Math. Phys. 15:1675(1974); B. Sutherland, Phys. Rev. B12:3795(1975).Google Scholar
  51. 51.
    J.B. Parkinson, J. Phys. C20:L1029(1987); ibid. C21:3793(1988); M.N. Barber and M.T. Bachelor, Phys. Rev. B40:4621(1989); A. Klümper, J. Phys. A23:809(1990).Google Scholar
  52. 52.
    A.V. Chubukov, J. Phys.: Condens. Matter 2:1593(1990); Phys. Rev. B47:872(1993).Google Scholar
  53. 53.
    G. Fáth and J. Sólyom, Phys. Rev. B44:11836(1991); ibid. B47:872(1993).Google Scholar
  54. 54.
    T. Tonegawa and I. Harada, J. Phys. Soc. Japan 56:2153(1987).Google Scholar
  55. 55.
    A.B. Harris and R.V. Lange, Phys. Rev. 157:279(1967); W.F. Brinkman and T.M. Rice, Phys. Rev. B2:1324(1970); J.E. Hirsch, Phys. Rev. Lett. 54:1317(1985).Google Scholar
  56. 56.
    V.J. Emery, Phys. Rev. Lett. 58:2794(1987); J.E. Hirsch, ibid. 59:228(1987); P.B. Littlewood, C.M. Varma, and E. Abrahams, ibid. 60:379(1987).Google Scholar
  57. 57.
    F.C. Zhang and T.M. Rice, Phys. Rev. B37:3759(1988); ibid. B41:7243(1990).Google Scholar
  58. 58.
    V.J. Emery and G. Reiter, Phys. Rev. B38:11938(1988); ibid. B41:7247(1990).Google Scholar
  59. 59.
    E. Dagotto, Rev. Mod. Phys. 66:763(1994).Google Scholar
  60. 60.
    P.W. Anderson, in: “Valence Fluctuations in Solids,” L.M. Falicov, W. Hanke, and M.M. Maple, ed., North-Holland, Amsterdam (1981), p. 451; C.M. Varma, Comments Solid State Phys. 11:221(1985).Google Scholar
  61. 61.
    L. Yu, “Solitons and Polarons in Conducting Polymers,” World Scientific, Singapore (1988).Google Scholar
  62. 62.
    D. Jerome and H.J. Schultz, Adv. Phys. 31:299(1982).Google Scholar
  63. 63.
    E.H. Lieb and F.Y. Wu, Phys. Rev. Lett. 20:1445(1968).Google Scholar
  64. 64.
    F.D.M. Haldane, Phys. Rev. Lett. 45:1358(1980); J. Phys. C14:2585(1981); Phys. Lett. 81A:153(1981).Google Scholar
  65. 65.
    J.M. Luttinger, J. Math. Phys. 4:1154(1963).MathSciNetGoogle Scholar
  66. 66.
    K.G. Wilson, Phys. Rev. D10:2445(1974).Google Scholar
  67. 67.
    J. Kogut and L. Susskind, Phys. Rev. D11:395(1975).Google Scholar
  68. 68.
    S. Elitzur, Phys. Rev. D12:3978(1975).Google Scholar
  69. 69.
    A.M. Polyakov, Phys. Lett. B72:477(1978).Google Scholar
  70. 70.
    E. Fradkin and L. Susskind, Phys. Rev. D17:2637(1978).MathSciNetGoogle Scholar
  71. 71.
    S.D. Drell, H.R. Quinn, B. Svetitsky, and M. Weinstein, Phys. Rev. D19:619(1979).Google Scholar
  72. 72.
    R. Balian, J.M. Drouffe, and C. Itzykson, Phys. Rev. D10:3376(1974); ibid. D11:2098(1975); ibid. D11:2104(1975).Google Scholar
  73. 73.
    D. Horn, M. Weinstein, and S. Yankielowicz, Phys. Rev. D19:3715(1979).Google Scholar
  74. 74.
    F.J. Wegner, J. Math. Phys. 12:2259(1971).MathSciNetGoogle Scholar
  75. 75.
    S.F. Edwards and P.W. Anderson, J. Phys. F5:965(1975).Google Scholar
  76. 76.
    A.A. Migdal, Soy. Phys. - JETP 42:413(1976); ibid. 42:743(1976).Google Scholar
  77. 77.
    R.F. Bishop, N.J. Davidson, and Y. Xian, article in this volume.Google Scholar
  78. 78.
    J.B. Kogut, Phys. Rep. 67:67(1980).MathSciNetADSCrossRefGoogle Scholar
  79. 79.
    H.G. Dosch and V.F. Müller, Fortschr. Phys. 27:547(1979).Google Scholar
  80. 80.
    I. Montvay and G. Münster, “Quantum Fields on a Lattice,” Cambridge Univ. Press (1994).CrossRefGoogle Scholar
  81. 81.
    C.J. Hamer, J. Oitmaa, and W.H. Zheng, Phys. Rev. D43:1978(1991).Google Scholar
  82. 82.
    S.B. Liang, B. Doucot, and P.W. Anderson, Phys. Rev. Lett. 61:365(1988).Google Scholar
  83. 83.
    D. Horn and M. Weinstein, Phys. Rev. D30:1256(1984).Google Scholar
  84. 84.
    K.G. Wilson, Nucl. Phys. B (Proc. Suppl.) 17:82(1990).Google Scholar
  85. 85.
    R.J. Bartlett, J. Phys. Chem. 93:1697(1989); Theor. Chim. Acta 80:71(1991).Google Scholar
  86. 86.
    R. Jastrow, Phys. Rev. 98:1479(1955).Google Scholar
  87. 87.
    F. Iwamoto and M. Yamada, Prog. Theor. Phys. 17:543(1957).Google Scholar
  88. 88.
    J.W. Clark and P. Westhaus, J. Math. Phys. 9:131(1968); P. Westhaus and J.W. Clark, ibid. 9:149(1968).Google Scholar
  89. 89.
    M. Gaudin, J. Gillespie, and G. Ripka, Nucl. Phys. A176:237(1971); S. Fantoni and S. Rosati, Nuovo Cim. 20A:179(1974); G. Ripka, Phys. Rep. 56:1(1979); Nucl. Phys. A314:115(1979).Google Scholar
  90. 90.
    S. Fantoni and S. Rosati, Lett. Nuovo Cim. 10:545(1974); Nuovo Cim. 25A:593 (1975);E. Krotscheck and M.L. Ristig, Phys. Lett. 48A:17(1974); Nucl. Phys. A242:389(1975).Google Scholar
  91. 91.
    J.W. Clark, in: “Progress in Particle and Nuclear Physics,” Vol. 2, D.H. Wilkinson, Pergamon, Oxford (1979), p. 89.Google Scholar
  92. 92.
    S. Rosati and M. Viviani, in: “First International Course on Condensed Matter,” D. Prosperi, S. Rosati, and G. Violini, ed., (ACIF Series, Vol. 8), World Scientific, Singapore (1988), p. 231.Google Scholar
  93. 93.
    (a) R.F. Bishop, in:“Recent Progress in Many-Body Theories,” Vol. 1, A.J. Kallio, E. Pajanne, and R.F. Bishop, ed., Plenum, New York (1988), p. 385; (b) A. Fabrocini and S. Rosati, in: “First International Course on Condensed Matter,” D. Prosperi, S. Rosati, and G. Violini, ed., (ACIF Series, Vol. 8), World Scientific, Singapore (1988), p. 87.Google Scholar
  94. 94.
    J.W. Clark and M.L. Ristig, Phys. Rev. 7:1792(1973); M.L. Ristig and J.W. Clark, Nucl. Phys. A199:351(1973).Google Scholar
  95. 95.
    L.R. Mead and J.W. Clark, Phys. Lett. 90B:331(1980).ADSCrossRefGoogle Scholar
  96. 96.
    J. Arponen, Ann. Phys. (NY) 151:311(1983).Google Scholar
  97. 97.
    K. Emrich, Nucl. Phys. A351:379(1981); ibid. A351:397(1981).Google Scholar
  98. 98.
    A.D. Jackson, A. Lande, and R.A. Smith, Phys. Rev. Lett. 54:1469(1985); E. Krotscheck, R.A. Smith, and A.D. Jackson, Phys. Rev. A33:3535(1986).Google Scholar
  99. 99.
    P.W. Kastelijn, Physica 18:104(1952).Google Scholar
  100. 100.
    W. Marshall, Proc. R. Soc. London Ser. A 232:48(1955).Google Scholar
  101. 101.
    H. Taketa and T. Nakamura, J. Phys. Soc. Japan 11:919(1956); R. Bartkowski, Phys. Rev. B5:4536(1972); M. Suzuki and S. Miyashita, Can. J. Phys. 56:902 (1978); R.B. Pearson, Phys. Rev. A18:2655(1978).Google Scholar
  102. 102.
    P. Horsch and W. von der Linden, Z. Phys. B72:181(1988); D.A. Huse and V. Elser, Phys. Rev. Lett. 60:2531(1988); Z. Liu and E. Manousakis, Phys. Rev. B40:11437(1989).Google Scholar
  103. 103.
    S. Sachdev, Phys. Rev. B39:12232(1989); T. Pang, ibid. B43:3362(1991).Google Scholar
  104. 104.
    E. Manousakis, Phys. Rev. B40:4904(1989).Google Scholar
  105. 105.
    P.W. Anderson, Phys. Rev. 86:694(1952); R. Kubo, ibid. 87:568(1952); T. Oguchi, ibid. 117:117(1960).Google Scholar
  106. 106.
    J.E. Hirsch and S. Tang, Phys. Rev. B40:4769(1989); S. Tang, M.E. Lazzouni, and J.E. Hirsch, ibid. B40:5000(1989).Google Scholar
  107. 107.
    M. Takahashi, Phys. Rev. B40:2494(1989).Google Scholar
  108. 108.
    M. Roger and J.H. Hetherington, Phys. Rev. B41:200(1990).Google Scholar
  109. 109.
    M. Roger and J.H. Hetherington, Europhys. Lett. 11:255(1990).Google Scholar
  110. (110a).
    R.F. Bishop, J.B. Parkinson, and Y. Xian, Phys. Rev. B43:13782(1991);ADSCrossRefGoogle Scholar
  111. (110b).
    R.F. Bishop, J.B. Parkinson, and Y. Xian, Theor. Chim. Acta 80:181(1991);CrossRefGoogle Scholar
  112. (110c).
    R.F. Bishop, J.B. Parkinson, and Y. Xian, Phys. Rev. B44:9425(1991);ADSCrossRefGoogle Scholar
  113. (110d).
    R.F. Bishop, J.B. Parkinson, and Y. Xian, in: “Recent Progress in Many Body Theories,” Vol. 3, T.L. Ainsworth, C.E. Campbell, B.E. Clements, and E. Krotscheck, ed., Plenum, New York (1992), p. 117;CrossRefGoogle Scholar
  114. (110e).
    R.F. Bishop, J.B. Parkinson, and Y. Xian, J. Phys.: Condens. Matter 4:5783(1992).ADSCrossRefGoogle Scholar
  115. 111.
    R.F. Bishop, J.B. Parkinson, and Y. Xian, Phys. Rev. B46:880(1992).Google Scholar
  116. 112.
    R.F. Bishop, J.B. Parkinson, and Y. Xian, J. Phys.: Condens. Matter 5:9169(1993).Google Scholar
  117. 113.
    F.E. Harris, Phys. Rev. B47:7903(1993).Google Scholar
  118. (114a).
    D.J.J. Farnell and J.B. Parkinson, J. Phys.: Condens. Matter 6:5521(1994);ADSCrossRefGoogle Scholar
  119. (114b).
    Y. Xian, ibid. 6:5965(1994).Google Scholar
  120. 115.
    F. Cornu, Th. Jolicoeur, and J.C. Le Guillou, Phys. Rev. B49:9548(1994).Google Scholar
  121. 116.
    R.F. Bishop, R.G. Hale, and Y. Xian, Phys0. Rev. Lett. 73:3157(1994); UMIST preprint (1994).CrossRefGoogle Scholar
  122. 117.
    T. Barnes, D. Kotchan, and E.S. Swanson, Phys. Rev. B39:4357(1989).Google Scholar
  123. 118.
    R.R.P. Singh, Phys. Rev. B39:9760(1989); R.R.P. Singh and D.A. Huse, ibid. B40:7247(1989); W. Zheng, J. Oitmaa, and C.J. Hamer, ibid. B43:8321(1991).Google Scholar
  124. 119.
    B.I. Shraiman and E.D. Siggia, Phys. Rev. Lett. 61:467(1988); ibid. 62:1564(1989).Google Scholar
  125. 120.
    H. Yokoyama and H. Shiba, J. Phys. Soc. Japan 56:3570(1987); C. Gros, R. Joynt, and T.M. Rice, Phys. Rev. B36:381(1987); S. Fantoni, X.Q. Wang, E. Tosatti, and L. Yu, Physica C153–155:1255(1988).Google Scholar
  126. 121.
    X.Q. Wang, S. Fantoni, E. Tosatti, L. Yu, and M. Viviani, Phys. Rev. B41:11479 (1990).ADSCrossRefGoogle Scholar
  127. 122.
    X.Q. Wang, S. Fantoni, E. Tosatti, and L. Yu, in: “Condensed Matter Theories,” Vol. 5, V.C. Aguilera-Navarro, ed., Plenum, New York (1990), p. 203; X.Q. Wang, S. Fantoni, E. Tosatti, L. Yu, and X.Q.G. Wang, in: “Recent Progress in Many-Body Theories,” Vol. 2, Y. Avishai, ed., Plenum New York (1990), p. 297.Google Scholar
  128. 123.
    T.A. Kaplan, P. Horsch, and P. Fulde, Phys. Rev. Lett. 49:889(1982); P. Fazekas and K. Penc, Int. J. Mod. Phys. B1:1021(1988).Google Scholar
  129. 124.
    C.H. Yang and J.W. Clark, Nucl. Phys. A174:49(1971); E. Krotscheck and J.W. Clark, ibid. A333:77(1980); T.C. Paulick and G.E. Campbell, Phys. Rev. B16:2000(1977); S. Fantoni, Nucl. Phys. A363:381(1981).Google Scholar
  130. 125.
    C.F. Lo, E. Manousakis, and Y.L. Wang, Phys. Lett. 156A:42(1991).ADSCrossRefGoogle Scholar
  131. 126.
    F. Petit and M. Roger, Phys. Rev. B49:3453(1994).Google Scholar
  132. 127.
    R.F. Bishop, Y. Xian, and C. Zeng, Int. J. Quantum Chem. - in press.Google Scholar
  133. 128.
    J.R. Schrieffer, X.G. Wen, and S.C. Zhang, Phys. Rev. B39:11663(1989).Google Scholar
  134. 129.
    A. Dabringhaus and M.L. Ristig, in: “Condensed Matter Theories,” Vol. 6, S. Fantoni and S. Rosati, ed., Plenum, New York (1991), p. 291; A. Dabringhaus, M.L. Ristig, and J.W. Clark, Phys. Rev. D43:1978(1991); M.L. Ristig and A. Dabringhaus, in:“Recent Progress in Many-Body Theories,” Vol. 3, T.L. Ainsworth, C.E. Campbell, B.E. Clements, and E. Krotscheck, ed., Plenum, New York (1992), p. 107.Google Scholar
  135. 130.
    R.F. Bishop, A.S. Kendall, L.Y. Wong, and Y. Xian, Phys. Rev. D48:887(1993); in: “Condensed Matter Theories,” Vol. 8, L. Blum and F.B. Malik, ed., Plenum, New York (1993), p. 269.Google Scholar
  136. 131.
    R.F. Bishop and Y. Xian, Acta Phys. Pol. 24:541(1993); Nucl. Phys. B (Prot. Suppl.) 34:808(1994).Google Scholar
  137. 132.
    C.H. Llewellyn Smith and N.J. Watson, Phys. Lett. B302:463(1993).Google Scholar
  138. 133.
    J.P. Greensite, Nucl. Phys. B166:113(1980).Google Scholar
  139. 134.
    C.E. Campbell, K.E. Kürten, M.L. Ristig, and G. Senger, Phys. Rev. B30:3728(1984); G. Senger, M.L. Ristig, K.E. Kiirten, and C.E. Campbell, Phys. Rev. B33:7562(1986); G. Senger and M.L. Ristig, in: “Condensed Matter Theories,” Vol. 5, V.C. Aguilera-Navarro, ed., Plenum, New York (1990), p. 133; C.E. Campbell, in: “Condensed Matter Theories,” Vol. 8, L. Blum and F.B. Malik, ed., Plenum, New York (1992), p. 149.Google Scholar
  140. 135.
    M. Altenbokum, K. Emrich, H. Kümmel, and J.G. Zabolitzky, in: “Condensed Matter Theories,” Vol. 2, P. Vashishta, R.K. Kalia, and R.F. Bishop, ed., Plenum, New York (1987), p. 389; G. Sanyal, S.H. Mandal, and D. Mukherjee, Chem. Phys. Lett. 192:55(1992); G. Sanyal, S.H. Mandal, S. Guha, and D. Mukherjee, Phys. Rev. E48:3373(1993).Google Scholar

Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • Raymond F. Bishop
    • 1
  1. 1.Department of MathematicsUMIST (University of Manchester Institute of Science and Technology)ManchesterUK

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