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Theory and phenomenology for a variety of classical and quantum phase transitions

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Abstract

In this review, we first introduce recent progress in the mathematical structure of the three-dimensional Ising model, from the points of view of topologic, algebraic and geometric aspects. Then we discuss in turn Anderson localization due to disorder and then first- and second-order metal-insulator transitions, depending on electron correlation, with and without a magnetic field. Finally, we make intimate contact with the phase diagram showing the equilibrium between low temperature regimes of the magnetically induced Wigner electron solid and the so-called Laughlin electron liquid in the two-dimensional case.

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Notes

  1. A Reidemeister move refers to one of three local moves on a link diagram. Each move operates on a small region of the diagram and is one of three types: I. Twist and untwist in either direction. II. Move one loop completely over another. III. Move a string completely over or under a crossing.

References

  1. L. Onsager, Phys. Rev. 65, 117 (1944)

    Article  CAS  Google Scholar 

  2. L.P. Kadanoff, W. Götze, D. Hamblen, R. Hecht, E.A.S. Lewis, V.V. Palciauskas, M. Rayl, J. Swift, D. Aspnes, J. Kane, Rev. Mod. Phys. 39, 395 (1967)

    Article  CAS  Google Scholar 

  3. B. Widom, J. Chem. Phys. 43, 3898 (1965)

    Article  Google Scholar 

  4. K.G. Wilson, J. Kogut, Phys. Rep. 12, 75 (1974)

    Article  Google Scholar 

  5. C. Domb, in Phase Transitions and Critical Phenomena, vol. 3, ed. by C. Domb, M.S. Green (Academic Press, London, 1974)

  6. M.E. Fisher, Rep. Prog. Phys. 30, 615 (1967)

    Article  CAS  Google Scholar 

  7. K. Binder, E. Luijten, Phys. Rep. 344, 179 (2001)

    Article  CAS  Google Scholar 

  8. A. Pelissetto, E. Vicari, Phys. Rep. 368, 549 (2002)

    Article  CAS  Google Scholar 

  9. Z.D. Zhang, Phil. Mag. 87, 5309 (2007)

    Article  CAS  Google Scholar 

  10. D.J. Klein, N.H. March, Phys. Lett. A 372, 5052 (2008)

    Article  CAS  Google Scholar 

  11. F.Y. Wu, B.M. McCoy, M.E. Fisher, L. Chayes, Phil. Mag. 88, 3093 (2008)

    Article  CAS  Google Scholar 

  12. F.Y. Wu, B.M. McCoy, M.E. Fisher, L. Chayes, Phil. Mag. 88, 3103 (2008)

    Article  CAS  Google Scholar 

  13. J.H.H. Perk, Phil. Mag. 89, 761 (2009)

    Article  CAS  Google Scholar 

  14. J.H.H. Perk, Phil. Mag. 89, 769 (2009)

    Article  CAS  Google Scholar 

  15. J.H.H. Perk, Bull. Soc. Sci. Lett. Lódź Sér. Rech. Déform. 62 (2012) (in press); arXiv:1209.0731v1

  16. J.H.H. Perk, Bull. Soc. Sci. Lett. Lódź Sér. Rech. Déform. 63 (2013) (in press); arXiv:1209.0731v2

  17. Z.D. Zhang, Phil. Mag. 88, 3097 (2008)

    Article  CAS  Google Scholar 

  18. Z.D. Zhang, Phil. Mag. 89, 765 (2009)

    Article  CAS  Google Scholar 

  19. Z.D. Zhang, N.H. March, Bull. Soc. Sci. Lett. Lódź Sér. Rech. Déform. 62 (2012) (in press); arXiv:1209.3247v1

  20. Z.D. Zhang, N.H. March, Bull. Soc. Sci. Lett. Lódź Sér. Rech. Déform. 63 (2013) (in press); arXiv:1209.3247v5

  21. Z.D. Zhang, Chin. Phys. B 22, 030513 (2013)

    Article  Google Scholar 

  22. Ya.G. Sinai, Theory of Phase Transitions: Rigorous Results, Chapter II (Pergamon Press, Oxford, 1982)

  23. J. Glimm, A. Jaffe, Quantum Physics, Chapters 18 and 20, 2nd edn. (Springer, New York, 1987)

  24. R.B. Israel, Commun. Math. Phys. 50, 245 (1976)

    Article  Google Scholar 

  25. M. Zahradnik, J. Stat. Phys. 47, 725 (1987)

    Article  Google Scholar 

  26. R.B. Griffiths, in Phase Transitions and Critical Phenomena, ed. by C. Domb, M.S. Green. Rigorous results and theorems, vol. 1, Chapter 2 (Academic Press, New York, 1972)

  27. S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, 1999)

    Google Scholar 

  28. D. Ruelle, Statistical Mechanics, in Rigorous Results (Benjamin, NY, 1969)

    Google Scholar 

  29. S. Miracle-Solé, Theorems on Phase Transitions with a Treatment for the Ising Model, in Lecture Notes in Physics, vol. 54 (Springer, Berlin, 1976), p. 189

  30. J.L. Lebowitz, O. Penrose, Commun. Math. Phys. 11, 99 (1968)

    Article  Google Scholar 

  31. G. Gallavotti, S. Miracle-Solé, Commun. Math. Phys. 7, 274 (1968)

    Article  Google Scholar 

  32. G. Gallavotti, S. Miracle-Solé, D.W. Robinson, Phys. Lett. 25A, 493 (1967)

    Google Scholar 

  33. G. Gallavotti, S. Miracle-Solé, Commun. Math. Phys. 5, 317 (1967)

    Article  Google Scholar 

  34. J. Ławrynowicz, S. Marchiafava, A. Niemczynowicz, Adv. Appl. Clifford Alg. 20, 733 (2010)

    Article  Google Scholar 

  35. J. Ławrynowicz, S. Marchiafava, M. Nowak-Kępczyk, in Proceedings of the 9th International Workshop on Complex Structures, Integrability and Vector Fields, Sofia, Bulgaria, 25–29 August 2008, ed. by K. Sekigawa, V.S. Gerdjikov, S. Dimiev. Trends in differential geometry, complex analysis and mathemtical physics (World Scientific, Singapore, 2008), pp. 156–166. doi: 10.1142/9789814277723_0018

  36. J. Lawrynowicz, M. Nowak-Kepczyk, O. Suzuki, Inter. J. Bifurcation Chaos 22, 1230003 (2012)

    Google Scholar 

  37. J. Lawrynowicz, O. Suzuki, A. Niemczynowicz, Adv. Appl. Clifford Alg. 22, 757 (2012)

    Article  Google Scholar 

  38. F.Y. Wu, Rev. Mod. Phys. 54, 235 (1982)

    Article  Google Scholar 

  39. L.H. Kauffman, Knots and Physics, 3rd edn. (World Scientific, Singapore, 2001)

  40. L.H. Kauffman, Rep. Prog. Phys. 68, 2829 (2005)

    Article  Google Scholar 

  41. C.N. Yang, Phys. Rev. Lett. 19, 1312 (1967)

    Article  Google Scholar 

  42. R.J. Baxter, Ann. Phys. 70, 193 (1972)

    Article  Google Scholar 

  43. R.J. Baxter, Exactly Solved Models in Statistics Mechanics (Academic Press, London, 1982)

    Google Scholar 

  44. A.B. Zamolodchikov, Sov. Phys. JETP 52, 325 (1980)

    Google Scholar 

  45. A.B. Zamolodchikov, Commun. Math. Phys. 79, 489 (1981)

    Article  Google Scholar 

  46. M.T. Jaekel, J.M. Maillard, J. Phys. A 15, 1309 (1982)

    Article  Google Scholar 

  47. YuG Stroganov, Theor. Math. Phys. 110, 141 (1997)

    Article  Google Scholar 

  48. R.M. Kashaev, Lett. Math. Phys. 35, 389 (1996)

    Article  Google Scholar 

  49. R.M. Kashaev, I.G. Korepanov, S.M. Sergeev, Theor. Math. Phys. 117, 1402 (1998)

    Article  Google Scholar 

  50. I.G. Korepanov, Theor. Math. Phys. 118, 319 (1999)

    Article  Google Scholar 

  51. V.V. Bazhanov, R.J. Baxter, J. Stat. Phys. 69, 453 (1992)

    Article  Google Scholar 

  52. V.V. Bazhanov, R.J. Baxter, J. Stat. Phys. 71, 839 (1993)

    Article  Google Scholar 

  53. P. Jordan, J. von Neumann, E. Wigner, Ann. Math. 35, 29 (1934)

    Article  Google Scholar 

  54. Z.D. Zhang, N.H. March, J. Math. Chem. 49, 1283 (2011)

    Article  CAS  Google Scholar 

  55. S. de Leo, W.A. Rodrigues Jr, Inter. J. Theor. Phys. 36, 2725 (1997)

    Article  Google Scholar 

  56. D. Finkelstein, J.M. Jauch, S. Schiminovich, D. Speiser, J. Math. Phys. 3, 207 (1962)

    Article  Google Scholar 

  57. S. Marchiafava, J. Rembieliński, J. Math. Phys. 33, 171 (1992)

    Article  Google Scholar 

  58. S.L. Adler, Quaternion Quantum Mechanics and Quantum Fields (Oxford University Press, New York, Oxford, 1995)

    Google Scholar 

  59. S. de Leo, J. Math. Phys. 37, 2955 (1996)

    Article  Google Scholar 

  60. Z.D. Zhang, N.H. March, Bull. Soc. Sci. Lett. Lódź Sér. Rech. Déform. 62 (2012) (in press); arXiv:1110.5527

  61. N.F. Mott, W.D. Twose, Adv. Phys. 10, 107 (1961)

    Article  CAS  Google Scholar 

  62. P.W. Anderson, Phys. Rev. 109, 1492 (1958)

    Article  CAS  Google Scholar 

  63. J. Dancz, S.F. Edwards, N.H. March, J. Phys. C: Solid State Phys. 6, 873 (1973)

    Article  CAS  Google Scholar 

  64. N.H. March, G.G.N. Angilella, J. Math. Chem. 46, 532 (2009)

    Article  CAS  Google Scholar 

  65. A.M. García-García, Phys. Rev. Lett. 100, 076404 (2008)

    Article  Google Scholar 

  66. Z.D. Zhang, N.H. March, J. Math. Chem. 49, 816 (2011)

    Article  CAS  Google Scholar 

  67. Z.D. Zhang, N.H. March, J. Phys. Chem. Solids 72, 1529 (2011)

    Article  CAS  Google Scholar 

  68. E. Wigner, Phys. Rev. 46, 1002 (1934)

    Article  CAS  Google Scholar 

  69. M. Parrinello, N.H. March, J. Phys. C. Solid State Phys. 9, L147 (1976)

    Article  Google Scholar 

  70. E.L. Polloek, J.P. Hansen, Phys. Rev. A 8, 3110 (1973)

    Article  Google Scholar 

  71. G. Toulouse, J. Physique 36, 1137 (1975)

    Article  Google Scholar 

  72. B. Sulherland, Phys. Rev. Lett. 35, 185 (1975)

    Article  Google Scholar 

  73. J. Durkan, R.J. Elliott, N.H. March, Rev. Mod. Phys. 40, 812 (1968)

    Article  CAS  Google Scholar 

  74. E.Y. Andrei, G. Deville, D.C. Glattli, F.I.B. Williams, E. Paris, B. Etienne, Phys. Rev. Lett. 60, 2765 (1988)

    Article  CAS  Google Scholar 

  75. H. Buhmann, W. Joss, K. von Klitzing, I.V. Kukushkin, A.S. Plaut, G. Martinez, K. Ploog, V.B. Timofeev, Phys. Rev. Lett. 66, 926 (1991)

    Article  CAS  Google Scholar 

  76. M.J. Lea, N.H. March, W. Sung, J. Phys. Conden. Matter 3, 4301 (1991)

    Article  Google Scholar 

  77. A.B. Pippard, Elements of Classical Thermodynamics (Cambridge University Press, Cambridge, 1966)

  78. Y.S. Wu, Y. Yu, Y. Hatsugai, M. Kohmoto, Phys. Rev. B 57, 9907 (1998)

    Article  CAS  Google Scholar 

  79. N.H. March, Phys. Chem. Liquids 38, 151 (2000)

    Article  CAS  Google Scholar 

  80. N.H. March, J. Phys. Condens. Matter 5, B149 (1993)

    Article  CAS  Google Scholar 

  81. N.H. March, M. Suzuki, M. Parrinello, Phys. Rev. B 19, 2027 (1979)

    Article  CAS  Google Scholar 

  82. W.F. Brinkman, T.M. Rice, Phys. Rev. B 2, 1324 (1970)

    Article  Google Scholar 

  83. R.G. Chapman, N.H. March, Phys. Rev. B 38, 792 (1988)

    Article  CAS  Google Scholar 

  84. J. Strecka, J. Dely, L. Canova, Physica A 388, 2394 (2009)

    Article  CAS  Google Scholar 

  85. N.H. March, Z.D. Zhang, Phys. Lett. A 373, 2075 (2009)

    Article  CAS  Google Scholar 

  86. N.H. March, Z.D. Zhang, Phys. Chem. Liquids 47, 693 (2009)

    Article  CAS  Google Scholar 

  87. N.H. March, Z.D. Zhang, J. Math. Chem. 47, 520 (2010)

    Article  CAS  Google Scholar 

  88. N.H. March, Z.D. Zhang, Phys. Chem. Liquids 48, 279 (2010)

    Article  CAS  Google Scholar 

  89. Z.D. Zhang, N.H. March, Phys. Chem. Liquids 48, 403 (2010)

    Article  CAS  Google Scholar 

  90. Z.D. Zhang, N.H. March, Phase Trans. 84, 299 (2011)

    Article  CAS  Google Scholar 

  91. Z.D. Zhang, N.H. March, Phys. Chem. Liquids 49, 270 (2011)

    Article  CAS  Google Scholar 

  92. Z.D. Zhang, N.H. March, Phys. Chem. Liquids 49, 684 (2011)

    Article  CAS  Google Scholar 

  93. Z.D. Zhang, N.H. March, J. Math. Chem. 50, 920 (2012)

    Article  CAS  Google Scholar 

  94. Z.D. Zhang, N.H. March, Phys. Chem. Liquids 51, 261 (2013)

    Article  CAS  Google Scholar 

  95. J. Kaupužs, Can. J. Phys. 90, 373 (2012)

    Article  Google Scholar 

  96. J. Kaupužs, R.V.N. Melnik, J. Rimšāns, Commun. Comput. Phys. 14, 355 (2013)

    Google Scholar 

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Acknowledgments

Z.D.Z. acknowledges the support of the National Natural Science Foundation of China under Grant Number 50831006. N.H.M. wishes to acknowledge that his contribution to the present article was brought to fruition during a visit to ICTP, Trieste. Thanks are due to Professor V. E. Kravtsov for generous hospitality and for the very stimulating atmosphere afforded. NHM. also wishes to thank Professors D. Lamoen and C. Van Alsenoy for making possible his continuing affiliation with the University of Antwerp. Finally, Prof. P. M. Echenique has supported NHM most generously at Donostia International Physics Center (DIPC), San Sebastián.

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Authors and Affiliations

  1. Donostia International Physics Centre, San Sebastián, Spain

    N. H. March

  2. Department of Physics, University of Antwerp, Antwerp, Belgium

    N. H. March

  3. Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

    N. H. March

  4. Oxford University, Oxford, UK

    N. H. March

  5. Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang, 110016, People’s Republic of China

    Z. D. Zhang

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March, N.H., Zhang, Z.D. Theory and phenomenology for a variety of classical and quantum phase transitions. J Math Chem 51, 1694–1711 (2013). https://doi.org/10.1007/s10910-013-0181-4

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