RSS feeds
РусскийEnglish
Issue 5, 2024
Volume year page
Issues Authors PACS Subscription For authors

Issues

 / 

2006

 / 

July

←  →



Knots and links in the order parameter distributions of strongly correlated systems


Federal Research Center A.V. Gaponov-Grekhov Institute of Applied Physics of the Russian Academy of Sciences, ul. Ulyanova 46, Nizhny Novgorod, 603000, Russian Federation

Research on the coherent distribution of order parameters determining phase existence regions in the two-component Ginzburg-Landau model is reviewed. A major result of this research, obtained by formulating this model in terms of gauged order parameters (the unit vector field n, the density ρ2, and the particle momentum c), is that some of the universal phase and field configuration properties are determined by topological features related to the Hopf invariant Q and its generalizations. For sufficiently low densities, a ring-shaped density distribution may be favored over stripes. For an L < Q phase (L being the mutual linking index of the n and c field configurations), a gain in free energy occurs when a transition to a nonuniform current state occurs. A universal mechanism accounting for decorrelation with increasing charge density is discussed. The second part of the review is concerned with implications of non-Abelian field theory for knotted configurations. The key properties of semiclassical configurations arising in the Yang-Mills theory and the Skyrme model are discussed in detail, and the relation of these configurations to knotted distributions is scrutinized.

Fulltext pdf (484 KB)
Fulltext is also available at DOI: 10.1070/PU2006v049n07ABEH006022
PACS: 02.40.−k, 11.15.−q, 11.27.+d, 74.20.De (all)
DOI: 10.1070/PU2006v049n07ABEH006022
URL: https://ufn.ru/en/articles/2006/7/a/
000242169200001
2-s2.0-33751345507
2006PhyU...49..667P
Citation: Protogenov A P "Knots and links in the order parameter distributions of strongly correlated systems" Phys. Usp. 49 667–691 (2006)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Оригинал: Протогенов А П «Узлы и зацепления распределений параметров порядка в сильно коррелированных системах» УФН 176 689–715 (2006); DOI: 10.3367/UFNr.0176.200607a.0689

References (153) ↓ Cited by (11)

  1. Faddeev L D Preprint No. IAS-75-QS70 (Princeton, NJ: Institute for Advanced Study, 1975); Faddeev L D Lett. Math. Phys. 1 289 (1976)
  2. Katritch V et al. Nature 384 142 (1996)
  3. Stasiak A, Katritch V, Kauffman L H (Eds) Ideal Knots (Ser. on Knots and Everything, Vol. 19) (Singapore: World Scientific, 1998); Kawauchi A A Survey Of Knot Theory (Basel: Birkhäuser Verlag, 1996)
  4. Wilczek F (Ed.) Fractional Statistics And Anyon Superconductivity (Singapore: World Scientific, 1990)
  5. Protogenov A P Usp. Fiz. Nauk 162 (7) 1 (1992); Protogenov A P Sov. Phys. Usp. 35 535 (1992)
  6. Dagotto E Rev. Mod. Phys. 66 763 (1994)
  7. Brenig W Phys. Rep. 251 153 (1995)
  8. Tsuei C C, Kirtley J R Rev. Mod. Phys. 72 969 (2000)
  9. Lee P A, Nagaosa N, Wen X-G Rev. Mod. Phys. 78 17 (2006) cond-mat/0410445; Lee P A, Nagaosa N, Wen X-G cond-mat/0410445
  10. Babaev E, Faddeev L D, Niemi A J Phys. Rev. B 65 100512 (2002)
  11. Kugel K I, Pogosov W V, Rakhmanov A L Physica C 339 10 (2000)
  12. Mel’nikov A S et al. cond-mat/0001186
  13. Belyavskii V I, Kopaev Yu V Zh. Eksp. Teor. Fiz. 127 45 (2005)
  14. Belyavskii V I, Kopaev Yu V, Smirnov M Yu Zh. Eksp. Teor. Fiz. 128 525 (2005)
  15. Isaev L S, Protogenov A P Zh. Eksp. Teor. Fiz. 123 1297 (2003)
  16. Volovik G E, Gor’kov L P Zh. Eksp. Teor. Fiz. 88 1412 (1985)
  17. Mineev V P, Samokhin K V Zh. Eksp. Teor. Fiz. 105 747 (1994)
  18. Mermin N D, Ho T-L Phys. Rev. Lett. 36 594 (1976)
  19. Anderson P W Phys. Rev. Lett. 96 017001 (2006)
  20. Belyavskii V I, Kopaev Yu V Usp. Fiz. Nauk 176 457 (2006); Belyavskii V I, Kopaev Yu V Phys. Usp. 49 441 (2006)
  21. Ioffe L B, Wiegmann P B Phys. Rev. Lett. 65 653 (1990)
  22. Isaev L S, Protogenov A P Phys. Rev. B 69 012401 (2004)
  23. Faddeev L, Niemi A J Nature 387 58 (1997)
  24. Gladikowski J, Hellmund M Phys. Rev. D 56 5194 (1997)
  25. Battye R A, Sutcliffe P M Phys. Rev. Lett. 81 4798 (1998)
  26. Hietarinta J, Salo P Phys. Lett. B 451 60 (1999)
  27. Vakulenko A F, Kapitanskii L V Dokl. Akad. Nauk SSSR 246 840 (1979)
  28. Kundu A, Rybakov Yu P J. Phys. A 15 269 (1982)
  29. Ward R S Nonlinearity 12 241 (1999); Ward R S hep-th/9811176
  30. Derrick G H J. Math. Phys. 5 1252 (1964)
  31. Ruutu V M H et al. Pis’ma ZhETF 60 659 (1994)
  32. Makhlin Yu G, Misirpashaev T Sh Pis’ma ZhETF 61 48 (1995)
  33. Lin F, Yang Y Commun. Math. Phys. 249 273 (2004)
  34. Atiyah M The Geometry And Physics Of Knots (Cambridge: Cambridge Univ. Press, 1990)
  35. Sossinsky A Knots: Mathematics With A Twist (Cambridge: Harvard Univ. Press, 2002)
  36. The KnotPlot site, http://www.pims.math.ca/knotplot/
  37. Ropers knots page, http://www.realknots.com/
  38. Mathematical knots, http://www.cs.ubc.ca/nest/imager/contributions/scharein/knot-theory/
  39. The knot theory MA3F2 page, http://www.maths.warwick.ac.uk/~bjs/MA3F2-page.html
  40. Trefoil knot pages, http://www.jcu.edu/math/ictcm99/animations/knot.htm; Trefoil knot pages, http://www.unca.edu/~mcmcclur/java/LiveMathematica/trefoil.html
  41. Knots, links, braids, http://www.win.tue.nl/~aeb/at/algtop-5.html
  42. Knots and surfaces, http://www.ai.sri.com/~goldwate/math/knots/
  43. Abanov A G, Wiegmann P B hep-th/0105213
  44. Arnold V I, Khesin B A Topological Methods In Hydrodynamics (Applied Mathematical Sciences, Vol. 125) (New York: Springer-Verlag, 1998), Ch. 3
  45. Moffatt H K J. Fluid Mech. 35 117 (1969)
  46. Gordin V A, Petviashvili V I Zh. Eksp. Teor. Fiz. 95 1711 (1989)
  47. Zakharov V E, Kuznetsov E A Usp. Fiz. Nauk 167 1137 (1997); Zakharov V E, Kuznetsov E A Phys. Usp. 40 1087 (1997)
  48. Read N, Green D Phys. Rev. B 61 10267 (2000)
  49. Kamchatnov A M Zh. Eksp. Teor. Fiz. 82 117 (1982)
  50. Larkin A I, Ovchinnikov Yu N Zh. Eksp. Teor. Fiz. 47 1136 (1964)
  51. Fulde P, Ferrell R A Phys. Rev. 135 A550 (1964)
  52. Buzdin A I, Kachkachi H Phys. Lett. A 225 341 (1997)
  53. Houzet M, Buzdin A Europhys. Lett. 50 375 (2000)
  54. Kuznetsov E A, Mikhailov A V Phys. Lett. A 77 37 (1980)
  55. Kohsaka Y et al. Phys. Rev. Lett. 93 097004 (2004)
  56. Kohsaka Y et al. Physica C 388 - 389 283 (2003)
  57. Polyakov A M Zh. Eksp. Teor. Fiz. 68 1975 (1975)
  58. Niemi A J Phys. Rev. D 67 106004 (2003)
  59. van Baal P Nucl. Phys. B: Proc. Suppl. 108 3 (2002); van Baal P hep-th/0109148
  60. Volkov B A i dr. Zh. Eksp. Teor. Fiz. 81 729 (1981)
  61. Volkov B A i dr. Zh. Eksp. Teor. Fiz. 81 1904 (1981)
  62. Kotliar G Phys. Rev. B 37 3664 (1988)
  63. Affleck I, Marston J B Phys. Rev. B 37 3774 (1988); Marston J B, Affleck I Phys. Rev. B 39 11538 (1989)
  64. Nersesyan A, Luther A Preprint FTT-9 (Tbilisi: Institute of Physics, 1988)
  65. Nersesyan A A, Vachnadze G E J. Low Temp. Phys. 77 293 (1989)
  66. Nersesyan A A, Japaridze G I, Kimeridze I G J. Phys. Condens. Matter 3 3353 (1991)
  67. Hsu T C, Marston J B, Affleck I Phys. Rev. B 43 2866 (1991)
  68. Halperin B I, Rice T In Solid State Physics Vol. 21 (Eds F Seitz, D Turnbull, H Ehrenreich) (New York: Academic Press, 1968) p. 115, 1
  69. Schulz H J Phys. Rev. B 39 2940 (1989)
  70. Varma C M Phys. Rev. Lett. 83 3538 (1999)
  71. Varma C M Phys. Rev. B 55 14554 (1997)
  72. Wen X-G, Lee P A Phys. Rev. Lett. 76 503 (1996)
  73. Lee P A et al. Phys. Rev. B 57 6003 (1998)
  74. Nayak C Phys. Rev. B 62 4880 (2000)
  75. Chakravarty S et al. Phys. Rev. B 63 094503 (2001)
  76. Volkov B A, Gorbatsevich A A, Kopaev Yu V Zh. Eksp. Teor. Fiz. 86 1870 (1984)
  77. Volkov B A, Ginzburg V L, Kopaev Yu V Pis’ma ZhETF 27 221 (1978)
  78. Dubovik V M, Tosunyan L A Fiz. Elem. Chast. At. Yad. 14 1193 (1983)
  79. Dubovik V M, Tugushev V V Phys. Rep. 187 145 (1990)
  80. Miller M A Usp. Fiz. Nauk 142 147 (1984); Miller M A Sov. Phys. Usp. 27 69 (1984); Miller M A Izv. Vuzov. Radiofizika 29 991 (1986)
  81. Ginzburg V L et al. Solid State Commun. 50 339 (1984)
  82. Landau L D, Lifshits E M Elektrodinamika Sploshnykh Sred (M.: Nauka, 1982) p. 249
  83. Liu W V, Wilczek F Phys. Rev. Lett. 90 047002 (2003)
  84. Adler R J The Geometry Of Random Fields (Chichester: J. Wiley, 1981)
  85. Worsley K J Chance 9 27 (1996)
  86. Schmalian J "Are there gels in the quantum world?" Aspen Intern. Workshop: Frontiers In Correlated Matter, Aspen, CO, USA, 5 – 8 August 2004; Schmalian J http://frontiers.physics.rutgers.edu/qcrit3_files/schmallvision.pdf
  87. Metzler R et al. Phys. Rev. Lett. 88 188101 (2002); Metzler R et al. cond-mat/0110266
  88. Gantmakher V F Elektrony v Neuporyadochennykh Sredakh (M.: Fizmatlit, 2003) p. 160
  89. Emery V J, Kivelson S A Nature 374 434 (1995)
  90. Mihailovic D, Kabanov V V, Müller K A Europhys. Lett. 57 254 (2002)
  91. Tallon J L et al. Phys. Rev. B 51 R12911 (1995)
  92. Tallon J L et al. Phys. Rev. Lett. 79 5294 (1997)
  93. Boebinger G S et al. Phys. Rev. Lett. 77 5417 (1996)
  94. Faddeev L, Niemi A J Phys. Rev. Lett. 82 1624 (1999); Faddeev L, Niemi A J Phys. Lett. B 449 214 (1999); Faddeev L, Niemi A J Phys. Lett. B 464 90 (1999)
  95. Cho Y M Phys. Rev. D 21 1080 (1980); Cho Y M Phys. Rev. D 23 2415 (1981); Cho Y M Phys. Rev. D 62 074009 (2000); Cho Y M Phys. Rev. Lett. 44 1115 (1980)
  96. Cho Y M Phys. Rev. Lett. 46 302 (1981)
  97. Wu T T, Yang C N In Properties Of Matter Under Unusual Conditions; In Honor Of Edward Teller’s 60th Birthday (Eds H Mark, S Fernbach) (New York: Interscience Publ., 1969) p. 349
  98. De Alfaro V, Fubini S, Furlan G Phys. Lett. B 65 163 (1976)
  99. De Alfaro V, Fubini S, Furlan G Phys. Lett. B 72 203 (1977)
  100. Jacobs L, Rebbi C Phys. Rev. D 18 1137 (1978)
  101. Protogenov A P Phys. Lett. B 87 80 (1979)
  102. Protogenov A P Phys. Lett. B 67 62 (1977)
  103. Witten E Phys. Rev. Lett. 38 121 (1977)
  104. ’t Hooft G Nucl. Phys. B 153 141 (1979); ’t Hooft G Nucl. Phys. B 190 455 (1981)
  105. Polyakov A M Nucl. Phys. B 120 429 (1977)
  106. Langmann E, Niemi A J Phys. Lett. B 463 252 (1999); Langmann E, Niemi A J hep-th/9905147
  107. Bae W S, Cho Y M, Kimm S W Phys. Rev. D 65 025005 (2002); Bae W S, Cho Y M, Kimm S W hep-th/0105163
  108. Shabanov S V Phys. Lett. B 463 263 (1999); Shabanov S V hep-th/9907182; Shabanov S V hep-th/0004135
  109. Gies H Phys. Rev. D 63 125023 (2001)
  110. Belavin A A et al. Phys. Lett. B 59 85 (1975)
  111. ’t Hooft G Phys. Rev. D 14 3432 (1976)
  112. Isaev L S, Protogenov A P in Proc. of Intern. Workshop Nonlinear Physics: Theory and Experiment. III, Lecce, Italy, 2004
  113. Witten E Phys. Lett. B 117 324 (1982)
  114. van Baal P, Wipf A Phys. Lett. B 515 181 (2001); van Baal P, Wipf A hep-th/0105141
  115. Dittmann L, Heinzl T, Wipf A Nucl. Phys. B: Proc. Suppl. 106 - 107 649 (2002); Dittmann L, Heinzl T, Wipf A hep-lat/0110026
  116. Jackiw R, Nair V P, Pi S-Y Phys. Rev. D 62 085018 (2000); Jackiw R, Nair V P, Pi S-Y hep-th/0004084
  117. Cho Y M, Lee H, Pak D G hep-th/9905215
  118. Witten E Commun. Math. Phys. 121 351 (1989)
  119. Hikami K, Kirillov A N Phys. Lett. B 575 343 (2003); Hikami K, Kirillov A N hep-th/0308152
  120. Ward R S hep-th/0108082
  121. Skyrme T H R Proc. R. Soc. London A 260 127 (1961)
  122. Skyrme T H R Nucl. Phys. 31 556 (1962)
  123. Atiyah M F, Manton N S Phys. Lett. B 222 438 (1989)
  124. Manton N, Sutcliffe P Topological Solitons (Cambridge: Cambridge Univ. Press, 2004)
  125. Manton N S Commun. Math. Phys. 111 469 (1987)
  126. Ward R S Phys. Rev. D 70 061701 (2004); Ward R S hep-th/0407245
  127. ’t Hooft G, unpublished
  128. Ward R S Nonlinearity 14 1543 (2001); Ward R S hep-th/0108082
  129. Jackiw R, Nohl C, Rebbi C Phys. Rev. D 15 1642 (1977)
  130. Atiyah M F, Manton N S Commun. Math. Phys. 153 391 (1993)
  131. Faddeev L, Niemi A J Phys. Lett. B 525 195 (2002); Faddeev L, Niemi A J hep-th/0101078
  132. Bolokhov T A, Faddeev L D Teor. Mat. Fiz. 139 276 (2004)
  133. Faddeev L, Niemi A J Phys. Lett. B 449 214 (1999); Faddeev L, Niemi A J Phys. Lett. B 464 90 (1999); Faddeev L, Niemi A J hep-th/9907180
  134. Nasir S, Niemi A J hep-th/0106009
  135. Faddeev L Philos. Trans. R. Soc. London A 359 1399 (2001)
  136. Kondo K-I, Taira Y Prog. Theor. Phys. 104 1189 (2000); Kondo K-I, Taira Y hep-th/9911242
  137. Picken R F J. Math. Phys. 31 616 (1990)
  138. Perelomov A M Phys. Rep. 146 135 (1987)
  139. Boya L J, Perelomov A M, Santander M J. Math. Phys. 42 5130 (2001)
  140. Karabali D, Nair V P, Randjbar-Daemi S hep-th/0407007
  141. Shabanov S V J. Math. Phys. 43 4127 (2002); Shabanov S V hep-th/0202146
  142. Atiyah M, Witten E Adv. Theor. Math. Phys. 6 1 (2003); Atiyah M, Witten E hep-th/0107177
  143. Ranicki A High-dimensional Knot Theory. Algebraic Surgery In Codimension 2 (New York: Springer, 1998)
  144. Bär O, Imboden M, Wiese U-J Nucl. Phys. B 686 347 (2004); Bär O, Imboden M, Wiese U-J cond-mat/0310353
  145. Buniy R V, Kephart T W Phys. Lett. B 576 127 (2003)
  146. Buniy R V, Kephart T W Int. J. Mod. Phys. A 20 1252 (2005); Buniy R V, Kephart T W hep-ph/0408025
  147. Abanov A G, private communication
  148. Pontrjagin L Rec. Math. Moscou (9(51)) 331 (1941)
  149. Polyakov A M Mod. Phys. Lett. A 3 325 (1988)
  150. Abramyan L A, Protogenov A P, Verbus V A Pis’ma ZhETF 69 839 (1999)
  151. Protogenov A P Pis’ma ZhETF 73 292 (2001)
  152. Ladyzhenskaya O A The Mathematical Theory Of Viscous Incompressible Flow 2nd ed. (New York: Gordon and Breach, 1969)
  153. Rybakov Yu P Vestnik RUDN. Ser. Mat. 2 (2) 35 (1995)
© 1918–2024 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions