Issues

 / 

2017

 / 

September

  

Reviews of topical problems


Berezinskii—Kosterlitz—Thouless transition and two-dimensional melting

, , ,
Institute for High Pressure Physics, Russian Academy of Sciences, Troitsk, Moscow, Russian Federation

The main aspects of the theory of phase transitions in two-dimensional degenerate systems (Berezinskii—Kosterlitz—Thouless, or BKT, transitions) are reviewed in detail, including the transition mechanism, the renormalization group as a tool for describing the transition, and how the transition scenario can possibly depend on the core energy of topological defects (in particular, in thin superconducting films). Various melting scenarios in two-dimensional systems are analyzed, and the current status of actual experiments and computer simulations in the field is examined. Whereas in three dimensions melting always occurs as a single first-order transition, in two dimensions, as shown by Halperin, Nelson and Young, melting via two continuous BKT transitions with an intermediate hexatic phase characterized by quasi-long range orientational order is possible. There is also a possibility, however, for a first-order phase transition to occur. Recently, one further melting scenario, different from that occurring in the Berezinskii—Kosterlitz—Thouless—Halperin—Nelson—Young (BKTHNY) theory, has been proposed, according to which a solid can melt in two stages — a continuous BKT type solid-hexatic transition and then a first order hexatic phase—isotropic liquid one. Particular attention is given to the melting scenario as a function of the potential shape and to the random pinning effect on two-dimensional melting. In particular, it is shown that random pinning can alter the melting scenario fundamentally in the case of a first-order transition. Also considered is the melting of systems with potentials having a negative curvature in the repulsion region — potentials that are successfully used in describing the anomalous properties of water in two dimensions.

Fulltext is available at IOP
Keywords: two-dimensional systems, Berezinskii—Kosterlitz—Thouless transition, superfluid films, superconducting films, XY model, two-dimensional crystals, topological defects, vortices, dislocations, disclinations, hexatic phase, two-dimensional melting, Berezinskii—Kosterlitz—Thouless—Halperin—Nelson—Young theory, first order transition
PACS: 02.70.Ns, 05.70.Ln, 64.10.+h, 64.60.Ej, 64.70.D− (all)
DOI: 10.3367/UFNe.2017.06.038161
URL: https://ufn.ru/en/articles/2017/9/a/
Citation: Ryzhov V N, Tareyeva E E, Fomin Yu D, Tsiok E N "Berezinskii—Kosterlitz—Thouless transition and two-dimensional melting" Phys. Usp. 60 857–885 (2017)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Received: 15th, May 2017, revised: 23rd, June 2017, 29th, June 2017

Оригинал: Рыжов В Н, Тареева Е Е, Фомин Ю Д, Циок Е Н «Переход Березинского—Костерлица—Таулеса и двумерное плавление» УФН 187 921–951 (2017); DOI: 10.3367/UFNr.2017.06.038161

References (313) ↓ Cited by (42) Similar articles (20)

  1. Berezinskii V L Zh. Eksp. Teor. Fiz. 59 907 (1970); Berezinskii V L Sov. Phys. JETP 32 493 (1971)
  2. Berezinskii V L Zh. Eksp. Teor. Fiz. 61 1144 (1971); Berezinskii V L Sov. Phys. JETP 34 610 (1972)
  3. Kosterlitz J M, Thouless D J J. Phys. C 6 1181 (1973)
  4. Kosterlitz J M Rep. Prog. Phys. 79 026001 (2016)
  5. Ambegaokar V Usp. Fiz. Nauk 178 1359 (2008); Ambegaokar V Phys. Usp. 51 1287 (2008)
  6. Landau L D Zh. Eksp. Teor. Fiz. 7 627 (1937); Landau L D Sobranie Trudov Vol. 1 (M.: Nauka, 1937) p. 253; Landau L Phys. Z. Sowjetunion 11 545 (1937)
  7. Patashinskii A Z, Pokrovskii V L Zh. Eksp. Teor. Fiz. 46 994 (1964); Patashinskii A Z, Pokrovskii V L Sov. Phys. JETP 19 677 (1964)
  8. Patashinskii A Z, Pokrovskii V L Zh. Eksp. Teor. Fiz. 50 439 (1966); Patashinskii A Z, Pokrovskii V L Sov. Phys. JETP 23 292 (1966)
  9. Kadanoff L P Physics 2 263 (1966)
  10. Fisher M E Rep. Prog. Phys. 30 615 (1967)
  11. Wilson K G Phys. Rev. B 4 3174 (1971); Wilson K G Phys. Rev. B 4 3184 (1971)
  12. Wilson K G, Fisher M E Phys. Rev. Lett. 28 240 (1972)
  13. Wilson K G Phys. Rev. Lett. 28 548 (1972)
  14. Patashinskii A Z, Pokrovskii V L Fluktuatsionnaya Teoriya Fazovykh Perekhodov (M.: Nauka, 1982); Per. na angl. yaz. 1-go russk. izd., Patashinskii A Z, Pokrovskii V L Fluctuation Theory Of Phase Transitions (Oxford: Pergamon Press, 1979)
  15. Kagiwada R S et al. Phys. Rev. Lett. 22 338 (1969)
  16. Chester M, Yang L C, Stephens J B Phys. Rev. Lett. 29 211 (1972)
  17. Henkel R P, Smith E N, Reppy J D Phys. Rev. Lett. 23 1276 (1969)
  18. Kukich G, Henkel R P, Reppy J D Phys. Rev. Lett. 21 197 (1968)
  19. Clow J R, Reppy J D Phys. Rev. Lett. 19 291 (1967)
  20. Amit D J Phys. Lett. A 26 448 (1968)
  21. Amit D J Phys. Lett. A 26 466 (1968)
  22. Alder B J, Wainwright T E J. Chem. Phys. 27 1208 (1957)
  23. Alder B J, Wainwright T E J. Chem. Phys. 31 459 (1959)
  24. Alder B J, Wainwright T E J. Chem. Phys. 33 1439 (1960)
  25. Alder B J, Wainwright T E Phys. Rev. 127 359 (1962)
  26. Wood W W, Jacobson J D J. Chem. Phys. 27 1207 (1957)
  27. Stanley H E, Kaplan T A Phys. Rev. Lett. 17 913 (1966)
  28. Stanley H E Phys. Rev. Lett. 20 589 (1968)
  29. Moore M A Phys. Rev. Lett. 23 861 (1969)
  30. Kosterlitz J M, Thouless D J J. Phys. C 5 L124 (1972)
  31. Kosterlitz J M J. Phys. C 7 1046 (1974)
  32. Stishov S M Usp. Fiz. Nauk 114 3 (1974); Stishov S M Sov. Phys. Usp. 18 625 (1975)
  33. Bogolyubov N N "Kvazisrednie v zadachakh statisticheskoi mekhaniki" Sobranie Nauchnykh Trudov v 12-ti Tomakh Vol. 6 (M.: Nauka, 2006) p. 236; Per. na angl. yaz., Bogolubov N N (Jr.) "Quasi-averages in problems of statistical mechanics" Quantum Statistical Mechanics: Selected Works (Hackensack, NJ: World Scientific, 2015)
  34. Sadovnikov B I, Fedyanin V K Teor. Mat. Fiz. 16 368 (1973); Sadovnikov B I, Fedyanin V K Theor. Math. Phys. 16 901 (1973)
  35. Sadovnikov B I, Sorokina E M Dokl. Akad. Nauk SSSR 188 788 (1969); Sadovnikov B I, Sorokina E M Sov. Phys. Dokl. 14 968 (1970)
  36. Mermin N D Phys. Rev. 176 250 (1968)
  37. Halperin B I, Nelson D R Phys. Rev. Lett. 41 121 (1978)
  38. Nelson D R, Halperin B I Phys. Rev. B 19 2457 (1979)
  39. Young A P Phys. Rev. B 19 1855 (1979)
  40. Berezinskii V L "Nizkotemperaturnye svoistva dvumernykh sistem s nepreryvnoi gruppoi simmetrii" Diss. ... kand. fiz.-mat. nauk (M.: Institut teoreticheskoi fiziki im. L.D. Landau, 1971); Berezinskii V L Nizkotemperaturnye Svoistva Dvumernykh Sistem s Nepreryvnoi Gruppoi Simmetrii (M.: Fizmatlit, 2007)
  41. José J V (Ed.) 40 Years Of Berezinskii—Kosterlitz—Thouless Theory (Singapore: World Scientific, 2013)
  42. Ryzhov V N Usp. Fiz. Nauk 187 125 (2017); Ryzhov V N Phys. Usp. 60 114 (2017)
  43. Minnhagen P Rev. Mod. Phys. 59 1001 (1987)
  44. Nelson D R, Kosterlitz J M Phys. Rev. Lett. 39 1201 (1977)
  45. Bishop D J, Reppy J D Phys. Rev. Lett. 40 1727 (1978)
  46. Rudnick I Phys. Rev. Lett. 40 1454 (1978)
  47. Young A P J. Phys. C 11 L453 (1978)
  48. Levin Y, Li X, Fisher M E Phys. Rev. Lett. 73 2716 (1994)
  49. Ohta T Prog. Theor. Phys. 60 968 (1978)
  50. Wiegmann P B J. Phys. C 11 1583 (1978)
  51. Amit D J, Goldschmidt Y Y, Grinstein S J. Phys. A 13 585 (1980)
  52. Minnhagen P, Nylén M Phys. Rev. B 31 5768 (1985)
  53. Minnhagen P, Weber H Phys. Rev. B 32 3337(R) (1985)
  54. Weber H, Minnhagen P Phys. Rev. B 38 8730 (1988)
  55. Minnhagen P Phys. Rev. B 27 2807 (1983)
  56. Minnhagen P Phys. Rev. B 28 2463 (1983)
  57. Minnhagen P Phys. Rev. B 29 1440 (1984)
  58. Minnhagen P Phys. Rev. B 32 3088 (1985)
  59. Minnhagen P Phys. Rev. Lett. 54 2351 (1985)
  60. Minnhagen P Phys. Rev. B 32 7548 (1985)
  61. Minnhagen P, Wallin M Phys. Rev. B 36 5620 (1987)
  62. Minnhagen P, Wallin M Phys. Rev. B 40 5109 (1989)
  63. Jonsson A, Minnhagen P, Nylén M Phys. Rev. Lett. 70 1327 (1993)
  64. Théron R et al. Phys. Rev. Lett. 71 1246 (1993)
  65. Thijssen J M, Knops H J F Phys. Rev. B 38 9080 (1988)
  66. Lee J-R, Teitel S Phys. Rev. Lett. 64 1483 (1990)
  67. Lee J-R, Teitel S Phys. Rev. Lett. 66 2100 (1991)
  68. Lee J-R, Teitel S Phys. Rev. B 46 3247 (1992)
  69. Caillol J M, Levesque D Phys. Rev. B 33 499 (1986)
  70. Xu B-W, Zhang Y-M Phys. Rev. B 50 18651 (1994)
  71. Ding G-H, Xu B-W Phys. Rev. B 51 12653 (1995)
  72. Hansen J P, Viot P J. Stat. Phys. 38 823 (1985)
  73. Mayer J E, Mayer M G Statistical Mechanics (New York: Willey, 1977)
  74. Deutsch C, Lavaud M Phys. Rev. A 9 2598 (1974)
  75. Everts H U, Koch W Z. Phys. B 28 117 (1977)
  76. Høye J S, Olaussen K Physica A 104 447 (1980)
  77. Høye J S, Olaussen K Physica A 107 241 (1981)
  78. Salzberg A M, Prager S J. Chem. Phys. 38 2587 (1963)
  79. Beasley M R, Mooij J E, Orlando T P Phys. Rev. Lett. 42 1165 (1979)
  80. Pearl J Proc. of the 9th Intern. Conf. on Low Temperature Physics, LT-9, 1964 (Eds J G Daunt et al.) (New York: Plenum Press, 1965) p. 566
  81. Irz D Yu, Ryzhov V N, Tareyeva E E Phys. Lett. A 207 374 (1995)
  82. Tinkham M Introduction To Superconductivity (New York: McGraw-Hill, 1975)
  83. Halperin B I, Nelson D R J. Low Temp. Phys. 36 599 (1979)
  84. Garland J C, Lee H J Phys. Rev. B 36 3638 (1987)
  85. Hsu J W P, Kapitulnik A Phys. Rev. B 45 4819 (1992)
  86. Epstein K, Goldman A M, Kadin A M Phys. Rev. Lett. 47 534 (1981)
  87. Fiory A T, Hebard A F, Glaberson W I Phys. Rev. B 28 5075 (1983)
  88. Jeanneret B et al. Phys. Rev. B 40 11374(R) (1989)
  89. Wallin M Phys. Rev. B 41 6575 (1990)
  90. Turneaure S J, Lemberger T R, Graybeal J M Phys. Rev. Lett. 84 987 (2000)
  91. Crane R W et al. Phys. Rev. B 75 094506 (2007)
  92. Liu W et al. Phys. Rev. B 84 024511 (2011)
  93. Kamlapure A et al. Appl. Phys. Lett. 96 072509 (2010)
  94. Mondal M et al. Phys. Rev. Lett. 107 217003 (2011)
  95. Lin Y-H, Nelson J, Goldman A M Phys. Rev. Lett. 109 017002 (2012)
  96. Misra S et al. Phys. Rev. Lett. 110 037002 (2013)
  97. Baity P G et al. Phys. Rev. B 93 024519 (2016)
  98. Daptary G N et al. Phys. Rev. B 94 085104 (2016)
  99. Maccari I, Benfatto L, Castellani C arXiv:1702.05930
  100. Ryzhov V N, Tareeva E E Teor. Mat. Fiz. 96 425 (1993); Ryzhov V N, Tareeva E E Theor. Math. Phys. 96 1062 (1993)
  101. Ryzhov V N, Tareyeva E E Phys. Rev. B 48 12907 (1993)
  102. Ryzhov V N, Tareyeva E E Phys. Rev. B 49 6162 (1994)
  103. Irz D Yu, Ryzhov V N, Tareeva E E Teor. Mat. Fiz. 107 100 (1996); Irz D Y, Ryzhov V N, Tareeva E E Theor. Math. Phys. 107 499 (1996)
  104. Irz D Yu, Ryzhov V N, Tareyeva E E Phys. Rev. B 54 3051 (1996)
  105. Nelson D R Phys. Rev. Lett. 60 1973 (1988)
  106. Nelson D R, Seung H S Phys. Rev. B 39 9153 (1989)
  107. Fetter A L, Hohenberg P C Superconductivity (Ed. R D Parks) (New York: M. Dekker, 1969) p. 817
  108. Feynman R P, Hibbs A R Quantum Mechanics And Path Integrals (New York: McGraw-Hill, 1965)
  109. Balescu R Equilibrium And Nonequilibrium Statistical Mechanics (New York: Wiley, 1975); Per. na russk. yaz., Balesku R Ravnovesnaya i Neravnovesnaya Statisticheskaya Mekhanika Vol. 1 (M.: Mir, 1978)
  110. Strandburg K J (Ed.) Bond-Orientational Order In Condensed Matter Systems (New York: Springer-Verlag, 1992)
  111. Brinkman W F, Fisher D S, Moncton D E Science 217 693 (1982)
  112. Strandburg K J Rev. Mod. Phys. 60 161 (1988)
  113. Kleinert H Gauge Fields In Condensed Matter Vol. 2 (Singapore: World Scientific, 1989)
  114. Glaser M A, Clark N A Adv. Chem. Phys. 83 543 (1993)
  115. Stishov S M Usp. Fiz. Nauk 154 93 (1988); Stishov S M Sov. Phys. Usp. 31 52 (1988)
  116. Ryzhov V N, Tareeva E E Teor. Mat. Fiz. 73 463 (1987); Ryzhov V N, Tareeva E E Theor. Math. Phys. 73 1344 (1987)
  117. Ryzhov V N, Tareyeva E E J. Phys. C 21 819 (1988)
  118. Ryzhov V N Teor. Mat. Fiz. 80 107 (1989); Ryzhov V N Theor. Math. Phys. 80 745 (1989)
  119. Ryzhov V N J. Phys. Condens. Matter 2 5855 (1990)
  120. Levine D, Steinhardt P J Phys. Rev. Lett. 53 2477 (1984)
  121. Steinhardt P J, Nelson D R, Ronchetti M Phys. Rev. B 28 784 (1983)
  122. Landau L D Zh. Eksp. Teor. Fiz. 7 19 (1937); Landau L D Phys. Z. Sowjetunion 11 26 (1937)
  123. Peierls R E Helv. Phys. Acta 7 81 (1934)
  124. Peierls R E Ann. Inst. Henri Poincará 5 177 (1935)
  125. Landau L D, Lifshits E M Statisticheskaya Fizika Vol. 1 (M.: Nauka, 1976); Per. na angl. yaz., Landau L D, Lifshitz E M Statistical Physics Vol. 1 (Oxford: Pergamon Press, 1980)
  126. Brock J D Bond-Orientational Order In Condensed Matter Systems (Ed. K J Strandburg) (New York: Springer-Verlag, 1992) p. 1
  127. Landau L D, Lifshits E M Teoriya Uprugosti (M.: Nauka, 1987); Per. na angl. yaz., Landau L D, Lifshitz E M Theory Of Elasticity (Oxford: Pergamon Press, 1986)
  128. Barker J A, Henderson D Rev. Mod. Phys. 48 587 (1976)
  129. Ryzhov V N Zh. Eksp. Teor. Fiz. 100 1627 (1991); Ryzhov V N Sov. Phys. JETP 73 899 (1991)
  130. Nelson D R, Pelcovits R A Phys. Rev. B 16 2191 (1977)
  131. Saito Y Phys. Rev. B 26 6239 (1982)
  132. Strandburg K J Phys. Rev. B 34 3536(R) (1986)
  133. Saito Y Phys. Rev. Lett. 48 1114 (1982)
  134. Ryzhov V N, Tareyeva E E Phys. Lett. A 158 321 (1991)
  135. Ryzhov V N, Tareeva E E Teor. Mat. Fiz. 92 331 (1992); Ryzhov V N, Tareeva E E Theor. Math. Phys. 92 922 (1992)
  136. Prestipino S, Saija F, Giaquinta P V J. Chem. Phys. 137 104503 (2012)
  137. Prestipino S, Saija F, Giaquinta P V Phys. Rev. Lett. 106 235701 (2011)
  138. Dudalov D E et al. J. Phys. Conf. Ser. 510 012016 (2014)
  139. Dudalov D E et al. J. Chem. Phys. 141 18C522 (2014)
  140. Dudalov D E et al. Soft Matter 10 4966 (2014)
  141. Tsiok E N et al. Phys. Rev. E 92 032110 (2015)
  142. Tsiok E N, Fomin Yu D, Ryzhov V N arXiv:1701.03643
  143. Krott L B, Barbosa M C J. Chem. Phys. 138 084505 (2013)
  144. Krott L B, Barbosa M C Phys. Rev. E 89 012110 (2014)
  145. Krott L B, Bordin J R, Barbosa M C J. Phys. Chem. B 119 291 (2015)
  146. Almudallal A M, Buldyrev S V, Saika-Voivod I J. Chem. Phys. 137 034507 (2012)
  147. Algara-Siller G et al. Nature 519 443 (2015)
  148. Zhao J et al. Science 343 1228 (2014)
  149. Meng Q et al. Phys. Rev. B 90 020509(R) (2014)
  150. Miller W L, Cacciuto A Soft Matter 7 7552 (2011)
  151. Zhang F, Liu Y, Yan H J. Am. Chem. Soc. 135 7458 (2013)
  152. Chui S T Phys. Rev. B 28 178 (1983)
  153. Chui S T Phys. Rev. Lett. 48 933 (1982)
  154. Kleinert H Phys. Lett. A 95 381 (1983)
  155. Kleinert H Phys. Lett. A 96 302 (1983)
  156. Janke W, Kleinert H Phys. Lett. A 105 134 (1984)
  157. Nelson D R Phys. Rev. B 26 269 (1982)
  158. Janke W, Kleinert H Phys. Lett. A 114 255 (1986)
  159. Strandburg K J, Solla S A, Chester G V Phys. Rev. B 28 2717 (1983)
  160. Kleinert H Phys. Lett. A 130 443 (1988)
  161. Janke W, Kleinert H Phys. Rev. Lett. 61 2344 (1988)
  162. Kleinert H Phys. Lett. A 136 468 (1989)
  163. Janke W, Kleinert H Phys. Rev. B 41 6848 (1990)
  164. Ryzhov V N Teor. Mat. Fiz. 88 449 (1991); Ryzhov V N Theor. Math. Phys. 88 990 (1991)
  165. Cha M-C, Fertig H A Phys. Rev. Lett. 73 870 (1994)
  166. Grier D G et al. Phys. Rev. Lett. 66 2270 (1991)
  167. Fisher D S, Halperin B I, Morf R Phys. Rev. B 20 4692 (1979)
  168. Lozovik Yu E, Farztdinov V M Solid State Commun. 54 725 (1985)
  169. Lozovik Yu E et al. Phys. Lett. A 112 61 (1985)
  170. Ryzhov V N, Tareyeva E E Phys. Rev. B 51 8789 (1995)
  171. Ryzhov V N, Tareeva E E Zh. Eksp. Teor. Fiz. 108 2044 (1995); Ryzhov V N, Tareeva E E JETP 81 1115 (1995)
  172. Ryzhov V N, Tareyeva E E Physica C 205 55 (1993)
  173. Ryzhov V N, Tareyeva E E Physica A 314 396 (2002)
  174. Pomirchi L M, Ryzhov V N, Tareeva E E Teor. Mat. Fiz. 130 119 (2002); Pomirchi L M, Ryzhov V N, Tareyeva E E Theor. Math. Phys. 130 101 (2002)
  175. Chumakov E S et al. Physica A 432 279 (2015)
  176. Ryzhov V N i dr. Teor. Mat. Fiz. 191 424 (2017); Ryzhov V N et al. Theor. Math. Phys. 191 842 (2017)
  177. Ryzhov V N, Tareyeva E E Phys. Lett. A 75 88 (1979)
  178. Ryzhov V N, Tareeva E E Teor. Mat. Fiz. 48 416 (1981); Ryzhov V N, Tareeva E E Theor. Math. Phys. 48 835 (1981)
  179. Ryzhov V N Teor. Mat. Fiz. 55 128 (1983); Ryzhov V N Theor. Math. Phys. 55 399 (1983)
  180. Nelson D R Phys. Rev. B 27 2902 (1983)
  181. Sachdev S, Nelson D R J. Phys. C 17 5473 (1984)
  182. Dzyubenko A B, Lozovik Yu E Zh. Eksp. Teor. Fiz. 102 284 (1992); Dzyubenko A B, Lozovik Yu E Sov. Phys. JETP 75 149 (1992)
  183. Cha M-C, Fertig H A Phys. Rev. Lett. 74 4867 (1995)
  184. Herrera-Velarde S, von Grünberg H H Soft Matter 5 391 (2009)
  185. Deutschländer S et al. Phys. Rev. Lett. 111 098301 (2013)
  186. Horn T et al. Phys. Rev. E 88 062305 (2013)
  187. Qi W, Dijkstra M Soft Matter 11 2852 (2015)
  188. Mansoori G A, Rice S A Adv. Chem. Phys. 156 197 (2015)
  189. Grimes C C, Adams G Phys. Rev. Lett. 42 795 (1979)
  190. Mehrotra R, Guenin B M, Dahm A J Phys. Rev. Lett. 48 641 (1982)
  191. Gallet F et al. Phys. Rev. Lett. 49 212 (1982)
  192. Deville G et al. Phys. Rev. Lett. 53 588 (1984)
  193. Mehrotra R et al. Phys. Rev. B 29 5239(R) (1984)
  194. Stan M A, Dahm A J Phys. Rev. B 40 8995 (1989)
  195. Guo C J et al. Phys. Rev. Lett. 51 1461 (1983)
  196. Morf R H Phys. Rev. Lett. 43 931 (1979)
  197. Bedanov V M, Gadiak G V, Lozovik Yu E Zh. Eksp. Teor. Fiz. 88 1622 (1985); Bedanov V M, Gadiyak G V, Lozovik Yu E Sov. Phys. JETP 61 967 (1985)
  198. Gann R C, Chakravarty S, Chester G V Phys. Rev. B 20 326 (1979)
  199. Taub H et al. Phys. Rev. B 16 4551 (1977)
  200. McTague J P et al. Phys. Rev. B 25 7765 (1982)
  201. Specht E D et al. Phys. Rev. B 30 1589(R) (1984)
  202. Heiney P A et al. Phys. Rev. Lett. 48 104 (1982)
  203. Heiney P A et al. Phys. Rev. B 28 6416 (1983)
  204. Dimon P et al. Phys. Rev. B 31 437 (1985)
  205. Rosenbaum T F et al. Phys. Rev. Lett. 50 1791 (1983)
  206. Nagler S E et al. Phys. Rev. B 32 7373 (1985)
  207. Greiser N et al. Phys. Rev. Lett. 59 1706 (1987)
  208. Koch S W, Abraham F F Phys. Rev. B 27 2964 (1983)
  209. Abraham F F Phys. Rev. Lett. 50 978 (1983)
  210. Abraham F F Phys. Rev. B 29 2606 (1984)
  211. Dahm A J Phys. Rev. B 29 484 (1984)
  212. Jin A J, Bjustrom M R, Chan M H W Phys. Rev. Lett. 62 1372 (1989)
  213. Migone A D, Li Z R, Chan M H W Phys. Rev. Lett. 53 810 (1984)
  214. Murray C A Bond-Orientational Order In Condensed Matter Systems (Ed. K J Strandburg) (New York: Springer-Verlag, 1992) p. 137
  215. Murray C A, Van Winkle D H Phys. Rev. Lett. 58 1200 (1987)
  216. Murray C A, Wenk R A Phys. Rev. Lett. 62 1643 (1989)
  217. Murray C A, Van Winkle D H, Wenk R A Phase Trans. 21 93 (1990)
  218. Murray C A, Sprenger W O, Wenk R A Phys. Rev. B 42 688 (1990)
  219. Armstrong A J, Mockler R C, O’Sullivan W J J. Phys. Condens. Matter 1 1707 (1989)
  220. Tang Y et al. Phys. Rev. Lett. 62 2401 (1989)
  221. Tang Y et al. Phase Trans. 21 75 (1990)
  222. Skjeltorp A T Phys. Rev. Lett. 51 2306 (1983)
  223. Kalia R K, Vashishta P J. Phys. C 14 L643 (1981)
  224. Broughton J Q, Gilmer G H, Weeks J D Phys. Rev. B 25 4651 (1982)
  225. Bladon P, Frenkel D Phys. Rev. Lett. 74 2519 (1995)
  226. Chou T, Nelson D R Phys. Rev. E 53 2560 (1996)
  227. Helgesen G, Skjeltorp A T Physica A 170 488 (1991)
  228. Kusner R E et al. Phys. Rev. Lett. 73 3113 (1994)
  229. Kusner R E et al. Phys. Rev. B 51 5746 (1995)
  230. Gasser U et al. ChemPhysChem 11 963 (2010)
  231. Zahn K, Maret G Phys. Rev. Lett. 85 3656 (2000)
  232. Keim P, Maret G, von Grünberg H H Phys. Rev. E 75 031402 (2007)
  233. von Grünberg H H et al. Phys. Rev. Lett. 93 255703 (2004)
  234. Dillmann P, Maret G, Keim P J. Phys. Condens. Matter 20 404216 (2008)
  235. Assoud L et al. Phys. Rev. Lett. 102 238301 (2009)
  236. Deutschländer S et al. Phys. Rev. Lett. 113 127801 (2014)
  237. Kusner R E, Mann J A, Dahm A J Phys. Rev. B 49 9190 (1994)
  238. Seshadri R, Westervelt R M Phys. Rev. Lett. 66 2774 (1991)
  239. Seshadri R, Westervelt R M Phys. Rev. B 46 5142 (1992)
  240. Seshadri R, Westervelt R M Phys. Rev. B 46 5150 (1992)
  241. Chudnovsky E M Phys. Rev. B 40 11355(R) (1989)
  242. Rice S A Chem. Phys. Lett. 479 1 (2009)
  243. Naidoo K J, Schnitker J J. Chem. Phys. 100 3114 (1994)
  244. Frenkel D, McTague J P Phys. Rev. Lett. 42 1632 (1979)
  245. Zollweg J A, Chester G V, Leung P W Phys. Rev. B 39 9518 (1989)
  246. Chen K, Kaplan T, Mostoller M Phys. Rev. Lett. 74 4019 (1995)
  247. Bagchi K, Andersen H C, Swope W Phys. Rev. Lett. 76 255 (1996)
  248. Bagchi K, Andersen H C, Swope W Phys. Rev. E 53 3794 (1996)
  249. Zollweg J A, Chester G V Phys. Rev. B 46 11186(R) (1992)
  250. Lee J, Strandburg K J Phys. Rev. B 46 11190(R) (1992)
  251. Weber H, Marx D Europhys. Lett. 27 593 (1994)
  252. Allen M P et al. J. Chem. Phys. 78 4206 (1983)
  253. Novaco A D, Shea P A Phys. Rev. B 26 284 (1982)
  254. Abraham F F Phys. Rev. Lett. 44 463 (1980)
  255. Barker J A, Hendersen D, Abraham F F Physica A 106 226 (1981)
  256. Toxvaerd S Phys. Rev. Lett. 44 1002 (1980)
  257. Broughton J Q, Gilmer G H, Weeks J D Phys. Rev. B 25 4651 (1982)
  258. Weber H, Marx D, Binder K Phys. Rev. B 51 14636 (1995)
  259. Frenkel D, Smit B Understanding Molecular Simulation: From Algorithms To Applications (Computational Science Series) Vol. 1 2nd ed. (San Diego: Academic Press, 2002)
  260. Mayer J E, Wood W W J. Chem. Phys. 42 4268 (1965)
  261. Lee J, Kosterlitz J M Phys. Rev. B 43 3265 (1991)
  262. Hoover W G, Ree F H J. Chem. Phys. 49 3609 (1968)
  263. Lee J, Kosterlitz J M Phys. Rev. Lett. 65 137 (1990)
  264. Fernández J F, Alonso J J, Stankiewicz J Phys. Rev. Lett. 75 3477 (1995)
  265. Binder K, Sengupta S, Nielaba P J. Phys. Condens. Matter 14 2323 (2002)
  266. Mak C H Phys. Rev. E 73 065104(R) (2006)
  267. Jaster A Europhys. Lett. 42 277 (1998)
  268. Jaster A Physica A 264 134 (1999)
  269. Jaster A Phys. Rev. E 59 2594 (1999)
  270. Bernard E P, Krauth W Phys. Rev. Lett. 107 155704 (2011)
  271. Engel M et al. Phys. Rev. E 87 042134 (2013)
  272. Alonso J J, Fernández J F Phys. Rev. E 59 2659 (1999)
  273. Weeks J D Phys. Rev. B 24 1530 (1981)
  274. Wierschem K, Manousakis E Phys. Rev. B 83 214108 (2011)
  275. Naidoo K, Schnitker J, Weeks J D Mol. Phys. 80 1 (1993)
  276. Vaulina O S, Koss (Adamovich) X G Phys. Lett. A 373 3330 (2009)
  277. Vaulina O S, Vasilieva E V Phys. Lett. A 378 719 (2014)
  278. Koss X G, Vaulina O S J. Phys. Conf. Ser. 653 012103 (2015)
  279. Kapfer S C, Krauth W Phys. Rev. Lett. 114 035702 (2015)
  280. Qi W, Gantapara A P, Dijkstra M Soft Matter 10 5449 (2014)
  281. Qi W-K et al. J. Phys. Condens. Matter 20 245102 (2008)
  282. Zu M et al. Phys. Rev. Lett. 117 085702 (2016)
  283. Zhang H et al. Phys. Rev. E 89 062410 (2014)
  284. Huberman B A, Doniach S Phys. Rev. Lett. 43 950 (1979)
  285. Fisher D S Phys. Rev. B 22 1190 (1980)
  286. Conen E, Schmid A J. Low Temp. Phys. 17 331 (1974)
  287. Gammel P L, Hebard A F, Bishop D J Phys. Rev. B 40 7354(R) (1989)
  288. Hebard A F, Fiory A T Physica B+C 109-110 1637 (1982)
  289. Berghuis P, van der Slot A L F, Kes P H Phys. Rev. Lett. 65 2583 (1990)
  290. Hikami S, Fujita A, Larkin A I Phys. Rev. B 44 10400(R) (1991)
  291. Tešanović Z, Xing L Phys. Rev. Lett. 67 2729 (1991)
  292. Kato Y, Nagaosa N Phys. Rev. B 47 2932(R) (1993)
  293. Kato Y, Nagaosa N Phys. Rev. B 48 7383 (1993)
  294. Hu J, MacDonald A H Phys. Rev. Lett. 71 432 (1993)
  295. Šášik R, Stroud D Phys. Rev. B 49 16074(R) (1995)
  296. Franz M, Teitel S Phys. Rev. B 51 6551 (1995)
  297. O’Neil J A, Moore M A Phys. Rev. Lett. 69 2582 (1992)
  298. O’Neil J A, Moore M A Phys. Rev. B 48 374 (1993)
  299. Moore M A Phys. Rev. B 39 136 (1989)
  300. Moore M A Phys. Rev. B 45 7336 (1992)
  301. Guillamon I et al. Nature Phys. 5 651 (2009)
  302. Guillamon I et al. Nature Phys. 10 851 (2014)
  303. Fomin Yu D et al. J. Chem. Phys. 129 064512 (2008)
  304. Gribova N V et al. Phys. Rev. E 79 051202 (2009)
  305. Fomin Yu D, Tsiok E N, Ryzhov V N J. Chem. Phys. 135 234502 (2011)
  306. Fomin Yu D, Tsiok E N, Ryzhov V N J. Chem. Phys. 135 124512 (2011)
  307. Ryltsev R E, Chtchelkatchev N M, Ryzhov V N Phys. Rev. Lett. 110 025701 (2013)
  308. Fomin Yu D, Tsiok E N, Ryzhov V N Phys. Rev. E 87 042122 (2013)
  309. Fomin Yu D et al. J. Chem. Phys. 141 034508 (2014)
  310. Fomin Yu D, Ryzhov V N, Gribova N V Phys. Rev. E 81 061201 (2010)
  311. Fomin Yu D, Tsiok E N, Ryzhov V N J. Chem. Phys. 134 044523 (2011)
  312. LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator), http://lammps.sandia.gov/
  313. Thorneywork A L et al. Phys. Rev. Lett. 118 158001 (2017)

© 1918–2022 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions