Phase transitions in frustrated Ising models (Review)
Murtazaev A.K. 1, Ramazanov M.K. 1
1Institute of Physics, Dagestan Federal Research Center, Russian Academy of Sciences, Makhachkala, Russia
Email: akai2005@mail.ru, sheikh77@mail.ru

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The current state of research on phase transitions and critical phenomena in frustration is considered modeled Ising models. The results obtained on the basis of highly efficient algorithms are discussed. Monte Carlo method for Ising models on lattices of various dimensions and types. A main attention given to the results of studying phase transitions, critical and thermodynamic properties of models Ising, taking into account the exchange interactions of the first and second neighbors, as well as the external magnetic fields. Phase diagrams of the dependence of the critical temperature on the value of the exchange interactions of second neighbors and an external magnetic field. The regions of observation of phase transitions of the first and second kind. The universality classes of critical behavior are established. For some models have ranges with non-universal critical behavior. Discussed features of the critical behavior of frustrated Ising models. Keywords: Ising model, frustrations, phase transitions, Monte Carlo method.
  1. W. Lenz. Z. Phys. 21, 613 (1920)
  2. E. Ising. Z. Phys. 31, 253 (1925)
  3. L. Onsager. Phys. Rev. 65, 117 (1944)
  4. T.D. Lee, C.N. Yang. Phys. Rev. 87, 410 (1952)
  5. J.C. Wheeler. Ann. Rev. Phys. Chem. 28, 411 (1977)
  6. M. Blume, V.J. Emery, R.B. Griffiths. Phys. Rev. A 4, 1071 (1971)
  7. C.J. Thompson. Biopolymers 6, 1101 (1968)
  8. S. Torquatto. Phys. Biol. 8, 015017 (2011)
  9. E. Alvarez-Lacalle, B. Echebarria, J. Spalding, Y. Shiferaw. Phys. Rev. Lett. 114, 108101 (2015)
  10. A. Mailhot, M.L. Plumer, A. Caille. Phys. Rev. B 50, 6854 (1994-II)
  11. K. Binder, E. Luijten. Phys. Rep. 344, 179 (2001)
  12. D.P. Landau. Physica A 205, 41 (1994)
  13. I.P. Beloborov, R.S. Gekht, V.A. Ignatchenko. ZhETF 84, 1097 (1983). (in Russian)
  14. D. Loison, A.I. Sokolov, B. Delamotte, S.A. Antonenko, K.D. Schotte, H.T. Diep. JETP Lett. 72, 337 (2000)
  15. J. Snyder, J.S. Slusky, R.J. Cava, P. Schiffer. Nature 413, 48 (2001)
  16. J.S. Gardner, A. Keren, G. Ehlers, C. Stock, Eva Segal, J.M. Roper, B. F k, M.B. Stone, P.R. Hammar, D.H. Reich, B.D. Gaulin. Phys. Rev. B 68, 180401 (2003)
  17. G. Moller, R. Moessner. Phys. Rev. Lett. 96, 237202 (2006)
  18. F.A. Kassan-Ogly, B.N. Filippov, A.K. Murtazaev, M.K. Ramazanov, M.K. Badiev. JMMM 324, 3418 (2012)
  19. S.E. Korshunov. UFN, 176, 233 (2006). (in Russian)
  20. R. Baxter. Tochno reshaemye modeli v statisticheskoy mekhanike / Pod red. A.M. Brodskogo. Mir, M., (1985). 486 p. (in Russian)
  21. L.D. Fosdik. Bull. Am. Phys. Soc. 2, 239 (1957)
  22. D.P. Landau. Phys. Rev. B 13, 2997 (1976)
  23. D.P. Landau. Phys. Rev. B 14, 255 (1976)
  24. D.P. Landau. Phys. Rev. B 16, 4164 (1977)
  25. K. Binder. Phys. Status Solidi B 46, 567 (1971)
  26. L. Hua, J.W. Tucker. JMMM 140-144, 1509 (1995)
  27. D. Farsal, M. Snina, M. Bennai, M. Badia. J. Supercond. Nov. Magn. 30, 2187 (2017)
  28. D. Farsal, M. Bennai, M. Badia. SPIN 8, 1850010 (2018)
  29. T. Balcerzak, K. Sza owski, A. Bobak, M. vZukovivc. Phys. Rev. E 98, 022123 (2018)
  30. Y. Aoyama, W. Chen, M. Tanaka. J. Phys. Soc. Jpn. 66, 272 (1997)
  31. M.E.J. Newman, G.T. Barkema. Phys. Rev. E 53, 393 (1996)
  32. E.T. Gawlinski, S. Kumar, M. Grant, J.D. Gunton, K. Kaski. Phys. Rev. B 32, 1575 (1985)
  33. C. Dekker, B.J. Dikken, A.F.M. Arts. Solid State Commun. 54, 887 (1985)
  34. J.D. Shore, G.M. Thurston. Phys. Rev. E 92, 062123 (2015)
  35. H.-J. Wu, W. Wang, B.-C. Li, Q. Li, J.-H. Xu, F. Wang. J. Phys. Chem. Solids 136, 109174 (2020)
  36. O. Nagai, Y. Yamada, K. Nishino, Y. Miyatake. Phys. Rev. B 35, 3425 (1987)
  37. R. Bidaux, N. Boccara. Phys. Rev. B 34, 4881 (1986)
  38. M. Danino. Solid State Commun. 52, 885 (1984)
  39. S.N. Coppersmith. Phys. Rev. B 32, 1594 (1985)
  40. J.D. Kimel, S. Black, P. Carter, Y.L. Wang. Phys. Rev. B 35, 3347 (1987)
  41. S.V. Belim, I.B. Larionov. Vestn. Moskovsk. un-ta. Ser. 3. Fizika. Astronomiya, 4 (58), (2018). (in Russian)
  42. T. Arh, B. Sana, M. Pregelj, P. Khuntia, Z. Jaglivcic, M.D. Le, P.K. Biswas, P. Manuel, L. Mangin-Thro, A. Ozarowski, A. Zorko. Nature Mater. 21, 416 (2022)
  43. K. Dutta, D. Talukdar. JMMM 556, 169344 (2022)
  44. Y. Shokef, A. Souslov, T.C. Lubensky. Proc. Natl. Acad. Sci. U.S.A. (PNAS) 108, 11804 (2011)
  45. W. Kerler, P. Rehberg. Phys. Rev. B 49, 9688 (1994)
  46. A.K. Murtazaev, I.K. kamilov, M.K. Ramazanov. FTT, 47, 1125 (2005). (in Russian)
  47. M. Schmidt, G.L. Kohlrausch, F.M. Zimmer. Physica A 596, 127126 (2022)
  48. E. Jurvcivsinova, M. Jurvcivsin. Physica A 603, 127731 (2022)
  49. R.M. Liu, W.Z. Zhuo, X.B. Lu, X.S. Gao, M.H. Qin, J.-M. Liu, S. Dong. Phys. Rev. E 93, 032114 (2016)
  50. R. Moessner, S.L. Sondhi. Phys. Rev. B 63, 224401 (2001)
  51. A.G. Cavaliere, A. Pelissetto. J. Phys. A 52, 174002 (2019)
  52. D.N. Yasinskaya, V.A. Ulitko, Yu.D. Panov. FTT, 63, 1350 (2021). (in Russian)
  53. E. Jurvcivsinova, M. Jurvcivsin. Phys. Rev. E 104, 044121 (2021)
  54. E.S. Tsuvarev, F.A. Kassan-Ogly, A.I. Proshkin. ZhETF 158, 504 (2020). (in Russian)
  55. J. Wu, D.C. Mattis. Phys. Rev. B 67, 224414 (2003)
  56. T. Yoshioka, A. Koga, N. Kawakami. J. Phys. Soc. Jpn. 73, 1805 (2004)
  57. N. De La Espriella, G.M. Buendia, J.C. Madera. J. Phys. Commun. 2, 025006 (2018)
  58. H. Bouda, L. Bahmad, R. Masrour, A. Benyoussef. J. Supercond. Nov. Magn. 32, 1837 (2019)
  59. A. Jabar, R. Masrour. Physica A 515, 270 (2019)
  60. R. Masrour, A. Jabar. Physica A 539, 122878 (2020)
  61. E. Jurvcivsinova, M. Jurvcivsin. Phys. Rev. B 101, 214443 (2020)
  62. E. Jurvcivsinova, M. Jurvcivsin. Physica A 583, 126338 (2021)
  63. E. Jurvcivsinova, M. Jurvcivsin. Phys. Lett. A 388, 127043 (2021)
  64. S.V. Belim, I.B. Larionov, R.V. Solonetsky. Fizika metallov i metallovedenie, 117, 1115 (2016). (in Russian)
  65. P.F. Godoy, M. Schmidt, M.F. Zimmer. Phys. Lett. A 384, 126687 (2020)
  66. H. Li, L.-P. Yang. Phys. Rev. E 104, 024118 (2021)
  67. A.O. Sorokin. Physica A 602, 127621 (2022)
  68. Y. Hu, P. Charbonneau. Phys. Rev. B 104, 144429 (2021)
  69. A.V. Zarubin, F.A. Kassan-Ogly, A.I. Proshkin. JMMM 514, 167144 (2020)
  70. F.A. Kassan-Ogly, A.I. Proshkin. Phys. Met. Metallogr. 120, 1359 (2019)
  71. A.I. Proshkin, F.A. Kassan-Ogly. Phys. Met. Metallogr 120, 1366 (2019)
  72. G. Toulouse. Commun. Phys. 2, 115 (1977)
  73. V.S. Dotsenko. UFN, 163, 1 (1993). (in Russian)
  74. A.P. Ramirez. Annu. Rev. Mater. Sci. 24, 453 (1994)
  75. V.S. Dotsenko. UFN 165, 481 (1995). (in Russian)
  76. I.K. Kamilov, A.K. Murtazaev, Kh.K. Aliev. UFN, 169, 773 (1999). (in Russian)
  77. K. Binder. Metody Monte-Carlo v statisticheskoy fizike / Pod. red. G.I. Marchuk, G.A. Mikhailov. Mir, M., (1982). 400 p. (in Russian)
  78. A. Mitsutake, Y. Sugita, Y. Okamoto. Biopolymers (Peptide Science) 60, 96 (2001)
  79. F. Wang, D.P. Landau. Phys. Rev. Lett. 86, 2015 (2001)
  80. M.N. Barber. J. Phys. A 12, 679 (1979)
  81. D.P. Landau, K. Binder. Phys. Rev. B 31, 5946 (1985)
  82. J.A. Plascak. Physica A 183, 563 (1992)
  83. D.P. Landau, K. Binder. Monte Carlo Simulations in Statistical Physics. Cambridge University Press, Cambridge (2000)
  84. M.P. Nightingale. Phys. Lett. A 59, 486 (1977)
  85. R. H. Swendsen, S. Krinsky. Phys. Rev. Lett. 43, 177 (1979)
  86. K. Binder, D.P. Landau. Phys. Rev. B 21, 1941 (1980)
  87. K. Minami, M. Suzuki. J. Phys. A 27, 7301 (1994)
  88. J.L. Morian-Liopez, F. Aguilera-Granja, J.M. Sanchez. Phys. Rev. B 48, 3519 (1993)
  89. J.L. Moran-Lopez, F. Aguilera-Granja, J.M. Sanchez. J. Phys.: Condens. Matter 6, 9759 (1994)
  90. E. Lopez-Sandoval, J.L. Moran-Lopez, F. Aguilera-Granja. Solid State Commun. 112, 437 (1999)
  91. C. Buzano, M. Pretti. Phys. Rev. B 56, 636 (1997)
  92. A.K. Murtazaev, M.K. Ramazanov, M.K. Badiyev, FNT 37, 1258 (2011). (in Russian)
  93. K. Binder, J.-Sh. Wang. J. Status Phys. 55, 87 (1989)
  94. Junqi Yin, D.P. Landau. Phys. Rev. E 80, 051117 (2009)
  95. K. Bimder, D.V. Kheerman. Modelirovanie metodom Monte-Carlo v statisticheskoy fizike. Nauka, M., (1995). (in Russian)
  96. A.R. Anjos, J.R. Viana, J.R. Sousa. Phys. Lett. A 372, 1180 (2008)
  97. A. Kalz, A. Honecker. Phys. Rev. B 86, 134410 (2012)
  98. S. Jin, A. Sen, A.W. Sandvik. Phys. Rev. Lett. 108, 045702 (2012)
  99. A.E. Ferdinand, M.E. Fisher. Phys. Rev. 185, 832 (1969)
  100. M.E. Fisher, M.N. Barber. Phys. Rev. Lett. 28, 1516 (1972)
  101. P. Peczak, A.M. Ferrenberg, D.P. Landau. Phys. Rev. B 43, 6087 (1991)
  102. D. Staufer, A. Aharony. Introduction to Percolation Theory. Taylor \& Francis, London (1994)
  103. A.K. Murtazaev, I.K.Kamilov, M.K. Ramazanov. FNT 32, 323 (2006). (in Russian)
  104. A.K. Murtazaev, M.K. Ramazanov. Phys. Rev. B 76, 174421 (2007)
  105. Ch. Holm, W. Janke. Phys. Rev. B 48, 936 (1993)
  106. A. Malakis, P. Kalozoumis, N. Tyraskis. Eur. Phys. J. B 50, 63 (2006)
  107. A.K. Murtazaev, M.K. Ramazanov, F.A. Kassan-Ogly, M.K. Badiyev, ZhETF 144, 1239 (2013). (in Russian)
  108. F.A. Kassan-Ogly, A.K. Murtazaev, A.K. Zhuravlev, M.K. Ramazanov, A.I. Proshkin. JMMM 384, 247 (2015)
  109. A.K. Murtazaev, M.K. Ramazanov, M.K. Badiev. Physica B: Condens. Matter 476, 1 (2015)
  110. M.K. Ramazanov, A.K. Murtazaev, M.A. Magomedov. Solid State Commun. 233, 35 (2016)
  111. P. Chandra, P. Coleman, I. Ritchey. J. de Physique 33, 591 (1993)
  112. J.T. Chalker, P.C.W. Holdsworth, E.F. Shender. Phys. Rev. Lett. 68, 855 (1992)
  113. A.B. Harris, C. Kallin, A.J. Berlinsky. Phys. Rev. B 45, 2899 (1992)
  114. R.S. Gekht, I.N. Bondarenko. ZhETF 113, 2209 (1998). (in Russian)
  115. M. Wolf, K.D. Schotte. J. Phys. A 21, 2195 (1988)
  116. R.S. Gekht, V.I. Ponomarev. Phase Transitions 20, 27 (1990)
  117. M. Stahn, R.E. Lechner, H. Dachs, H.E. Jacobs. J. Phys. C 16, 5073 (1983)
  118. M.G. Townsend, G. Longworth, E. Roudaut. Phys. Rev. B 33, 4919 (1986)
  119. T. Matsuo, H. Suga. Rev. Inorg. Chem. 3, 371 (1981)
  120. M. Takano, T. Shinjo, T. Takada. J. Phys. Soc. Jpn. 30, 1049 (1971)
  121. T. Takagi, M. Mekata. J. Phys. Soc. Jpn. 62, 3943 (1993)
  122. M.K. Ramazanov. Pis'ma v ZhETF 94 (335), (2011). (in Russian)
  123. M.K. Ramazanov, A.K. Murtazaev, M.A. Magomedov, M.K. Badiev. Phase Transitions 91, 610 (2018)
  124. A.R. Anjos, J.R. Viana, J.R. de Sousa, J.A. Plascak. Phys. Rev. E 76, 022103 (2007)
  125. D.R.O. Salmon, N. Crokidakis, A.M. Neto, T.I. Padilha, J.R. Viana, J.R. de Sousa. Int. J. Mod. Phys. B 27, 1350162 (2013)
  126. M.K. Ramazanov, A.K. Murtazaev. Pis'ma v ZhETF 101 793 (2015). (in Russian)
  127. M.K. Ramazanov, A.K. Murtazaev. Pis'ma v ZhETF 103 522 (2016). (in Russian)
  128. J.C. Le Guillou, J. Zinn-Justin. Phys. Rev. B 21, 3976 (1980)
  129. H.T. Diep, P. Lallemand, O. Nagai. J. Phys. C 18, 1067 (1985)
  130. L.W. Bernardi, K. Hukushima, H. Takayama. J. Phys. A 32, 1787 (1999)
  131. M.K. Ramazanov, A.K. Murtazaev. FTT, 59, 1797 (2017). (in Russian)
  132. M.K. Ramazanov, A.K. Murtazaev. Phase Transitions 91, 83 (2018)
  133. G.H. Wannier. Phys. Rev. 79, 357 (1950)
  134. G.F. Newell. Phys. Rev. 79, 876 (1950)
  135. R.M.F. Houtapel. Physica 16, 425 (1950)
  136. J. Hose, K. Kadanoff, S. Kirkpatricks, D.R. Nelson. Physica 16, 1217 (1977)
  137. D. Blankschtein, M. Ma, A. N. Berker, G.S. Grest, C.M. Soukoulis. Phys. Rev. B 29, 5250 (1984)
  138. G.A. Baker, Jr., B.G. Nickel, D.I. Meiron. Phys. Rev. B 17, 1365 (1978)
  139. O. Heinonen, R.G. Petschek. Phys. Rev. B 40, 9052 (1989)
  140. M.L. Plumer, A. Mailhot, R. Ducharme, A. Caille, H.T. Diep. Phys. Rev. B 47, 312 (1993)
  141. A. Bunker, B.D. Gaulin, C. Kallin. Phys. Rev. B 48, 15 861 (1993)
  142. N. Todoroki, S. Miyashita. J. Phys. Soc. Jpn. 73, 412 (2004)
  143. H. Shiba. Prog. Teor. Phys. 64, 466 (1980)
  144. M. Kaburagi, T. Tonegawa, J. Kanamori. J. Phys. Soc. Jpn. 51, 3857 (1982)
  145. P. Matsubara, S. Ikeda. Phys. Rev. B 28, 4064 (1983)
  146. P. Matsubara, S. Inawashira. J. Phys. Soc. Jpn. 53, 4373 (1984)
  147. H. Yoshizawa, K. Hirakawa. J. Phys. Soc. Jpn. 46, 448 (1980)
  148. E. Rastelli, S. Regina, A. Tassi. Phys. Rev. 71, 174406 (2005)
  149. A.K. Murtazaev, M.K. Ramazanov, M.K. Badiev. Physica A 507, 210 (2018)
  150. F. Wang, D.P. Landau. Phys. Rev. E 64, 056101 (2001)
  151. A.K. Murtazaev, M.K. Ramazanov, M.K. Badiyev. FNT 45, 1493 (2019)
  152. A.K. Murtazaev, M.K. Ramazanov, M.K. Badiyev. FTT, 61, 1898 (2019). (in Russian)
  153. M.K. Badiyev, A.K. Murtazaev, M.K. Ramazanov. ZhETF 150, 722 (2016). (in Russian)
  154. P.H. Lundow, K. Markstrom, A. Rosengren. Phil. Mag. 89, 2042 (2009)
  155. P. Butera, M. Comi. Phys. Rev. B 65, 144431 (2002)
  156. P. Butera, M. Comi. Phys. Rev. B 72, 014442 (2005)
  157. M. Plischke, J. Oitmaa. Phys. Rev. B 19, 487 (1979)
  158. J.R. Banavar, D. Jasnow, D.P. Landau. Phys. Rev. B 20, 3820 (1979)
  159. M.J. Velgakis, M. Ferer. Phys. Rev. B 27, 401 (1983)
  160. A.K. Murtazaev, M.K. Ramazanov, F.A. Kassan-Ogly, D.R. Kurbanova. ZhETF 147, 127 (2015). (in Russian)
  161. A.K. Murtazaev, M.K. Ramazanov, D.R. Kurbanova, M.K. Badiev, Ya.K. Abuev. FTT, 59, 1082 (2017). (in Russian)
  162. A.K. Murtazaev, M.K. Ramazanov, M.A. Magomedov, D.R. Kurbanova. FTT, 60, 1798 (2018). (in Russian)
  163. A.K. Murtazaev, M.A. Magomedov, M.K. Ramazanov. Pis'ma v ZhETF 107, 265 (2018). (in Russian)
  164. A.K. Murtazaev, M.K. Ramazanov, D.R. Kurbanova, M.A. Magomedov, K.Sh. Murtazaev. Mater.Lett. 236, 669 (2019)
  165. Y.-H. Song, Y. Kim, N. Li, B.-N. Lu, R. He, D. Lee. Phys. Rev. C 104, 044304 (2021)
  166. M.Yu. Lavrentiev, D. Nguyen-Manh, S.L. Dudarev. Phys. Rev. B 81, 184202 (2010)
  167. R. Leidl, H.W. Diehl. Phys. Rev. B 57, 1908 (1998)
  168. P. Villars, K. Cenzual, R. Gladyshevskii. Handbook of inorganic substances. Berlin (2017). P. 1955
  169. E. Mengotti, L.J. Heyderman, A. Bisig, A. Fraile Rodr guez, L. Le Guyader, F. Nolting, H.B. Braun. J. Appl. Phys. 105, 113113 (2009)
  170. A. Smerald, F. Mila. Phys. Rev. Lett. 115, 147202 (2015)
  171. S. Mahmoudian, L. Rademaker, A. Ralko, S. Fratini, V. Dobrosavljevic. Phys. Rev. Lett. 115, 025701 (2015)
  172. A.K. Murtazaev, M.K. Badiev, M.K. Ramazanov, M.A. Magomedov. Physica A 555, 124530 (2020)
  173. M.K. Badiyev, A.K. Murtazaev, M.K. Ramazanov, M.A. Magomedov. FNT 46, 824 (2020). (in Russian)
  174. A.K. Murtazaev, M.A. Magomedov, M.K. Ramazanov, M.K. Badiev. Physica E 142, 115320 (2022)
  175. M.K. Badiyev, A.K. Murtazaev, M.K. Ramazanov, M.A. Magomedov. ZhETF 161, 753 (2022). (in Russian)
  176. A.K. Murtazaev, M.K. Ramazanov, K.Sh. Murtazaev, M.A. Magomedov, M.K. Badiev. FTT, 62, 229 (2020). (in Russian)
  177. K.S. Murtazaev, A.K. Murtazaev, M.K. Ramazanov, M.A. Magomedov, A.A. Murtazaeva. Low Temperature Phys. 47, 515 (2021)
  178. K.S. Murtazaev, M.A. Magomedov, A.K. Murtazaev, M.K. Ramazanov. Physica E 148, 115646 (2023).
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