Geodesic
Some peculiarities in the behavior of non-Ising spin glasses
Teoretičeskaâ i matematičeskaâ fizika, Tome 182 (2015) no. 3, pp. 500-512 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This paper is a review. We outline the main directions in the modern theory of spin glasses. The main content is based on our recent papers, devoted to studying replica symmetry breaking in non-Ising spin glasses. Studying a series of generalized models showed a certain uniformity of the behavior of these generalized spin glasses. Essentially, we observe a significant difference between their behavior and the behavior of the known systems with random couplings of Ising spins—the Sherrington–Kirkpatrick model and the corresponding $p$-spin model. We find the bifurcation point for the solution with the first replica symmetry breaking, study the form and stability of the solution near the bifurcation point, and show in which cases the transition to the glass state occurs continuously and in which cases, with a jump of the order parameters.
Keywords: spin glass, replica method, Potts model
Mots-clés : quadrupole glass, solution bifurcation. 
@article{TMF_2015_182_3_a6,
     author = {E. E. Tareeva and T. I. Schelkacheva and N. M. Chtchelkachev},
     title = {Some peculiarities in the~behavior of {non-Ising} spin glasses},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {500--512},
     year = {2015},
     volume = {182},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2015_182_3_a6/}
}
TY  - JOUR
AU  - E. E. Tareeva
AU  - T. I. Schelkacheva
AU  - N. M. Chtchelkachev
TI  - Some peculiarities in the behavior of non-Ising spin glasses
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2015
SP  - 500
EP  - 512
VL  - 182
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2015_182_3_a6/
LA  - ru
ID  - TMF_2015_182_3_a6
ER  - 
%0 Journal Article
%A E. E. Tareeva
%A T. I. Schelkacheva
%A N. M. Chtchelkachev
%T Some peculiarities in the behavior of non-Ising spin glasses
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2015
%P 500-512
%V 182
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2015_182_3_a6/
%G ru
%F TMF_2015_182_3_a6
E. E. Tareeva; T. I. Schelkacheva; N. M. Chtchelkachev. Some peculiarities in the behavior of non-Ising spin glasses. Teoretičeskaâ i matematičeskaâ fizika, Tome 182 (2015) no. 3, pp. 500-512. http://geodesic.mathdoc.fr/item/TMF_2015_182_3_a6/

[1] D. Sherrington, S. Kirkpatrick, Phys. Rev. Lett., 35:26 (1975), 1792–1796 ; S. Kirkpatrick, D. Sherrington, Phys. Rev. B, 17:11 (1978), 4384–4403 | DOI | DOI

[2] G. Parisi, J. Phys. A: Math. Gen., 13:4 (1980), L115–L121 | DOI

[3] M. Mézard, G. Parisi, M. Virasoro, Spin Glass Theory and Beyond, World Sci., Singapore, 1987 | MR | Zbl

[4] E. Gardner, Nucl. Phys. B, 257:6 (1985), 747–765 | DOI | MR

[5] D. Gross, M. Mézard, Nucl. Phys. B, 240:4 (1984), 431–452 | DOI | MR

[6] B. Derrida, Phys. Rev. Lett., 45:2 (1980), 79–82 ; Phys. Rev. B, 24:5 (1981), 2613–2626 | DOI | MR | DOI | MR | Zbl

[7] E. E. Tareeva, T. I. Schelkacheva, N. M. Schelkachev, TMF, 160:2 (2009), 385–400 | DOI | DOI | MR

[8] T. I. Schelkacheva, E. E. Tareyeva, N. M. Chtchelkatchev, Phys. Rev. B, 82:13 (2010), 134208, 5 pp., arXiv: 1005.4923 | DOI

[9] A. Crisanti, L. Leuzzi, T. Rizzo, Phys. Rev. B, 71:9 (2005), 094202, 11 pp., arXiv: cond-mat/0406649 | DOI

[10] A. Crisanti, L. Leuzzi, G. Parisi, T. Rizzo, Phys. Rev. Lett., 92:12 (2004), 127203, 4 pp., arXiv: cond-mat/0307543 | DOI

[11] A. Crisanti, L. Leuzzi, Phys. Rev. B, 76:18 (2007), 184417, 8 pp., arXiv: 0705.3175 | DOI

[12] A. Crisanti, L. Leuzzi, Phys. Rev. B, 73:1 (2006), 014412, 20 pp. | DOI

[13] A. Crisanti, L. Leuzzi, Phys. Rev. Lett., 93:21 (2004), 217203, 4 pp., arXiv: cond-mat/0407129 | DOI

[14] F. Antenucci, C. Conti, A. Crisanti, L. Leuzzi, General phase-diagram of multimodal ordered and disordered lasers in closed and open cavities, arXiv: 1406.7826

[15] N. V. Gribova, V. N. Ryzhov, E. E. Tareyeva, Phys. Lett. A, 353:2–3 (2006), 226–229 | DOI

[16] E. E. Tareeva, TMF, 168:3 (2011), 584–592 | DOI | DOI

[17] T. Kirkpatrick, D. Thirumalai, Phys. Rev. Lett., 58:20 (1987), 2091–2094 | DOI | MR

[18] A. Crisanti, H. Horner, H.-J. Sommers, Z. Phys. B, 92:2 (1993), 257–271 | DOI | MR

[19] L. Bouchaud, J. P. Cugliandolo, J. Kurchan, M. Mézard, Physica A: Statist. Mech. Appl., 226:3 (1996), 243–273, arXiv: cond-mat/9511042 | DOI

[20] F. Caltagirone, U. Ferrari, L. Leuzzi, G. Parisi, F. Ricci-Tersenghi, T. Rizzo, Phys. Rev. Lett., 108:8 (2012), 085702, 5 pp., arXiv: 1111.6420 | DOI

[21] A. Crisanti, Nucl. Phys. B, 796:3 (2008), 425–456, arXiv: 0710.0601 | DOI | MR

[22] A. Crisanti, L. Leuzzi, A simple spin model for three steps relaxation and secondary processes in glass formers, arXiv: 1405.6832

[23] P. Charbonneau, J. Kurchan, G. Parisi, P. Urbani, F. Zamponi, Nature Commun., 5 (2014), 3725, 6 pp., arXiv: 1404.6809 | DOI

[24] T. Rizzo, Phys. Rev. E, 88:3 (2013), 032135, 12 pp., arXiv: 1305.2070 | DOI

[25] G. Parisi, F. Ricci-Tersenghi, T. Rizzo, J. Stat. Mech., 4 (2014), P04013, 21 pp. | DOI | MR

[26] G. Parisi, F. Zamponi, Rev. Modern Phys., 82:1 (2010), 789–845 | DOI

[27] P. G. Wolynes, V. Lubchenko, Structural Glasses and Supercooled Liquids: Theory, Experiment, and Applications, Wiley, New York, 2012

[28] L. Berthier, G. Biroli, Rev. Modern Phys., 83:2 (2011), 587–645, arXiv: 1011.2578 | DOI

[29] D. Panchenko, Ann. Probab., 42:3 (2014), 946–958 | DOI | MR | Zbl

[30] M. Talagrand, Mean Field Models for Spin Glasses, v. 1, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 54, Basic Examples, Springer, Berlin, 2011 | DOI | MR | Zbl

[31] F. Caltagirone, U. Ferrari, L. Leuzzi, G. Parisi, T. Rizzo, Phys. Rev. B, 83:10 (2011), 104202, 8 pp., arXiv: 1010.5000 | DOI

[32] K. Walasek, J. Phys. A: Math. Gen., 28:18 (1995), L497–L501 | DOI | MR | Zbl

[33] T. I. Schelkacheva, E. E. Tareyeva, N. M. Chtchelkatchev, Phys. Rev. E, 79:2 (2009), 021105, 7 pp., arXiv: 0809.0877 | DOI

[34] T. I. Schelkacheva, N. M. Chtchelkatchev, J. Phys. A: Math. Theor., 44:44 (2011), 445004, 14 pp., arXiv: 1103.4308 | DOI | Zbl

[35] E. E. Tareyeva, T. I. Schelkacheva, N. M. Chtchelkatchev, J. Phys. A: Math. Theor., 47:7 (2014), 075002, 12 pp., arXiv: 1309.4292 | DOI | MR | Zbl

[36] T. I. Schelkacheva, E. E. Tareyeva, N. M. Chtchelkatchev, Phys. Rev. E, 89:4 (2014), 042149, 5 pp., arXiv: 1309.6987 | DOI

[37] D. J. Gross, I. Kanter, H. Sompolinsky, Phys. Rev. Lett., 55:3 (1985), 304–307 | DOI

[38] E. De Santis, G. Parisi, F. Ritort, J. Phys. A: Math. Gen., 28:11 (1995), 3025–3041 | DOI | MR

[39] V. Janiš, A. Klíč, Phys. Rev. B, 84:6 (2011), 064446, 10 pp. | DOI

[40] N. V. Gribova, T. I. Schelkacheva, E. E. Tareyeva, J. Phys. A: Math. Theor., 43:49 (2010), 495006, 9 pp. | DOI | MR | Zbl

[41] M. M. Vainberg, V. A. Trenogin, Teoriya vetvleniya reshenii nelineinykh uravnenii, Nauka, M., 1969 | MR | MR | Zbl

[42] M. G. Vasin, E. E. Tareeva, T. I. Schelkachëva, N. M. Schelkachëv, TMF, 174:2 (2013), 228–242 | DOI | DOI | Zbl

[43] E. E. Tareeva, T. I. Schelkachëva, N. M. Schelkachëv, TMF, 155:2 (2008), 356–368 | DOI | DOI | MR | Zbl

[44] V. N. Ryzhov, E. E. Tareeva, T. I. Schelkachëva, N. M. Schelkachëv, TMF, 141:1 (2004), 131–140 | DOI | DOI | MR | Zbl

[45] J. R. L. de Almeida, D. J. Thouless, J. Phys. A, 11:5 (1978), 983–990 | DOI

[46] E. A. Luchinskaya, E. E. Tareeva, TMF, 87:3 (1991), 473–480 | DOI | MR | Zbl

[47] E. A. Luchinskaya, E. E. Tareeva, TMF, 91:1 (1992), 157–167 | DOI | MR

[48] E. A. Lutchinskaia, E. E. Tareyeva, Europhys. Lett., 17:2 (1992), 109–112 | DOI | MR

[49] N. V. Gribova, V. N. Ryzhov, E. E. Tareyeva, Phys. Rev. E, 68:6 (2003), 067103, 4 pp., arXiv: cond-mat/0307042 | DOI