Cavity method for quantum spin glasses on the Bethe lattice

Physical Review B - Condensed Matter and Materials Physics

Volumn 78, Issue 13, 2008, Pages

  Laumann, C ^a   Scardicchio, A ^a   Sondhi, S L ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 55349120166     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.78.134424     Document Type: Article

References (50)

1. D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35, 1792 (1975). 10.1103/PhysRevLett.35.1792
2. D. J. Thouless, P. W. Anderson, and R. G. Palmer, Philos. Mag. 35, 593 (1977). 10.1080/14786437708235992
3. It is interesting to notice that in the TAP approach (Ref.), the free energy was found as a consequence of the mean-field equations, reversing the usual order of reasoning. This was mended by Anderson shortly thereafter (Ref.) and in many other papers later. However this unusual secondary role of the free energy in the problem at hand has survived until today and seems to be a peculiarity of spin glasses.
4. A. J. Bray and M. A. Moore, J. Phys. C 13, L469 (1980). 10.1088/0022-3719/13/19/002
5. J. R. L. de Almeida and D. J. Thouless, J. Phys. A 11, 983 (1978). 10.1088/0305-4470/11/5/028
6. G. Parisi, J. Phys. A 13, L115 (1980) 10.1088/0305-4470/13/4/009
7. G. Parisi, J. Phys. A 13, 1101 (1980) 10.1088/0305-4470/13/3/042
8. G. Parisi, J. Phys. A 13, 1887 (1980) 10.1088/0305-4470/13/5/047
9. G. Parisi, Phys. Rev. Lett. 50, 1946 (1983). 10.1103/PhysRevLett.50.1946
10. P. W. Anderson, Lectures on Amorphous Systems, Proceedings of the 31st Session of the Les Houches Summer School, Les Houches, France, July-August, 1978 (unpublished), pp. 159-261.
11. D. R. Bowman and K. Levin, Phys. Rev. B 25, 3438 (1982). 10.1103/PhysRevB.25.3438
12. L. Viana and A. J. Bray, J. Phys. C 18, 3037 (1985). 10.1088/0022-3719/ 18/15/013
13. J. T. Chayes, L. Chayes, J. P. Sethna, and D. J. Thouless, Commun. Math. Phys. 106, 41 (1986) 10.1007/BF01210926
14. J. M. Carlson, J. T. Chayes, L. Chayes, J. P. Sethna, and D. J. Thouless, Europhys. Lett. 5, 355 (1988). 10.1209/0295-5075/5/4/013
15. C. Kwon and D. J. Thouless, Phys. Rev. B 37, 7649 (1988). 10.1103/PhysRevB.37.7649
16. P. Y. Lai and Y. Y. Goldschmidt, J. Phys. A 22, 399 (1989) 10.1088/0305-4470/22/4/009
17. Y. Y. Goldschmidt and P. Y. Lai, J. Phys. A 23, L775 (1990). 10.1088/0305-4470/23/15/014
18. D. S. Fisher and D. A. Huse, Phys. Rev. Lett. 56, 1601 (1986) 10.1103/PhysRevLett.56.1601
19. D. S. Fisher and D. A. Huse, Phys. Rev. B 38, 386 (1988). 10.1103/PhysRevB.38.386
20. R. Monasson, R. Zecchina, S. Kirkpatrick, B. Selman, and L. Troyansky, Nature (London) 400, 133 (1999) 10.1038/22055
21. M. Mezard, G. Parisi, and R. Zecchina, Science 297, 812 (2002) 10.1126/science.1073287
22. M. Mézard and R. Zecchina, Phys. Rev. E 66, 056126 (2002). 10.1103/PhysRevE.66.056126
23. B. Bollobas, Random Graphs (Cambridge University Press, Cambridge, 2001).
24. M. Mezard and G. Parisi, J. Stat. Phys. 111, 1 (2003) 10.1023/A:1022221005097
25. M. Mezard and G. Parisi, Eur. Phys. J. B 20, 217 (2000). 10.1007/PL00011099
26. S. Kirkpatrick, C. D. Gelatt, Jr., and M. P. Vecchi, Science 220, 671 (1983). 10.1126/science.220.4598.671
27. Y. Fu and P. Anderson, J. Phys. A 19, 1605 (1986). 10.1088/0305-4470/19/ 9/033
28. A. Montanari, F. Ricci-Tersenghi, and G. Semerjian, J. Stat. Mech.: Theory Exp. 2008, P04004. 10.1088/1742-5468/2008/04/P04004
29. E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, Science 292, 472 (2001). 10.1126/science.1057726
30. T. K. Kopeć and K. D. Usadel, Phys. Rev. Lett. 78, 1988 (1997) 10.1103/PhysRevLett.78.1988
31. T. K. Kopec and K. D. Usadel, Phys. Status Solidi B 243, 502 (2006). 10.1002/pssb.200541282
32. S. Knysh and V. N. Smelyanskiy, arXiv:cond-mat/0602257 (unpublished)
33. A. Boulatov and V. N. Smelyanskiy, Phys. Rev. A 68, 062321 (2003). 10.1103/PhysRevA.68.062321
34. A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996). 10.1103/RevModPhys.68.13
35. S. F. Edwards and P. W. Anderson, J. Phys. F: Met. Phys. 5, 965 (1975). 10.1088/0305-4608/5/5/017
36. In particular, the cavity field distribution will depend only on the depth of a site in the Cayley tree, assuming i.i.d. boundary fields. This distribution becomes depth independent sufficiently deep in the tree.
37. Readers, familiar with the literature, may note that this allows one to bypass the parametrization of the order parameter on the replica indices, a particularly thankless task on the Bethe lattice since an infinite sequence of order parameters qa,b, qa,b,c, qa,b,c,d,... is necessary (Ref.).
38. D. S. Fisher, Phys. Rev. B 50, 3799 (1994) 10.1103/PhysRevB.50.3799
39. B. M. McCoy and T. T. Wu, Phys. Rev. 176, 631 (1968) 10.1103/PhysRev.176. 631
40. O. Motrunich, S. C. Mau, D. A. Huse, and D. S. Fisher, Phys. Rev. B 61, 1160 (2000). 10.1103/PhysRevB.61.1160
41. C. De Dominicis and P. Mottishaw, J. Phys. A 20, L1267 (1987). 10.1088/0305-4470/20/18/009
42. K. D. Usadel, Solid State Commun. 58, 629 (1986) 10.1016/0038-1098(86) 90234-6
43. H.-J. Sommers and K. D. Usadel, Z. Phys. B: Condens. Matter 47, 63 (1982). 10.1007/BF01686185
44. G. E. Hinton and T. J. Sejnowski, in Parallel Distributed Processing, edited by, D. E. Rumelhart, and, J. L. McClelland, (MIT, Cambridge, 1986), p. 282.
45. We note that a rearrangement of the sum over configurations according to the number of kinks looks such as a natural perturbative expansion in Bt for the system and naively scales polynomially O (Ntk), where k is the maximum number of kinks summed. Unfortunately, little is gained from this rearrangement in terms of the accurately computable region of the phase diagram: the expected number of kinks for relevant rod configurations scales ∼2 Bt /τ and thus the sum is again exponential in Bt /τ.
46. M. C. Arnesen Phys. Rev. Lett. 87, 017901 (2001). 10.1103/PhysRevLett.87. 017901
47. M. B. Hastings, Phys. Rev. B 76, 201102 (R) (2007). 10.1103/PhysRevB.76. 201102
48. M. Leifer and D. Poulin, Ann. Phys. (N.Y.) 323, 1899 (2008). 10.1016/j.aop.2007.10.001
49. D. Poulin and E. Bilgin, Phys. Rev. A 77, 052318 (2008). 10.1103/PhysRevA.77.052318
50. F. Krzakala, A. Rosso, G. Semerjian, and F. Zamponi, arXiv:0807.2553, Phys. Rev. B (to be published 2008).