Spherical 2+p spin-glass model: An analytically solvable model with a glass-to-glass transition

Physical Review B - Condensed Matter and Materials Physics

Volumn 73, Issue 1, 2006, Pages

  Crisanti, A ^a   Leuzzi, L ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33144459945     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.73.014412     Document Type: Article

References (76)

1. T. R. Kirkpatrick and D. Thirumalai, Phys. Rev. B PRBMDO 0163-1829 10.1103/PhysRevB.36.5388 36, 5388 (1987).
2. D. Thirumalai and T. R. Kirkpatrick, Phys. Rev. B PRBMDO 0163-1829 10.1103/PhysRevB.38.4881 38, 4881 (1988).
3. A. Crisanti and H-.J. Sommers, Z. Phys. B: Condens. Matter ZPCMDN 0722-3277 10.1007/BF01309287 87, 341 (1992).
4. J. P. Bouchaud, L. F. Cugliandolo, J. Kurchan, and M. Mézard, in Spin Glasses and Random Fields, edited by, A. P. Young, (World Scientific, Singapore, 1997).
5. Th. M. Nieuwenhuizen, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.74.4289 74, 4289 (1995).
6. A. J. Bray and M. A. Moore, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.41.1068 41, 1068 (1978).
7. E. Pytte and J. Rudnick, Phys. Rev. B PRBMDO 0163-1829 10.1103/PhysRevB.19.3603 19, 3603 (1979).
8. S. Ciuchi and A. Crisanti, Europhys. Lett. EULEEJ 0295-5075 10.1209/epl/i2000-00215-y 49, 754 (2000).
9. W. Götze and L. Sjörgen, J. Phys.: Condens. Matter JCOMEL 0953-8984 10.1088/0953-8984/1/26/015 1, 4203 (1989).
10. M. Mézard and G. Parisi, J. Phys. I JPGCE8 1155-4304 10.1051/jp1:1991171 1, 809 (1991).
11. T. Giamarchi and P. Le Doussal, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.72.1530 72, 1530 (1994);
12. T. Giamarchi and P. Le Doussal, Phys. Rev. B PRBMDO 0163-1829 10.1103/PhysRevB.52.1242 52, 1242 (1995).
13. L. F. Cugliandolo, J. Kurchan, and P. Le Doussal, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.76.2390 76, 2390 (1996).
14. P. Le Doussal and K. J. Wiese, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.80.2362 80, 2362 (1998).
15. A. Crisanti and L. Leuzzi, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.93.217203 93, 217203 (2004).
16. O. B. Tsiok, V. V. Brazhkin, A. G. Lyapin, and L. G. Khvostantsev, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.80.999 80, 999 (1998).
17. L. Huang and J. Kieffer, Phys. Rev. B PRBMDO 0163-1829 10.1103/PhysRevB.69.224203 69, 224203 (2004).
18. S. K. Deb, M. Wilding, M. Somayazulu, and P. F. McMillan, Nature (London) NATUAS 0028-0836 10.1038/35107036 414, 528 (2001).
19. P. H. Poole, F. Sciortino, U. Essmann, and H. E. Stanley, Nature (London) NATUAS 0028-0836 10.1038/360324a0 360, 324 (1992);
20. P. H. Poole, U. Essmann, F. Sciortino, and H. E. Stanley, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.48.4605 48, 4605 (1993).
21. K. Dawson, G. Foffi, M. Fuchs, W. Gotze, F. Sciortino, M. Sperl, P. Tartaglia, T. Voigtmann, and E. Zaccarelli, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.63.011401 63, 011401 (2000).
22. E. Zaccarelli, G. Foffi, K. A. Dawson, F. Sciortino, and P. Tartaglia, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.63.031501 63, 031501 (2001).
23. F. Sciortino, Nature (London) NATUAS 0028-0836 10.1038/nmat752 1, 145 (2002).
24. S.-H. Chen, W. R. Chen, and F. Mallamace, Science SCIEAS 0036-8075 10.1126/science.1082364 300, 619 (2003).
25. T. Eckert and E. Bartsch, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.89.125701 89, 125701 (2002).
26. K. N. Pham, Science SCIEAS 0036-8075 10.1126/science.1068238 296, 104 (2002).
27. A. Caiazzo, A. Coniglio, and M. Nicodemi, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.93.215701 93, 215701 (2004).
28. A. Crisanti and Th. M. Nieuwenhuizen, J. Phys. I JPGCE8 1155-4304 6, 56 (1996).
29. S. F. Edwards and P. W. Anderson, J. Phys. F: Met. Phys. JPFMAT 0305-4608 10.1088/0305-4608/5/5/017 5, 965 (1975).
30. G. Parisi, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/13/4/009 13, L115 (1980);
31. G. Parisi, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/13/3/042 13, 1101 (1980);
32. G. Parisi, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/13/5/047 13, 1887 (1980).
33. A. Annibale, G. Gualdi, and A. Cavagna, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/37/47/001 37, 11311 (2004).
34. S. K. Ma, Modern Theory of Critical Phenomena, Vol. 46 of Frontiers in Physics Lecture Note Series (Benjamin/Cummings, New York, 1976).
35. M. Mezárd, G. Parisi, and M. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1987).
36. K. H. Fischer and J. A. Hertz, Spin Glasses (Cambridge University Press, Cambridge, England, 1991).
37. D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.35.1792 35, 1792 (1975).
38. The replica trick uses the identity ln x̄ = limn→0 ln xn ̄ n and hence Φ(n) should be known for a continuous set of values of n and not just for integers. Consequence of this is that the limit limn→0 Φ(n) is not the analytic continuation of Φ(n) down to n=0.
39. F. Guerra and F. L. Toninelli, Commun. Math. Phys. CMPHAY 0010-3616 10.1007/s00220-002-0699-y 230, 71 (2002).
40. M. Talagrand, C. R. Acad. Sci., Ser. I: Math. CASMEI 0764-4442 337, 111 (2003);
41. M. Talagrand, Ann. (to be published).
42. The calculation of the eigenvalues of the quadratic form 17 for 1RSB solution of the 2+p model does not differ much from that of the spherical p -spin model. As a consequence to help the reader interested into the details of the derivation we shall use for the eigenvalues the same notation as Ref..
43. J. R. L. de Almeida and D. J. Thouless, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/11/5/028 11, 983 (1978).
44. χ (q1) and χ (q0) are two of the eigenvalues of the overlap matrix qαβ in the 1RSB ansatz denoted, respectively, η0 and η1 in Ref..
45. For finite n>1 the saddle point calculation requires that G[q] be maximal at the stationary point. However, the dimension n(n-1)2 of the q space becomes negative for n<1 so that G[q] should be minimized in the limit n→0. Consistently the free energy functional is maximized for n→0.
46. The lowest value of the order parameter matrix qαβ is the minimum possible overlap between replicas. It is always zero in the absence of an external field.
47. A. Crisanti (unpublished).
48. C. De Dominicis and D. M. Carlucci, C. R. Acad. Sci. Paris 323, 263 (1996);
49. C. De Dominicis, D. M. Carlucci, and T. Temesvári, J. Phys. I JPGCE8 1155-4304 10.1051/jp1:1997128 7, 105 (1997).
50. G. Parisi, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.43.1754 43, 1754 (1979).
51. The most general form of the order parameter function in the 1RSB phase is q(x)= q0 θ(m-x)+ q1 θ(x-m). However, in absence of external fields, q0 =0.
52. To maintain the notation q1 for the overlap associated to the 1RSB part of the order parameter function we use the notation q0 for the largest value of the continuous part q(x) of the order parameter function. This notation should not be confusing since q(0)=0.
53. A. Crisanti, L. Leuzzi, and G. Parisi, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/35/3/303 35, 481 (2002).
54. C. De Dominicis and I. Kondor, Phys. Rev. B PRBMDO 0163-1829 10.1103/PhysRevB.27.606 27, 606 (1983);
55. I. Kondor and C. De Dominicis, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/16/2/005 16, L73 (1983).
56. A. Crisanti and T. Rizzo, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.65.046137 65, 046137 (2002).
57. In the expansion given in Ref. the exponent of the third term is erroneously given as x7.
58. T. R. Kirkpatrick and P. G. Wolynes, Phys. Rev. B PRBMDO 0163-1829 10.1103/PhysRevB.36.8552 36, 8552 (1987).
59. A. Crisanti, H. Horner, and H.-J. Sommers, Z. Phys. B: Condens. Matter ZPCMDN 0722-3277 10.1007/BF01312184 92, 257 (1993).
60. A. Crisanti and H-.J. Sommers, J. Phys. I JPGCE8 1155-4304 10.1051/jp1:1995164 5, 805 (1995).
61. A. Crisanti, L. Leuzzi, and T. Rizzo, Eur. Phys. J. B EPJBFY 1434-6028 10.1140/epjb/e2003-00325-x 36, 129 (2003).
62. T. Plefka, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/15/6/035 15, 1971 (1982);
63. T. Plefka, Europhys. Lett. EULEEJ 0295-5075 10.1209/epl/i2002-00457-7 58, 892 (2002).
64. T. R. Kirkpatrick and P. G. Wolynes, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.35.3072 35, 3072 (1987).
65. A. Crisanti and L. Leuzzi (unpublished).
66. R. Monasson, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.75. 2847 75, 2847 (1995).
67. L. F. Cugliandolo and J. Kurchan, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.71.173 71, 173 (1993).
68. D. J. Thouless, P. W. Anderson, and R. G. Palmer, Philos. Mag. PHMAA4 0031-8086 35, 593 (1977).
69. H. Horner, Z. Phys. B: Condens. Matter ZPCMDN 0722-3277 10.1007/BF01311653 66, 175 (1987).
70. H. Sompolinsky, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.47. 935 47, 935 (1981).
71. C. de Dominicis, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/15/1/010 15, L47 (1982).
72. S. Franz, M. Mezárd, G. Parisi, and L. Peliti, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.81.1758 81, 1758 (1998);
73. S. Franz, M. Mezárd, G. Parisi, and L. Peliti, J. Stat. Phys. JSTPBS 0022-4715 10.1023/A:1004602906332 97, 459 (1999).
74. Such phase can be built ad hoc also in the case of discrete variables but it turns out to be always unstable (Ref.).
75. H-.J. Sommers and W. Dupont, J. Phys. C JPSOAW 0022-3719 10.1088/0022-3719/17/32/012 17, 5785 (1984).
76. A. Crisanti, L. Leuzzi, G. Parisi, and T. Rizzo, Phys. Rev. B PRBMDO 0163-1829 10.1103/PhysRevB.70.064423 70, 064423 (2004).