Simplicity of state and overlap structure in finite-volume realistic spin glasses

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 57, Issue 2, 1998, Pages 1356-1366

  Newman, C M ^a   Stein, D L ^b  

^a NEW YORK UNIVERSITY   (United States)

^b UNIVERSITY OF ARIZONA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 4244055093     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.57.1356     Document Type: Article

References (59)

1. M. Mézard, G. Parisi, and M. A. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1987).
2. K. Binder and A. P. Young, Rev. Mod. Phys. 58, 801 (1986)
3. G. Parisi, Phys. Rev. Lett. 43, 1754 (1979)
4. G. Parisi, Phys. Rev. Lett. 50, 1946 (1983)
5. A. Houghton, S. Jain, and A. P. Young, J. Phys. C 16, L375 (1983)
6. M. Mézard, G. Parisi, N. Sourlas, G. Toulouse, and M. Virasoro, Phys. Rev. Lett. 52, 1156 (1984)
7. M. Mézard, G. Parisi, N. Sourlas, G. Toulouse, and M. Virasoro, J. Phys. (France) 45, 843 (1984)
8. R. Rammal, G. Toulouse, and M. A. Virasoro, Rev. Mod. Phys. 58, 765 (1986)
9. M. Mézard and G. Parisi, J. Phys. I 1, 809 (1991).
10. J.-P. Bouchaud, M. Mézard, and J. S. Yedidia, Phys. Rev. Lett. 67, 3840 (1991)
11. J. S. Yedidia, in 1992 Lectures in Complex Systems, edited by D. L. Stein (Addison-Wesley, Reading, MA, 1993), p. 299.
12. G. Parisi, Physica A 194, 28 (1993)
13. D. Badoni, J. C. Ciria, G. Parisi, F. Ritort, J. Pech, and J. J. Ruiz-Lorenzo, Europhys. Lett. 21, 495 (1993)
14. S. Franz, G. Parisi, and M. A. Virasoro, J. Phys. I 4, 1657 (1994).
15. F. Ritort, Phys. Rev. B 50, 6844 (1994)
16. E. Marinari, G. Parisi, and F. Ritort, J. Phys. A 27, 2687 (1994).
17. P. Le Doussal and T. Giamarchi, Phys. Rev. Lett. 74, 606 (1995)
18. E. Marinari, G. Parisi, J. J. Ruiz-Lorenzo, and F. Ritort, Phys. Rev. Lett. 76, 843 (1996)
19. E. Marinari, G. Parisi, and J. J. Ruiz-Lorenzo, in Spin Glasses and Random Fields, edited by A. P. Young (World Scientific, Singapore, in press).
20. C. M. Newman and D. L. Stein, Phys. Rev. Lett. 76, 515 (1996)
21. C. M. Newman and D. L. Stein, Phys. Rev. Lett. 76, 4821 (1996)
22. C. M. Newman and D. L. Stein, Phys. Rev. E 55, 5194 (1997).
23. C. M. Newman and D. L. Stein, in Mathematics of Spin Glasses and Neural Networks, edited by A. Bovier and P. Picco (Birkhäuser, Boston, in press).
24. C. M. Newman, Topics in Disordered Systems (Birkhäuser, Basel, 1997).
25. C. M. Newman and D. L. Stein, in Proceedings of the 1997 International Congress of Mathematical Physics, edited by D. De Wit, M. D. Gould, P. A. Pearce, and A. J. Bracken (World Scientific, Singapore, in press).
26. S. Edwards and P. W. Anderson, J. Phys. F 5, 965 (1975)
27. D. S. Fisher and D. A. Huse, Phys. Rev. Lett. 56, 1601 (1986)
28. D. A. Huse and D. S. Fisher, J. Phys. A 20, L997 (1987).
29. D. S. Fisher and D. A. Huse, J. Phys. A 20, L1005 (1987).
30. D. S. Fisher and D. A. Huse, Phys. Rev. B 38, 386 (1988)
31. W. L. McMillan, J. Phys. C 17, 3179 (1984)
32. A. J. Bray and M. A. Moore, Phys. Rev. Lett. 58, 57 (1987)
33. D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35, 1792 (1975)
34. J. D. Reger, R. N. Bhatt, and A. P. Young, Phys. Rev. Lett. 64, 1859 (1990)
35. S. Caracciolo, G. Parisi, S. Patarnello, and N. Sourlas, J. Phys. (France) 51, 1877 (1990)
36. C. M. Newman and D. L. Stein, Phys. Rev. B 46, 973 (1992)
37. We reemphasize a point made earlier in Sec. II, which in the current context implies that only boundary conditions chosen independently of the couplings should be used to construct the metastate. We note that there might exist pure states for a given J that would not appear in a metastate so constructed; i.e., they would not appear in the pure state decomposition of any of the Γ’s appearing in the metastate. A situation where such “invisible” pure states occur in the context of Ising ferromagnets is discussed in Sec. V. In the spin glass, such states would require special coupling-dependent boundary conditions in order to appear in the metastate. As indicated in our remarks at the beginning of Sec. II, these states, should they exist, could be of mathematical interest but would almost certainly play no physical role. [Arguments along these lines can also be found in Sec. 3 of A. C. D. van Enter and J. Frölich, Commun. Math. Phys. 98, 425 (1985)]. It should be understood throughout that when “pure states” are referred to (e.g., in Sec. IV), we generally mean only those that appear in a metastate constructed using coupling-independent boundary conditions
38. M. Aizenman and J. Wehr, Commun. Math. Phys. 130, 489 (1990)
39. It should be noted that currently existing proofs require not only a subsequence L1, L2,…, LN,… of cube sizes, but possibly also a subsequence of N’s when taking the histogram limit. However, the crucial point is that this subsequence of cube sizes, even if necessary in some instances, remains independent of J.
40. As shown in Ref. 20, the rigorous exclusion of the non-self-averaging property for PJ(q) also implies a lack of ultrametricity of distances among all of the pure states. That is, although not also rigorously excluded, the ultrametricity property was shown to be highly implausible. It is worth noting that an ultrametric structure in state space can appear in the ground state structure of models with deterministic (and hence trivially self-averaged) interactions, although in order to obtain this structure one has to make a very artificial choice of interaction. For details, see A. C. D. van Enter, A. Hof, and J. Miekisz, J. Phys. A 25, L1133 (1992).
41. C. M. Newman and D. L. Stein, Phys. Rev. Lett. 72, 2286 (1994)
42. C. M. Newman and D. L. Stein, J. Stat. Phys. 82, 1113 (1996)
43. M. Cieplak, A. Maritan, and J. R. Banavar, Phys. Rev. Lett. 72, 2320 (1994)
44. Note that free boundary conditions are not flip related to periodic and antiperiodic boundary conditions.
45. The uniform distribution in the case of the strongly disordered model corresponds to the sign of each tree in the invasion forest being chosen by the flip of a fair coin.
46. This is clearly a subset of all pairs of the ground states that are distinct within the cube ΛL; the reason for this restriction will be discussed in Sec. VI.
47. A. C. D. van Enter, J. Stat. Phys. 60, 275 (1990)
48. M. Lederman, R. Orbach, J. M. Hamann, M. Ocio, and E. Vincent, Phys. Rev. B 44, 7403 (1991)
49. 0Y. G. Joh, R. Orbach, and J. M. Hamann, Phys. Rev. Lett. 77, 4648 (1996)
50. In Appendix 1 of Ref. 19, for example, it is argued that the pure state, or thermodynamic, structure is merely a mathematical infinite-volume construct that has little or no physical relevance to real (finite-volume) systems such as spin glasses. We believe those arguments to be misleading, and indeed, misdirected in that the metastate approach precisely does connect the behavior of observable quantities in finite volumes with the thermodynamic structure of the system. (Moreover, the suggestion in that same reference that the Boltzmann-Gibbs probability distribution does not even exist in the infinite-volume limit for many disordered systems is simply incorrect.) It is, for example, a misconception that the behavior of correlation functions is more physical or less “metaphorical” (cf. Appendix 1 of Ref. 19) than thermodynamic states. Indeed, the two are simply different labels for the same object, in the same way that one can talk either of the probability distribution of a random variable or the set of its moments.
51. This should not be confused with the fact that, if many pure states are present, then changes in boundary conditions can change the state everywhere in the volume, including the region about the origin. In this situation, boundary conditions can select the thermodynamic state in the interior; but in order to see which state has been selected, one must still measure correlations in a region about the origin sufficiently far from the boundaries.
52. M. Aizenman, Commun. Math. Phys. 73, 83 (1980)
53. Y. Higuchi, in Random Fields, Esztergom (Hungary) 1979, edited by J. Fritz, J. L. Lebowitz, and D. Szász (North-Holland, Amsterdam), p. 517.
54. Such a non-translation-invariant pure state will occur in higher dimensions than two, below the roughening temperature.
55. A. C. D. van Enter (private communication).
56. Although all direct numerical computations of PJ(q) [and P(q)] of which we are aware compute overlaps in the full volume, at least one computation has been reported 1819 that does examine a type of overlap measure, called the Binder cumulant, constructed on restricted subvolumes. Although strictly speaking the measurement reported has a dynamical component, it may contain potentially interesting and currently unexplained information on the equilibrium spin glass. However, the limited nature of the measurements done to date seem to us insufficient grounds for ruling out the droplet-scaling picture, as asserted in 1819.
57. The possibility that finite size effects might be persistent in systems with quenched disorder was also noted in Ref. 19.
58. We should point out the special properties, under these arguments, of free boundary conditions. Free BC’s are not flip related to any others and our arguments in Secs. IV and V do not apply to them. We further note that in the SK model itself, free BC’s are in some sense the only natural boundary condition available. So could it be the case that the nonstandard SK picture might appear under free BC’s and no other? We do not find this to be a reasonable possibility because, unlike in the case of the infinite-ranged model, there is nothing particularly special about free BC’s in finite-dimensional short-ranged models. Although for technical reasons our arguments apply to BC’s such as periodic, antiperiodic, fixed, and so on, the crucial aspect of our arguments is more closely related to the property that these BC’s are chosen independently of the couplings. In this respect free BC’s for arbitrary volumes are no different from the others. In the highly disordered model, for example, we expect (but have not proved) that the periodic or antiperiodic BC metastate is identical to the free BC metastate (cf. Sec. IV).
59. We discussed in the Appendix to Ref. 22 various subtleties associated with the precise method of construction of the overlap distribution. In this paper we have referred only to the case where the overlap is computed in finite volumes using the replica measure ρJn(L) discussed in that paper. If replica nonindependence 2122 were present, as would be the case if the chaotic pairs picture were to hold, then one could construct a different infinite-volume overlap distribution by breaking replica symmetry after the infinite-volume limit is taken (cf. construction 2 of Ref. 20). This would be the replica overlap for the average ρJ of the metastate, and it would be the same not only for almost all flip-related boundary conditions but also, at the same time, for almost every J. Given that, the only reasonable possibilities for this overlap function within the chaotic pairs scenario would be either a single δ function at the origin, or (less likely, we believe) a continuous distribution between ±qEA with no δ-function spikes.