Abstract
The statistics of low energy states of the 2D Ising spin glass with +1 and −1 bonds are studied for L × L square lattices with L≤ 48, and p = 0.5, where p is the fraction of negative bonds, using periodic and/or antiperiodic boundary conditions. The behavior of the density of states near the ground state energy is analyzed as a function of L, in order to obtain the low temperature behavior of the model. For large finite L there is a range of T in which the heat capacity is proportional to T 5.33 ± 0.12. The range of T in which this behavior occurs scales slowly to T = 0 as L increases. Similar results are found for p = 0.25. Our results indicate that this model probably obeys the ordinary hyperscaling relation d ν = 2 − α, even though T c = 0. The existence of the subextensive behavior is attributed to long-range correlations between zero-energy domain walls, and evidence of such correlations is presented.
Similar content being viewed by others
References
D. J. Thouless, P. W. Anderson and R. G. Palmer, Phil. Mag. 35:593 (1977).
S. F. Edwards and P. W. Anderson, J. Phys. F 5:965 (1975).
P. W. Anderson, In: R. Balian, R. Maynard and G. Toulouse (eds.), Ill-Condensed Matter (North-Holland, Amsterdam, 1979), pp. 162–261.
H. Sompolinsky and A. Zippelius, Phys. Rev. Lett. 47:359 (1981).
H. Sompolinsky and A. Zippelius, Phys. Rev. B 25:6860 (1982).
R. Fisch, J. Stat. Phys. 125:793 (2006).
A. Galluccio, M. Loebl and J. Vondrák, Phys. Rev. Lett. 84:5924 (2000).
A. Galluccio, M. Loebl and J. Vondrák, Math. Program. Ser. A 90:273 (2001).
I. A. Campbell, A. K. Hartmann and H. G. Katzgraber, Phys. Rev. B 70:054429 (2004).
J. Lukic, A. Galluccio, E. Marinari, O. C. Martin and G. Rinaldi, Phys. Rev. Lett. 92:117202 (2004).
T. Jörg, J. Lukic, E. Marinari and O. C. Martin, Phys. Rev. Lett. 96:237205 (2006).
A. J. Bray and M. A. Moore, In: J. L. van Hemmen and I. Morgenstern (eds.), Heidelberg Colloquium on Glassy Dynamics (Springer, Berlin, 1986), pp. 121–153.
C. Amoruso, E. Marinari, O. C. Martin and A. Pagnani, Phys. Rev. Lett. 91:087201 (2003).
R. G. Palmer and J. Adler, Int. J. Mod. Phys. C 10:667 (1999).
J.-P. Bouchaud, F. Krzakala and O. C. Martin, Phys. Rev. B 68:224404 (2003).
L. Saul and M. Kardar, Phys. Rev. E 48:R3221 (1993).
L. Saul and M. Kardar, Nucl. Phys. B 432:641 (1994).
J. A. Blackman and J. Poulter, Phys. Rev. B 44:4374 (1991).
J.-S. Wang and R. H. Swendsen, Phys. Rev. B 38:4840 (1988).
B. Derrida and H. Hilhorst, J. Phys. C 14:L539 (1981).
F. Merz and J. T. Chalker, Phys. Rev. B 66:054413 (2002).
C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), pp. 453–455.
G. A. Baker, Jr. and J. C. Bonner, Phys. Rev. B 12:3741 (1975).
C. Amoruso, A. K. Hartmann, M. B. Hastings and M. A. Moore, Phys. Rev. Lett. 97:267202 (2006).
C. Wang, J. Harrington and J. Preskill, Ann. Phys. (N.Y.) 303:31 (2003).
R. Fisch, cond-mat/0703137.
Author information
Authors and Affiliations
Corresponding author
Additional information
PACS numbers: 75.10.Nr, 75.40.Mg, 75.60.Ch, 05.50.+q
Rights and permissions
About this article
Cite this article
Fisch, R. Subextensive Singularity in the 2D ± J Ising Spin Glass. J Stat Phys 128, 1113–1124 (2007). https://doi.org/10.1007/s10955-007-9336-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-007-9336-7