Publication: Correspondence between long-range and short-range spin glasses
Full text at PDC
Advisors (or tutors)
American Physical Society
We compare the critical behavior of the short-range Ising spin glass with a spin glass with long-range interactions which fall off as a power σ of the distance. We show that there is a value of σ of the long-range model for which the critical behavior is very similar to that of the short range model in four dimensions. We also study a value of σ for which we find the critical behavior to be compatible with that of the three-dimensional model, although we have much less precision than in the four-dimensional case.
© 2012 American Physical Society. We thank G. Parisi and M. Moore for discussions. A.P.Y. acknowledges support from the NSF through Grant No. DMR- 0906366 and a generous allocation of computer time from the Hierarchical Systems Research Foundation. The shortrange simulations, and part of the long-range simulations, were carried out on ARAGRID and BIFI computers. R.A.B., L.A.F., and V.M.M. acknowledge partial financial support from MICINN, Spain, Contract No. FIS2009 12648-C03. R.A.B. was also supported by the FPI Program (Diputación de Aragón, Spain). V.M.M. acknowledges the hospitality of the Physics Department of UCSC (visit funded by the Del Amo Foundation), where part of this work was performed.
1) K. Binder, A. P. Young, Rev. Mod. Phys., 58, 801 (1986). 2) Finite Size Scaling and Numerical Simulation of Statistical Systems, edited by V. Privman (World Scientific, Singapore, 1990). 3) D. Amit, V. Matín-Mayor, Field Theory, the Renormalization Group and Critical Phenomena (World Scientific, Singapore, 2005). 4) H. G. Katzgraber, A. P. Young, Phys. Rev. B, 67, 134410 (2003). 5) H. G. Katzgraber, A. P. Young, Phys. Rev. B, 68, 224408 (2003). 6) H. G. Katzgraber, A. P. Young, Phys. Rev. B, 72, 184416 (2005). 7) H. G. Katzgraber, D. Larson, A. P. Young, Phys. Rev. Lett., 102, 177205 (2009). 8) D. Larson, H. G. Katzgraber, M. A. Moore, A. P. Young, Phys. Rev. B, 81, 064415 (2010). 9) A. Sharma, A. P. Young, Phys. Rev. B, 83, 214405 (2011). 10) L. Leuzzi, G. Parisi, F. Ricci-Tersenghi, J. J. Ruiz Lorenzo, Phys. Rev. Lett., 101, 107203 (2008). 11) L. Leuzzi, G. Parisi, F. Ricci-Tersenghi, J. J. Ruiz Lorenzo, Phys. Rev. Lett., 103, 267201 (2009). 12) L. Leuzzi, G. Parisi, F. Ricci-Tersenghi, J. J. Ruiz Lorenzo, Philos. Mag., 91, 1917 (2011). 13) A. B. Harris, T. C. Lubensky, J.-H. Chen, Phys. Rev. Lett., 36, 415 (1976). 14) G. Kotliar, P. W. Anderson, D. L. Stein, Phys. Rev. B, 27, 602 (1983). 15) M. A. Moore (private communication). 16) G. Parisi (private communication). 17) M. E. Fisher, S.-k. Ma, B. G. Nickel, Phys. Rev. Lett., 29, 917 (1972). 18) M. Hasenbusch, A. Pelissetto, E. Vicari, Phys. Rev. B, 78, 214205 (2008). 19) M. E. J. Newman, G. T. Barkema, Monte Carlo Methods in Statistical Physics (Oxford University Press, New York, 1999). 20) Actually, some long-range models with power-law interactions can be simulated efficiently. Unfortunately, the cluster algorithms of Refs. 21 and 22 do not work for spin glasses. On the other hand, the stochastic cutoff method (Ref. 23) does not rely on a cluster update, but its efficiency for simulating spin glasses with long-range interactions is yet to be assessed. 21) E. Luijten and H. W. J. Blöte, Phys. Rev. Lett., 76, 1557 (1996). 22) K. Fukui, S. Todo, J. Comput. Phys., 228, 2629 (2009). 23) M. Sasaki, F. Matsubara, J. Phys. Soc. Jpn., 77, 024004 (2008). 24) F. Beyer, M. Weigel, M. A. Moore, Phys. Rev. B, 86, 014431 (2012). 25) M. Wittmann, A. P. Young, Phys. Rev. E, 85, 041104 (2012). 26) L. Viana, A. J. Bray, J. Phys. C, 18, 3037 (1985). 27) L. A. FernÁndez, V. MartÍn-Mayor, G. Parisi, B. Seoane, Phys. Rev. B, 81, 134403 (2010). 28) We run the Monte Carlo of the bonds for 106 sweeps. We are confident about graph equilibration because we compared the outcomes of widely differing starting points for the simulation: either a graph with the topology of a crystal with periodic boundary conditions, or the random graph described in the main text. For either type of starting point, we compared several graph properties, in particular the bond-length distribution and the “Hamiltonian” defined in Eq. (15). In all cases studied, we found that memory of the starting configuration was lost after 105 sweeps, but we simulated for a total 106 sweeps to be on the safe side. 29) B. Cooper, B. Freedman, D. Preston, Nucl. Phys. B, 210, 210 (1982). 30) M. Palassini, S. Caracciolo, Phys. Rev. Lett., 82, 5128 (1999). 31) H. G. Ballesteros, A. Cruz, L. A. Fernández, V. Martín-Mayor, J. Pech, J. J. Ruiz-Lorenzo, A. Tarancón, P. Téllez, C. L. Ullod, C. Ungil, Phys. Rev. B, 62, 14237 (2000). 32) K. Binder, Z. Phys. B, 43, 119 (1981). 33) H. G. Ballesteros, L. A. Fernández, V. Martín-Mayor, J. Pech, A. Muñoz Sudupe, Phys. Lett. B, 387, 125 (1996). 34) M. P. Nightingale, Physica A, 83, 561 (1976). 35) H. G. Ballesteros, L. A. Fernández, V. Martín-Mayor, A. Muñoz Sudupe, G. Parisi, J. J. Ruiz-Lorenzo, Phys. Rev. B, 58, 2740 (1998). 36) M. Weigel, W. Janke, Phys. Rev. Lett., 102, 100601 (2009). 37) K. Hukushima, K. Nemoto, J. Phys. Soc. Jpn., 65, 1604 (1996). 38) E. Marinari, F. Zuliani, J. Phys. A, 32, 7447 (1999). 39) T. Jörg, H. G. Katzgraber, Phys. Rev. B, 77, 214426 (2008).