Cross-over to Mean Field Behavior in Small-World Nanomaterials

Recently the suggestion was made to include small-world network effects into nanoscale materials [1,2]. Small-world connections lead to (slightly modified) mean-field behavior of model systems [3]. Consequently there is the possibility for a new class of nanomaterials governed by the mean-field fixed point. Here we describe finite-scale scaling investigations of ferromagnetic Ising models adding small-world connections to either $d=1$ or $d=2$ lattices with $V$ sites. In particular, we demonstrate how finite- size scaling works by scaling with a 'length' given by the logarithm of $V$ rather than a linear dimension of the system. Results are compared to studies of a related model with dilute long-range interactions [4]. \newline [1] M.A. Novotny and S.M. Wheeler, Brazilian J. Phys, vol. 34, p. 395 (2004). \newline [2] M.A. Novotny, et al, J. Appl. Phys, in press (preprint cond-mat/0410589). \newline [3] M.B. Hastings, Phys. Rev. Lett., vol. 91, 098701 (2003). \newline [4] R.T. Scalettar, Physica A, vol. 170, p. 282 (1991).