Critical behavior of ising models with random long-range (small-world) interactions

Abstract

The critical scaling behavior of Ising models with long range interactions is studied. These long-range interactions, when imposed in addition to interactions on a regular lattice, lead to small world graphs. Large-scale Monte Carlo simulations, together with finite-size scaling, is used to obtain the critical behavior of a number of different models. These include the z-model introduced by Scalettar, standard small-world bonds superimposed on a square lattice, and physical small-world bonds superimposed on a square lattice. These scaling results provide further evidence to support the existence of physical (quasi-) small-world nanomaterials.

The Ising universality class of kinetic exchange models of opinion dynamics

“…We show that at least in the infinite range limit of interactions between the agents in these two kinds of kinetic exchange models (of opinion formation dynamics), the steady state statistics clearly belong to the mean-field Ising universality class (β = 1/2, γ = 1 and dν = 2, satisfying Rushbrooke scaling 2β + γ = dν, giving ν = 1/2 for upper critical dimension d = 4). Additionally, the value of the Binder cumulant at the critical point (U * ) shows a value close to 0.30 consistently for these models, which is close to the field theoretically estimated value (≈ 0.27) for the Ising model in the mean field [20] and comparable with similar numerical estimates [21]. The numerical results for lower dimensional systems (see e.g., [19]) also indicate the same.…”

“…The value of the critical Binder cumulant (U * ) for all the noise amplitudes we studied (ζ = 0.3, 0.6, 0.9) show a value close to 0.30 (see Fig. 5), which is near the value (≈ 0.27) predicted for the critical Binder cumulant of the Ising model (using field theoretic ǫ expansion [20]; see also the numerical estimate ≈ 0.30 [21]). Numerical analysis here of the model gives dν ≃ 2, β ≃ 1/2 and γ ≃ 1 practically for all non-zero values of the annealed noise parameter ζ.…”

How Small-World Interactions Can Lead to Improved Quantum Annealer Designs

“…The small-world bonds can be added by starting, for example, with either a one-dimensional or a two-dimensional regular lattice. The consensus that has emerged is the addition of the small-world bonds gives a finite-temperature transition, even when one starts from a regular one-dimensional lattice that does not have a finite critical temperature, and furthermore the exponents associated with the critical behavior [32,44,45] have mean-field behavior; namely, the same as that associated with ferromagnetic models above the upper critical dimension d ferro upper = 4. For the ferromagnet, finite-size scaling of both the ferromagnetic Binder cumulant and ferromagnetic susceptibility gives the mean-field exponents ν ferro d≥4 = 2 and γ ferro d≥4 = 1.…”

Quantum transport through hierarchical structures

“…One is to understand how hierarchical models, in particular the Hanoi networks [23], affect quantum transport. The second is that such hierarchical models provide an intermediate between regular lattices and ones that have a small-world property [24], and would be of interest to understand quantum transport of nanomaterials that have a smallworld property [6,[25][26][27][28]. The third is that often phase transitions such as the metal-insulator transition or a ferromagnetic transition have some universal quantities that depend only on the dimension.…”