Replica symmetry breaking in the ‘small world’ spin glass

Abstract

We apply the cavity method to a spin glass model on a 'small world' lattice, a random bond graph super-imposed upon a 1-dimensional ferromagnetic ring. We show the correspondence with a replicated transfer matrix approach, up to the level of one step replica symmetry breaking (1RSB). Using the scheme developed by Mézard & Parisi for the Bethe lattice, we evaluate observables for a model with fixed connectivity and ±J long range bonds. Our results agree with numerical simulations significantly better than the replica symmetric (RS) theory.

Antiferromagnetic Ising model in small-world networks

“…In particular, instead of rewiring each bond with probability p, one can add shortcuts between pairs of sites taken at random, without removing bonds from the regular lattice. This procedure turns out to be more convenient for analytical calculations, but does not keep constant the mean connectivity k , which in this case increases with p. Spin glasses on such small-world networks, generated from a one-dimensional ring, have been studied earlier by replica symmetry breaking [46] and transfer matrix analysis [47].…”

Random-diluted triangular plaquette model: Study of phase transitions in a kinetically constrained model

“…According to the presence of both long and short-range plaquettes in the Random-Diluted TPM, the cavity equations (Appendix D) for its representation on the random graph are written by means of two different cavity fields, as is usually done for small-word networks 31 . The field u α→i determines the probability distribution p(σ i ) ∼ e uα→iσi when all the plaquettes attached to σ i but α are removed, and α is a long-range plaquette.…”

Communication and correlation among communities

“…In fact, for n ≥ 2, the free energy term L (Σ) is different from the original density functional of the model L (Σ) that lives in an enlarged space of the order parameters; the form of L (Σ) is equal to the form of L (Σ) only when calculated in a solution of the self-consistent system of Eqs. (36) or (40). In this sense, for n ≥ 2, the expression "Landau free energy density" for L (Σ) would be somehow inappropriate; the true Landau free energy density is represented by L (Σ) and is given in Sec.…”

“…We point out however that our results are completely novel. Notice in fact that, without any intention to be exhaustive in citing the large literature on the subject, the state of the art of analytical methods for disordered Ising models defined over Poissonian small-world graphs results nowadays as follows: i) in the case of no short-range couplings, J 0 = 0, and for one community, n = 1, modulo a large use of some population dynamics algorithm for low temperatures, the replica method and the cavity method [25,29,30,31] have established the base to solve exactly the model in any region of the phase diagram, even rigorously in the SK case [32,33] and in unfrustrated cases [34]; ii) for J 0 = 0 and n = 1 these methods have been successfully applied to the one-dimensional case [35,36] but a generalization to higher dimensions (except infinite dimensions [37]) seems impossible due to the presence of loops of any length [53]; on the other hand, even if it is exact only in the P region, the method we have presented in the Ref. [20], modulo solving analytically or numerically a non random Ising model, can be exactly applied in any dimension, and more in general to any underlying pure graph (L 0 , Γ 0 ); iii) for J 0 = 0 and n ≥ 2, the problem was solved only in the limit of infinite connectivity: exactly in the n = 2 CW case in its general form, which includes arbitrary sizes of the two communities, but with no coupling disorder [23]; and, within the replica-symmetric solution, in the generic n SK case, but only in the presence of a same mutual interaction among the n communities of same size [38,39].…”