Crossover phenomena in spin models with medium-range interactions and self-avoiding walks with medium-range jumps

Abstract

We study crossover phenomena in a model of self-avoiding walks with medium-range jumps, which corresponds to the limit N-->0 of an N-vector spin system with medium-range interactions. In particular, we consider the critical crossover limit that interpolates between the Gaussian and the Wilson-Fisher fixed point. The corresponding crossover functions are computed by using field-theoretical methods and an appropriate mean-field expansion. The critical crossover limit is accurately studied by numerical Monte Carlo simulations, which are much more efficient for walk models than for spin systems. Monte Carlo data are compared with the field-theoretical predictions for the critical crossover functions, finding good agreement. We also verify the predictions for the scaling behavior of the leading nonuniversal corrections. We determine phenomenological parametrizations that are exact in the critical crossover limit, have the correct scaling behavior for the leading correction, and describe the nonuniversal crossover behavior of our data for any finite range.

Critical properties of the three-dimensional equivalent-neighbor model and crossover scaling in finite systems

“…Interestingly, this result has been recovered in Ref. [11], where in addition it was found that the constant c 1 has a universal value −2/π ≈ −0.6366. Indeed, this agrees with the value −0.624 (7) obtained from an analysis of the available data for 25 < ∼ R 2 < ∼ 70.…”

Two-parameter model predictions and θ-point crossover for linear-polymer solutions

“…13 A second use of the TPM expressions is in the description of the crossover to the θ point. As we have explained in the introduction, the crossover functions defined in Eq.…”

“…Indeed, as discussed in Ref. 13 , the TPM describes the approach to the scaling limit when V m /l 3 ≪ 1, where l is the persistence length and V m is the volume occupied by a polymer blob of length l. In this paper we wish to compute the crossover functions for several quantities whose behavior in the good-solvent regime has been considered in Refs. 14, 15 . We compute numerically the TPM predictions for the second, third, and fourth virial coefficient, for the swelling factors, and the density corrections to the radius of gyration, the end-to-end distance, and the hydrodynamic radius.…”

Large- chiral transition in the Yukawa model

“…One can use field theory and compute the crossover functions by resumming the perturbative series [10,11,12,13]. Alternatively, one can use the fact that the field-theoretical crossover is equivalent to the critical crossover that occurs in models with medium-range interactions [14,15,13,16,17]. This allows one to use the wealth of results available for these spin systems [14,18,19,15,13,17].…”