Solution of a model of self-avoiding walks with multiple monomers per site on the Bethe lattice

Abstract

We solve a model of self-avoiding walks with up to two monomers per site on the Bethe lattice. This model, inspired in the Domb-Joyce model, was recently proposed to describe the collapse transition observed in interacting polymers [J. Krawczyk, Phys. Rev. Lett. 96, 240603 (2006)]. When immediate self-reversals are allowed (reversion-allowed model), the solution displays a phase diagram with a polymerized phase and a nonpolymerized phase, separated by a phase transition which is of first order for a nonvanishing statistical weight of doubly occupied sites. If the configurations are restricted forbidding immediate self-reversals (reversion-forbidden model), a richer phase diagram with two distinct polymerized phases is found, displaying a tricritical point and a critical end point.

Collapse transition in polymer models with multiple monomers per site and multiple bonds per edge

“…In contrast with these numerical results, the same thermodynamic behavior has been found for RA and RF models in exact solutions of them on hierarchical (Bethe and Husimi) lattices [42,43]. In such solutions, coil and globule phases are always separated by lines of continuous transitions, being a tricritical line in the region of β 2 < 0 (with β 1 > 0) and a line of critical-end-points (CEP) for β 1 < 0 (with β 2 > 0), both meeting at a multicritical point [43].…”

“…However, the rising of a stable ordered (crystalline) phase in the MMS models, without the addition of any local chain stiffness on them, seems quite unexpected. In fact, no evidence of the existence of a third phase in the canonical phase diagrams was found here or elsewhere [36,42,43].…”

“…Moreover, it is not possible to analyze the effect of dimensionality on the behavior of the models, because these lattices have infinite dimension. Anyhow, the results in [42,43] strongly suggest that the collapse transitions in MMS models: i) are not dependent on their details; and ii) are always continuous.…”

Grand-canonical and canonical solution of self-avoiding walks with up to three monomers per site on the Bethe lattice

“…A simple way to find the coexistence region [8] is just to iterate the recursion relations starting with 'natural' initial conditions, that is, using initial values for the ratios which correspond to a reasonable choice for the configurations at the surface of the Cayley tree. Actually, we used this procedure in our recent works on the K = 2 case of the present model [11,12]. Although this procedure has a considerable physical appeal, is simple and leads to reasonable results, we were not able to justify it starting from basic principles.…”

“…Recently, the RF and RA models in the grand-canonical ensemble were solved on the Bethe lattice [11] for the case K = 2. The parameter space for this model is defined by the statistical weights ω i , i = 1, 2, of sites occupied by i monomers (the weight of empty sites is equal to one).…”