Partition function zeros in two-dimensional lattice models of the polymer +θ-point

Abstract

We report studies of the partition function zeros in lattice models of interacting self-avoiding walh on the triangular and square lattices. The complex Boltzmann factor-plane zeros show a pattern similar to isotropic lattice spin models. For increasing length of the walks, zeros approach the real axis along a unique locus at the point which can be identified with the singularity, particularly by its location, which is consistent with the 0-point values found in other numerical studies. We report studies of the partition function zeros in the self-avoiding walk (SAW) model with nearest-neighbor bond interactions on the square (SQ) and triangular (TR) lattices. This lattice model' is believed to have a 0-point tran~ition.~,~ Let c(N,B) denote the number of N-step SAW having exactly B nearest-neigh- bor-site pairs. The partition function for fixed N is defined B where B runs from N to some N-dependent maximum value. The Boltzmann factor E > 0 is assigned per each nearest-neighbor bond contact in excess of the N bonds of the walk itself. It is expected that as N - m, thermo- dynamic quantities become singular at the @point value, et > 1. For 0 Et, the chains would be collapsed, globular. Near the borderline value, E,, scaling forms have been proposed2s3 and substantiated by several field-theoretical epsilon-expansion studies.*-' Numerical efforts to verify the theoretical picture have encountered technical difficulties. Since d = 3 is the upper critical dimension: factors complicate d = 3 scaling behaviors. Monte Carlo (MC) studies seem more or less consistent with the theory: see ref 8-10 and liter- ature cited therein. However, series analyses by Rapaport generally contradicted the theory: he found no exponent universality in the SAW and the collapsed regimes. More recently, attention has turned to the two- dimensional systems, which have no logarithmic com- plications. The interest in the physics of the 0-point in d = 2 is more than academic: d = 2 polymeric systems in the semicollapsed regime have been realized and studied experimenta1ly.l2-l5 It seems, however, that the experi- mental data available to date do not fit well within the theoretical picture. Numerical MC16-18 and transfer matrixlg studies of the SQ-lattice model of in- teracting SAW described above are generally consistent with tricritical scaling predictions. However, the exponent estimates'*J9 obtained by these techniques and also by series analysis methods20,21 are rather spread and incon- sistent with some of the experimental12 or analyticn values. Our study of the zeros in the complex-€ plane of the partition function, ZN(t), is intended mainly to establish that their pattern is of a typical form observed for several isotropic lattice spin models with a unique transition, in the complex temperature plane: see ref 23-29 and liter- ature cited therein. For a review of the Yang-Lee zeros in the complex-H plane, see ref 30 and 31. For anisotropic models, the pattern is distinctively different.32r33 For the interacting SAW model in three dimensions, Rapaport calculated location of zeros but could not draw definite