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Journal of Physics A Mathematical and General

A corner transfer matrix renormalization group investigation of the vertex-interacting self-avoiding walk model

Damien Paul Foster, C. Pinettes

Published Oct 1, 2003• 24 citations
Publisher

Abstract

A recently introduced extension of the corner transfer matrix renormalization group method useful for the study of self-avoiding walk-type models is presented in detail and applied to a class of interacting self-avoiding walks due to Blöte and Nienhuis. This model displays two different types of collapse transition depending on model parameters. One is the standard θ-point transition. The other is found to give rise to a first-order collapse transition despite being known to be in other respects critical.

DOI
10.1088/0305-4470/36/41/003

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A recently introduced extension of the corner transfer matrix renormalization group method useful for the study of self-avoiding walk-type models is presented in detail and applied to a class of interacting self-avoiding walks due to Blöte and Nienhuis. This model displays two different types of collapse transition depending on model parameters. One is the standard θ-point transition. The other is found to give rise to a first-order collapse transition despite being known to be in other respects critical.
A recently introduced extension of the Corner Transfer Matrix Renormalisation Group (CTMRG) method useful for the study of self-avoiding walk type models is presented in detail and applied to a class of interacting self-avoiding walks due to Blöte and Nienhuis. This model displays two different types of collapse transition depending on model parameters. One is the standard θθθ-point transition. The other is found to give rise to a first-order collapse transition despite being known to be in other respects critical.
The phase diagram for the bond-interacting self-avoiding walk is calculated using transfer matrices on finite strips. The model is shown to have a richer phase diagram than the related Θ\ThetaΘ-point model. In addition to the standard collapse transition, we conjecture the existence of a line of transitions in the Berezinskii--Kosterlitz--Thouless (BKT) universality class, terminating in a critical end point. Our results are in contradiction with a previous transfer matrix calculation by Machado K D, de Oliveira M J and Stilck J F 2001 {\em Phys Rev E} {\bf 64} 051810.
Research in recent years has shown that combining finite-size scaling theory with the transfer matrix technique yields a powerful tool for the investigation of critical behavior. In particular, the method has been used to study two-dimensional statistical mechanical and one-dimensional quantum mechanical systems. We review finite-size scaling theory from the general point of view of renormalization group theory for both continuous and first-order transitions (both for systems with discrete and continuous symmetries). We review applications where a comparison with exact results can be made. These include the Ising, Baxter, and q-state Potts models and the Ising model with a defect line. Various other applications such as quantum systems, the self-avoiding random walk, percolation, and Kosterlitz-Thouless transitions are briefly reviewed. The Kosterlitz-Thouless transitions and the critical fan in the antiferromagnetic 3-state Potts model are discussed at somewhat greater length.
Abstract The critical behavior of a dimer model with an interaction favoring parallel dimers in each plaquette of the square lattice is studied numerically using the corner transfer matrix renormalization group algorithm. The critical exponents are known to depend on the chemical potential of vacancies, or monomers. At large average density of the latter, the phase transition becomes the first-order. We compute the scaling dimensions of both the order parameter and temperature in the second-order regime and compare them with the conjecture that the critical behavior is the same as the Ashkin–Teller model on its self-dual critical line.