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arXiv:cond-mat/0210548 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Corner-transfer-matrix renormalization-group method for two-dimensional self-avoiding walks and other O(n) models

Damien Paul Foster, C. Pinettes

Published Apr 28, 2003• 14 citations
arXivHTMLPDF

Abstract

We present an extension of the corner-transfer-matrix renormalization-group (CTMRG) method to O(n) invariant models, with particular interest in the self-avoiding walk class of models [O(n=0)]. The method is illustrated using an interacting self-avoiding walk model. Based on the efficiency and versatility, when compared to other available numerical methods, we present the CTMRG as the method of choice for two-dimensional self-avoiding walk problems.

DOI
10.1103/physreve.67.045105

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We present an extension of the corner-transfer-matrix renormalization-group (CTMRG) method to O(n) invariant models, with particular interest in the self-avoiding walk class of models [O(n=0)]. The method is illustrated using an interacting self-avoiding walk model. Based on the efficiency and versatility, when compared to other available numerical methods, we present the CTMRG as the method of choice for two-dimensional self-avoiding walk problems.
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.
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.
This chapter introduces two kinds of RG methods for solving the leading eigenvalue and eigenvectors of a transfer matrix: TMRG (transfer matrix renormalization group) and CTMRG (corner transfer matrix renormalization group). These methods are developed to study the thermodynamic properties of two-dimensional classical statistical models. Furthermore, in the framework of MPS, the fixed-point equations of these methods are derived, and the steps for efficiently solving these equations are outlined.
In the context of tensor network states, we for the first time reformulate the corner transfer matrix renormalization group (CTMRG) method into a variational bilevel optimization algorithm. The solution of the optimization problem corresponds to the fixed-point environment pursued in the conventional CTMRG method, from which the partition function of a classical statistical model, represented by an infinite tensor network, can be efficiently evaluated. The validity of this variational idea is demonstrated by the high-precision calculation of the residual entropy of the dimer model, and is further verified by investigating several typical phase transitions in classical spin models, where the obtained critical points and critical exponents all agree with the best known results in literature. Its extension to three-dimensional tensor networks or quantum lattice models is straightforward, as also discussed briefly.