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Corner transfer matrices in statistical mechanics

R. J. Baxter

Published Jan 1, 2006• 7 citations
OpenAlex

Abstract

Corner transfer matrices are a useful tool in the statistical mechanics of simple two-dimensinal models.They can be very effective way of obtaining series expansions of unsolved models, and of calculating the order parameters of solved ones.Here we review these features and discuss the reason why the method fails to give the order parameter of the chiral Potts model..

Related papers

Corner transfer matrices in statistical mechanics

R. J. Baxter

Corner transfer matrices are a useful tool in the statistical mechanics of simple two-dimensinal models. They can be very effective way of obtaining series expansions of unsolved models, and of calculating the order parameters of solved ones. Here we review these features and discuss the reason why the method fails to give the order parameter of the chiral Potts model.

arXiv:cond-mat/0611167•Journal of Physics A Mathematical and Theoretical•Oct 2, 2007•11 citations

Corner transfer matrix derived variational methods in lattice statistical mechanics and quantum many-body systems

Chunhoa Chhenh

The concept of the corner transfer matrix (CTM) was first discovered by Baxter in 1968, when he derived a set of variational matrix equations, which for finite matrix sizes, could be numerically solved to obtain a sequence of approximations for the statistical mechanical properties of a system of monomers and dimers on a rectangular lattice [1]. It was not until 1978, however, in a seminal paper entitled "Variational Approximations for Square Lattice Models in Statistical Mechanics" [4], that Baxter outlined his CTM variational method, which brought to light the potential power of the former objects to obtain numerics and series expansions for unsolved models and to calculate the order parameter of solved ones. Subsequent numerical work led to the realisation that the method, though general, was not very efficient; and increasing efficiency required making model specific modifications, which restricted the transferability of the resulting algorithm to other models. The CTM variational method was thus not widely adopted, despite holding much promise. More recently, however, Nishino and Okunishi discovered that White's density matrix renormalisation group (DMRG) algorithm [55, 56] could be efficiently extended to study two-dimensional classical lattice models, if the density matrix was approximated by Baxters CTMs. Numerical tests of their algorithm, the corner transfer matrix renormalisation group (CTMRG) method, were met with much success. Notable among these is its implementation within the infinite projected entangled-pair states (iPEPS) algorithm of Orus and Vidal to simulate the ground state of infinite two-dimensional quantum systems. In this thesis we review CTM derived variational methods: Baxter's original CTM iterative method, and developments of the CTM concept within the DMRG algorithm by Nishino and Okunishi, which led to the CTMRG method to calculate critical phenomena of classical and quantum systems. This will begin with elucidating the theoretical formalisms of both methods in two dimensions and their generalisations to three dimensions; followed by review of important application, namely, within the iPEPS algorithm of Orus and Vidal to numerically study infinite planar quantum systems.

•Jan 1, 2011•0 citations

Series expansions from the corner transfer matrix renormalization group method: II. Asymmetry and high-density hard squares

Yao-ban Chan

The corner transfer matrix renormalization group method is a method, claimed to be subexponential, for numerically calculating physical quantities of statistical mechanical models. In a previous paper, we extended this method to generate series expansions. Here, we show how to extend both the original and series methods to deal with asymmetry in any model. We discuss the cases of rotational, translational and reflection asymmetry, and give some improvements to the method. This is demonstrated by an application of the method to generate series for the hard square model in the high-density regime, producing 51 terms of the partition function and 48 terms of the order parameter. These series are analysed, producing estimates of the critical point and exponents, and showing the likely presence of a confluent singularity with exponent 17/8 in the order parameter.

Journal of Physics A Mathematical and Theoretical•Mar 8, 2013•6 citations

Series expansions from the corner transfer matrix renormalization group method: the hard-squares model

Yao-ban Chan

The corner transfer matrix renormalization group method is an efficient method for evaluating physical quantities in statistical mechanical models. It originates from Baxter’s corner transfer matrix equations and method, and was developed by Nishino and Okunishi in 1996. In this paper, we review and adapt this method, previously used for numerical calculations, to derive series expansions. We use this to calculate 92 terms of the partition function of the hard-squares model. We use the resulting series to provide evidence supporting the claim that the method is subexponential in the number of generated terms, and briefly analyse the singularities of the function.

arXiv:1107.1677•Journal of Physics A Mathematical and Theoretical•Feb 9, 2012•17 citations

CHIRAL POTTS MODEL: CORNER TRANSFER MATRICES AND PARAMETRIZATIONS

R. J. Baxter

We consider the star-triangle relation and the form of its solutions. We present some simple parametrizations of the weight functions of the three-state chiral Potts model. This model does not have the “difference property”: we discuss the resulting difficulties in attempting to use the corner transfer matrix method for this model.

International Journal of Modern Physics B•Sep 30, 1993•11 citations

Infinite dimensional symmetry of corner transfer matrices

Brian Davies

We review some of the recent developments in two dimensional statistical mechanics in which corner transfer matrices provide the vital link between the physical system and the representation theory of quantum affine algebras. This opens many new possibilities, because the eigenstates may be described using the properties of q-vertex operators.

arXiv:hep-th/9312141•Dec 16, 1993•4 citations
Corner transfer matrices are a useful tool in the statistical mechanics of simple two-dimensinal models.They can be very effective way of obtaining series expansions of unsolved models, and of calculating the order parameters of solved ones.Here we review these features and discuss the reason why the method fails to give the order parameter of the chiral Potts model..
The concept of the corner transfer matrix (CTM) was first discovered by Baxter in 1968, when he derived a set of variational matrix equations, which for finite matrix sizes, could be numerically solved to obtain a sequence of approximations for the statistical mechanical properties of a system of monomers and dimers on a rectangular lattice [1]. It was not until 1978, however, in a seminal paper entitled "Variational Approximations for Square Lattice Models in Statistical Mechanics" [4], that Baxter outlined his CTM variational method, which brought to light the potential power of the former objects to obtain numerics and series expansions for unsolved models and to calculate the order parameter of solved ones. Subsequent numerical work led to the realisation that the method, though general, was not very efficient; and increasing efficiency required making model specific modifications, which restricted the transferability of the resulting algorithm to other models. The CTM variational method was thus not widely adopted, despite holding much promise. More recently, however, Nishino and Okunishi discovered that White's density matrix renormalisation group (DMRG) algorithm [55, 56] could be efficiently extended to study two-dimensional classical lattice models, if the density matrix was approximated by Baxters CTMs. Numerical tests of their algorithm, the corner transfer matrix renormalisation group (CTMRG) method, were met with much success. Notable among these is its implementation within the infinite projected entangled-pair states (iPEPS) algorithm of Orus and Vidal to simulate the ground state of infinite two-dimensional quantum systems. In this thesis we review CTM derived variational methods: Baxter's original CTM iterative method, and developments of the CTM concept within the DMRG algorithm by Nishino and Okunishi, which led to the CTMRG method to calculate critical phenomena of classical and quantum systems. This will begin with elucidating the theoretical formalisms of both methods in two dimensions and their generalisations to three dimensions; followed by review of important application, namely, within the iPEPS algorithm of Orus and Vidal to numerically study infinite planar quantum systems.
The corner transfer matrix renormalization group method is a method, claimed to be subexponential, for numerically calculating physical quantities of statistical mechanical models. In a previous paper, we extended this method to generate series expansions. Here, we show how to extend both the original and series methods to deal with asymmetry in any model. We discuss the cases of rotational, translational and reflection asymmetry, and give some improvements to the method. This is demonstrated by an application of the method to generate series for the hard square model in the high-density regime, producing 51 terms of the partition function and 48 terms of the order parameter. These series are analysed, producing estimates of the critical point and exponents, and showing the likely presence of a confluent singularity with exponent 17/8 in the order parameter.
The corner transfer matrix renormalization group method is an efficient method for evaluating physical quantities in statistical mechanical models. It originates from Baxter’s corner transfer matrix equations and method, and was developed by Nishino and Okunishi in 1996. In this paper, we review and adapt this method, previously used for numerical calculations, to derive series expansions. We use this to calculate 92 terms of the partition function of the hard-squares model. We use the resulting series to provide evidence supporting the claim that the method is subexponential in the number of generated terms, and briefly analyse the singularities of the function.