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Journal of Physics A Mathematical and Theoretical

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

Yao-ban Chan

Published Mar 8, 2013• 6 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8113/46/12/125009

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Comment on ‘Series expansions from the corner transfer matrix renormalization group method: the hard-squares model’

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Earlier this year Chan extended the low-density series for the hard-squares partition function $κ(z)$ to 92 terms. Here we analyse this extended series focusing on the behaviour at the dominant singularity $z_d$ which lies on on the negative fugacity axis. We find that the series has a confluent singularity of order 2 at $z_d$ with exponents $θ=0.83333(2)$ and $θ'= 1.6676(3)$. We thus confirm that the exponent $θ$ has the exact value $\frac56$ as observed by Dhar.

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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.
Earlier this year Chan extended the low-density series for the hard-squares partition function κ(z)κ(z)κ(z) to 92 terms. Here we analyse this extended series focusing on the behaviour at the dominant singularity zdz_dzd​ which lies on on the negative fugacity axis. We find that the series has a confluent singularity of order 2 at zdz_dzd​ with exponents θ=0.83333(2)θ=0.83333(2)θ=0.83333(2) and θ′=1.6676(3)θ'= 1.6676(3)θ′=1.6676(3). We thus confirm that the exponent θθθ has the exact value 56\frac5665​ as observed by Dhar.
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.
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.
The finite-lattice method of series expansion has been used to extend low-temperature series for the partition function, order parameter and susceptibility of the spin-1 Ising model on the square lattice. A new formalism is described which uses two distinct transfer-matrix approaches in order to significantly reduce computer memory requirements and which permits the derivation of the series to 79th order. Subsequent analysis of the series clearly confirms that the spin-1 model has the same dominant critical exponents as the spin- 1/2 Ising model. Accurate estimates for both the critical temperature and non-physical singularities are obtained. In addition, evidence for a non-analytic confluent correction with exponent Delta 1 =1.1+or-0.1 is found.
We report a real-space renormalization group (RSRG) algorithm, which is formulated through Baxter's corner transfer matrix (CTM), for two-dimensional (d = 2) classical lattice models. The new method performs the renormalization group transformation according to White's density matrix algorithm, so that variational free energies are minimized within a restricted degree of freedom m. As a consequence of the renormalization, spin variables on each corner of CTM are replaced by a m-state block spin variable. It is shown that the thermodynamic functions and critical exponents of the q = 2, 3 Potts models can be precisely evaluated by use of the renormalization group method.