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.
A singularity on the negative fugacity axis of the hard-square lattice gas is investigated in terms of numerical diagonalization of transfer matrices. The location of the singular point
zc− and the critical exponent
ν are accurately determined by the phenomenological renormalization technique as -0.11933888188(1) and 0.416667(1), respectively. It is also found that the central charge
c and the dominant scaling dimension
xσ are -4.399996(8) and -0.3999996(7), respectively. These results strongly support that this singularity belongs to the same universality class as the Yang-Lee edge singularity (
c=−22/5,xσ=−2/5andν=5/12).
A singularity on the negative-fugacity axis of the hard-core lattice gas is investigated in terms of numerical diagonalization of large-scale transfer matrices. For the hard-square lattice gas, the location of the singular point [Formula: see text] and the critical exponent ν are accurately determined by the phenomenological renormalization technique as -0.11933888188(1) and 0.416667(1), respectively. It is also found that the central charge c and the dominant scaling dimension xσ are -4.399996(8) and -0.3999996(7), respectively. Similar analyses for other hard-core lattice-gas models in two dimensions are also performed, and it is confirmed that the universality between these models does hold. These results strongly indicate that the present singularity belongs to the same universality class as the Yang–Lee edge singularity.
We have extended the series for the zero-field susceptibility of the spin-
s Ising model through tenth order in the reduced inverse temperature
K on the square, triangular, simple-cubic, body-centered-cubic, and face-centered-cubic lattices. The series coefficients
hn(s) are expressed as simple polynomials in
X=s(s+1). Using extended methods of analysis we have estimated the nature of the leading singularities on the face-centered lattice and conclude with good confidence that the susceptibility exponent ${\ensuremath{\gamma}}_{1}$ equals 1.25, independent of
s. The exponent of the leading correction term is estimated to be ${\ensuremath{\gamma}}_{2}\ensuremath{\simeq}0.75\ifmmode\pm\else\textpm\fi{}0.08$ in good agreement with renormalization-group theory. For
s=21 only, the amplitude of the confluent correction apparently vanishes. We have also studied the leading singularities on the triangular lattice and conclude that ${\ensuremath{\gamma}}_{1}=\frac{7}{4}$, independent of
s, with, however, much stronger corrections than in three dimensions. These results provide a very strong corroboration of the universality hypothesis.
The authors re-examine, using a confluent singularity analysis, existing low-temperature series for the mean size of finite clusters for the pure spin- 1 / 2 nearest-neighbour Ising model in two dimensions. Although the triangular lattice is problematical, results for the honeycomb and square lattices may be interpreted as indicating a dominant exponent theta =1.87+or-0.04 (consistent with the theoretical prediction theta = gamma + beta ), together with competing analytic and non-analytic ( Delta 1 =1.35+or-0.25) correction-to-scaling terms.