For systems with an infinite number of degrees of freedom, the Rayleigh-Ritz quotient of the standard variational method is analysed from an alternative point of view. Its numerator and denominator are treated as statistical models having the same correlation length. The approach is developed in the framework of the transfer matrix formalism and applied to Ising models. It allows the author to obtain exactly the correlation length along a row or column of the anisotropic square Ising model, for T>or=T c . Approximate applications of the method to the two-layer square and 3D simple cubic Ising models are also given. In these cases quite reliable critical curves are obtained through an essentially analytical procedure.
For systems with an infinite number of degrees of freedom, the Rayleigh-Ritz quotient of the standard variational method is analysed from an alternative point of view. Its numerator and denominator are treated as statistical models having the same correlation length. The approach is developed in the framework of the transfer matrix formalism and applied to Ising models. It allows the author to obtain exactly the correlation length along a row or column of the anisotropic square Ising model, for T>or=T c . Approximate applications of the method to the two-layer square and 3D simple cubic Ising models are also given. In these cases quite reliable critical curves are obtained through an essentially analytical procedure.
Within the framework of the transfer matrix formalism, a variational method is elaborated with a constraint related to the fundamental role of the correlation length near a critical point. The method is applied to a two-layer ferromagnetic film, made of two horizontal Ising planes interacting through a vertical coupling. The critical curve of the model, as well as the correlation length above and near T c are calculated. The critical exponent nu obtained agrees with the expected two-dimensional exact value. For the two particular values of the vertical coupling, for which there are numerical estimates of T c through series analysis or the Monte Carlo method, an essential agreement with these estimates is found. Analytical expressions are given in the weak and strong vertical coupling regimes.
An approximation for calculating the thermodynamic properties of spin-\textonehalf{} Ising models is described. The method is based on the formulation of Kramers and Wannier for the square lattice, and involves expressing the free energy in terms of the limiting maximum eigenvalue of a matrix sequence which represents the contribution from a single spin of the lattice. The application of this formulation to various two- and three-dimensional lattices shows that many of these are characterized by an identically structured matrix sequence with exactly two nonvanishing elements in each row and column. To explore the potential of using this similarity and its simplicity for analyzing Ising models we consider the two-dimensional square lattice in an external field and approximate the doubly infinite lattice by a semi-infinite one consisting of $n (\ensuremath{\le}10)$ infinitely long rows. Calculations are carried out and compared with (1) Onsager's formula for the free energy, (2) the Onsanger-Yang formula for the spontaneous magnetization, and (3) values calculated from series. Except for the region around the critical point where the correlation length exceeds n lattice spacings, substantial agreement---to better than 12 places of decimals in some cases---is obtained. As an additional measure of the potential of this method, a value for the critical exponent $\ensuremath{\delta}$ is computed by the extrapolation of the magnetization along the critical isotherm. The result is found to be in excellent agreement with the known value for $\ensuremath{\delta}$.
A transfer-matrix-scaling technique is developed for randomly diluted systems, and applied to the site-diluted Ising model on a square lattice in two dimensions. For each allowed disorder configuration between two adjacent columns, the contribution of the respective transfer matrix to the decay of correlations is considered only as far as the ratio of its two largest eigenvalues, allowing an economical calculation of a configuration-averaged correlation length. Standard phenomenological-renormalization procedures are then used to analyze aspects of the phase boundary which are difficult to assess accurately by alternative methods. For magnetic site concentration p close to pc, the extent of exponential behavior of the Tc\ifmmode\times\else\texttimes\fi{}p curve is clearly seen for over two decades of variation of p-pc. Close to the pure-system limit, the exactly known reduced slope is reproduced to a very good approximation, though with nonmonotonic convergence. The averaged correlation lengths are inserted into the exponent-amplitude relationship predicted by conformal invariance to hold at criticality. The resulting exponent \ensuremath{\eta} remains near the pure value (1/4) for all intermediate concentrations until it crosses over to the percolation value at the threshold.
The d-dimensional Ising transfer matrix is expressed exactly in terms of Fermi-type operators. In two dimensions this leads to the standard solution of the model. In three or more dimensions the transfer matrix is no longer a quadratic form in the Fermi variables, but rather contains quartic and higher-order terms. However, it can be expressed as a well-defined expansion about a quadratic form in Fermi operators. The zeroth-order transfer-matrix-called herein the "fermion model"---can be exactly diagonalized, and is well approximated in the critical region by ${V}^{(0)}\ensuremath{\approx}{[2sinh(2K)]}^{\frac{N}{2}}\mathrm{exp}\left[\ensuremath{-}N\ensuremath{\int}d\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}{\left({\ensuremath{\kappa}}^{2}+\ensuremath{\Sigma}{n=1}^{d\ensuremath{-}1}\ensuremath{\Sigma}{m=1}^{d\ensuremath{-}1}{q}_{n}{q}_{m}\right)}^{\frac{1}{2}}t(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})\right]$where N is the number of sites per layer, $\ensuremath{\mu}$ is the inverse correlation length, $t(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ takes values \ifmmode\pm\else\textpm\fi{}1/2, K is the reduced inverse temperature, and the integral extends over a Brillouin zone of the ($d\ensuremath{-}1$)-dimensional layer. The critical point of the model is located by the equation ${x}_{c}^{d}+{x}_{c}^{d\ensuremath{-}1}+{x}_{c}=1$, where ${x}_{c}={e}^{\ensuremath{-}2{K}_{c}}$. The critical exponent $\ensuremath{\nu}$ describing the divergence of the correlation length at Kc is found to take on its two-dimensional value, $\ensuremath{\nu}=1$, in all dimensions d. The possibility of using the fermion model as the basis for an expansion of the Ising critical exponents as a function of $\ensuremath{\delta}(=d\ensuremath{-}2)$ is pointed out.
The correlation length of the Ising model on the simple cubic lattice is calculated directly from the magnetization profiles which are determined in the pair, the square, and the cube approximations in the simplified cluster variation method. The critical exponent ν is obtained by applying the coherent anomaly method to the correlation length. The wave-number dependent magnetic susceptibility or the spin-pair correlation function is not required in obtaining the expression for the correlation length.
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.