Coherent Anomaly Method

The following sections are included:

• PREFACE

• CONTENTS

• ACKNOWLEDGMENTS

• LIST OF REPRINTS

The following sections are included:

• Introduction

• Spontaneous symmetry-breaking and long-range order

• Phenomenological theory of second-order phase transition

• Scaling law in critical phenomena

• Universality of critical exponents

• Coherent anomaly

• References

The following sections are included:

• Classical mean-field approximations

• Coherent-anomaly method

• References

The following sections are included:

• Overview of the chapter

• Super-effective-field theory

• Multi-effective-field theory

• Double-cluster approximation

• Cluster-variation method

• Power-series CAM theory

• Confluent transfer-matrix method

• CAM scheme in quantum Monte Carlo simulations

• Kosterlitz-Thouless transition

• Mean-field approximation and correlation functions on strips

• References

The following sections are included:

• Non-universality of the eight-vertex model

• Application of the CAM to the eight-vertex model

• Further applications

• References

The following sections are included:

• Experiments

• Edwards-Anderson theory

• Spin-glass susceptibility and nonlinear susceptibility

• Effective-field theory of spin glasses

• CAM estimates of the critical point and the exponent

• References

The following sections are included:

• Finite-temperature transition in the ferromagnetic Heisenberg model

• Ground-state phase transitions in quantum ferromagnets

• Ground-state phase transitions in quantum antiferromagnets

• Accurate estimation of KT-type transition point in quantum antiferromagnetic chains at the ground state

• References

The following sections are included:

• Percolation problem

• Self-avoiding walk

• Diffusion-limited aggregation

• References

The following sections are included:

• Introduction

• Stochastic Ising models

• Contact processes

• Branching annihilating random walk and other processes

• References

Please refer to full text.

A general method is proposed to study non-classical scaling behaviors of cooperative systems. This is based on the appearance of a “coherent anomaly” in mean-field-type or classical approximations of cooperative systems. The degree of approximation is analytically continued to construct a parameterized expression of a whole set of approximations, using Kubo's linear response theory. General relations between scaling (or critical) exponents and “coherent-anomaly exponents” are derived using the theory of envelope and are also obtained from a finite-degree-of-approximation scaling, which is inspired by Fisher's finite-size scaling. The present coherent anomaly method (CAM) is promising for calculating analytically non-classical scaling exponents in cooperative phenomena. A general scheme of CAM in critical phenomena is presented. It is shown that even the combination of the Weiss and Bethe approximations gives rather good estimates of static and dynamic critical exponents.

A coherent anomaly method (CAM) is formulated for critical phenomena on the basis of the general idea of CAM for cooperative phenomena proposed by Suzuki. The coherent anomalies of response functions are derived phenomenologically using Kubo's linear response theory and Fisher's scaling form of correlation functions. This yields the relation between the coherent anomaly exponents and non-classical critical exponents. The convergence of the CAM critical points is proven rigorously. Some explicit applications of the CAM theory based on systematic cluster-mean-field approximations and on the mean-field transfer-matrix method are presented to show how useful the CAM is in studying critical phenomena. The CAM theory of critical dynamics is also formulated to give the dynamical critical exponent Δ ≃ 1.84 in the two-dimensional kinetic Ising model.

Two kinds of systematic mean-field transfer-matrix methods are formulated in the 2-dimensional Ising spin system, by introducing Weiss-like and Bethe-like approximations. All the critical exponents as well as the true critical point can be estimated in these methods following the CAM procedure. The numerical results of the above system are T_c* = 2.271 (J/k_B), γ=γ’ ≃ 1.749, β≃0.131 and δ ≃ 15.1. The specific heat is confirmed to be continuous and to have a logarithmic divergence at the true critical point, i.e., α=α′=0. Thus, the finite-degree-of-approximation scaling ansatz is shown to be correct and very powerful in practical estimations of the critical exponents as well as the true critical point.

The systematic Weiss-like and Bethe-like approximations based on the mean-field transfer-matrix method are used to investigate the asymptotic behavior of the induced magnetization on a semi-infinite square lattice, and to investigate the wave-number dependence of the susceptibility in a nonuniform external field. The critical exponents ν, ν', ηⁱ and η are estimated following the general CAM prescription. A new scaling relation ν·ηⁱ=β is obtained in the framework of the finite-degree-of-approximation scaling. Together with previous papers, all the static critical exponents have been estimated by the CAM, and are shown to satisfy the well-known scaling relations.

A new scheme to study exotic phase transitions is formulated by introducing the concept of a super-effective field. A general mechanism of phase transitions is elucidated and a general criterion of order parameters is proposed on the basis of the newly formulated super-effective-field theory. An alternative formulation based on a decoupled density matrix is also given. It is easily shown using these formulations that a quantum chiral order appears in the antiferromagnetic XY model on the triangular lattice. A super-effective-field theory of spin glasses is also presented.

A multi-effective-field theory is formulated and applied to the two-dimensional Ising model from the viewpoint of the coherent-anomaly method (CAM). Two necessary conditions to construct the CAM canonical series are shown. Two series of approximations are derived on 3 × 3 - and 4 × 4-clusters and with the use of the CAM we estimate the critical exponent of the susceptibility within an error of 0.37 and 0.45 percent, respectively, for the exact value γ = 1.75 where the exact T_c* is assumed. This accuracy of the estimation shows the effectiveness of this theory. It is generally proved that certain kinds of approximations with different combinations of effective fields yield exactly the same approximate critical temperature and mean-field coefficient.

A systematic mean-field transfer-matrix method is formulated in the Ising spin systems by introducing a modified version of the Bethe approximation. The critical exponent γ and the true critical point T_C* are estimated by the coherent anomaly method. Results obtained for 2- and 3-dimensional Ising models (S = ½) are in excellent agreement with most methods existing so far. We also apphed our method to the flat Ising model (S = 1) with a positive biquadratic interaction; a very good agreement with Monte Carlo predictions was obtained.

It is shown phenomenologically that the double-cluster approximation is canonical in the sense of the CAM theory, namely that the coherent anomaly appearing in a series of double-cluster approximations gives asymptotically correct values of critical exponents without any logarithmic correction. This is a basic property desired in the CAM.

The cluster-variation method (CVM) proposed by Kikuchi is a general theory to give approximations, which are useful to discuss phase transitions qualitatively in many systems. Recently one of the present authors (M. S.) proposed the coherent-anomaly method (CAM). If we have a well-behaved series of approximations, which is called a canonical series, we can estimate critical exponents following the CAM. In this paper it is demonstrated by using the ferromagnetic Ising models that the CVM will provide a canonical series which shows coherent anomaly. Our result implies that combining the CVM with the CAM will give a powerful method to study phase transitions and critical phenomena.

The critical exponents of the bond percolation model are calculated in the D(= 2,3,…)-dimensional simple cubic lattice on the basis of Suzuki's coherent anomaly method (CAM) by making use of a series of the pair, the square-cactus and the square approximations of the cluster variation method (CVM) in the s-state Potts model. These simple approximations give reasonable values of critical exponents α, β, γ and ν in comparison with ones estimated by other methods. It is also shown that the results of the pair and the square-cactus approximations can be derived as exact results of the bond percolation model on the Bethe and the square-cactus lattice, respectively, in the presence of ghost field without recourse to the s→1 limit of the s-state Potts model.

The spin-pair correlation functions in the Ising model on the square lattice are obtained in three approximations of the simplified cluster variation method. We find good estimates of the critical exponents ην, μ and ν, by applying the coherent-anomaly method to the obtained spin-pair correlation functions.

The coherent anomaly method, introduced by Suzuki in 1986, provides, in principle, a remarkably simple approach to study critical phenomena within the framework of the mean-field approximation. Unlike the renormalization group techniques commonly employed to investigate critical phenomena, Suzuki's method exploits the systematic behavior of a sequence of mean-field approximations in order to extract critical temperature and exponents. Despite its conceptual simplicity, actual implementation of the CAM requires the ability to treat at least three levels of mean-field approximations belonging to what Suzuki has termed a “cardinal” sequence. Here we present a new method based on a continuous sequence of mean-field approximations from which the implementation of the CAM proceeds in a straightforward manner.

A new method to estimate critical exponents is proposed based on power-series expansions and the basic idea of the coherent-anomaly method (CAM). The coherent anomalies of physical quantities such as the susceptibility, χ₀, can be obtained by studying the zero for the first N polynomials of the power-series of the inverse of the relevant physical quantity, and by studying its critical coefficient near the zero point. The critical exponents are estimated on the basis of the general CAM theory. The Pade approximants are also combined with the CAM theory to estimate fractional critical exponents.

The phase diagram and the critical indices are investigated for the symmetric 16-vertex model on the square lattice by combining a variational series expansion and the coherent anomaly method. The nonuniversal critical exponents smoothly interpolate between two exactly solvable cases, namely the Baxter eight-vertex model and the Ising model.

The correlation function and susceptibility of the Ising model on regular lattices are shown to be studied from the CAM analysis of a systematic series of hierarchical models such as generalized cactus trees by introducing a new transfer matrix associated with the confluent transfer matrix. In this method, the singularity of the correlation length can be obtained. Each systematic solution yields the exponents ν = 1, γ = 1 and η = 1.

Layered square-lattice ferromagnetic Ising models (∞ × ∞ × n lattices) are studied by the Monte Carlo simulation for n = 3 to 7. The results show the dimensionality crossover from the two-dimensional square lattice Ising model to the three-dimensional simple cubic Ising model. Furthermore, the CAM analysis is performed to evaluate the critical exponent γ of the three-dimensional simple cubic ferromagnetic Ising model, which shows that γ = 1.21(4). The present analysis is also applicable to the study of the phase transition of quantum spin systems.

A series of effective-field approximations are formulated for the Kosterlitz-Thouless transition in the sine-Gordon model by means of the cumulant expansion and the variational method. Effective-field essential singularities are obtained for the correlation length as first derived by Saito in a single approximation. However, a systematic variance of the effective-field critical coefficient is found when the order of approximation n increases. The true critical exponent v̄ of the Kosterlitz-Thouless transition is thus revealed to be less than the effective-field exponent v̄₀, v̄ < v̄₀ = 1, from Suzuki's coherent-anomaly method. The phase transition in the two-dimensional XY model is studied from its relation to the sine-Gordon model. The critical exponent ŋ_c of the spin-spin correlation function at the critical point is found to be ŋ_c = 1/(4 + 1/π²).

The cluster variation method is applied to the ferromagnetic six-state clock model on the triangular lattice up to the hexagonal cluster approximation. The Kosterlitz-Thouless type phase transition of the model is studied by using the coherent anomaly method (CAM).

A transfer-matrix version of a mean-field approximation is proposed which leads to the exact critical temperatures for the S = 1/2 Ising model on the square, triangular, honeycomb, and centered square lattices. The estimations of critical temperatures for more complicated models are also shown to be very accurate.

It is shown that the two-dimensional square lattice Ising model with the nearest-neighbour ferromagnetic interaction J, the next-nearest-neighbour antiferromagnetic interaction J′ < 0 and the four-body interaction J_int shows non-universal critical behaviour when J is small. The estimations of critical temperature T_c and the critical exponent of susceptibility γ are performed within errors of 0.003% and 1.2%, respectively.

A new cluster-effective-field theory of spin glasses is formulated. Basic formulas for the spin-glass transition point and the spin-glass susceptibility in the high-temperature phase are obtained. The present theory combined with the coherent-anomaly method is shown to be useful to estimate the true critical point and the nonclassical critical exponent of a spin-glass transition. Concerning the two-dimensional ±J model, we have γ_s = 5.2(1) for T_SG = 0, which agrees well with the data by some other authors. As for the three-dimensional ±J model, the present tentative analysis gives T_SG = 1.2(1)(J/k_B) and γ_s = 4(1), but more extensive calculations are needed.

The coherent-anomaly method proposed by M. Suzuki is applied to the spin 1/2 ferromagnetic Heisenberg model to estimate the critical exponents. The effective Hamiltonian method to small-sized clusters for the simple cubic lattice, and the inverse temperature power series expansion of the susceptibility for the face centered cubic lattice are combined with the CAM. Both results show that the CAM is useful for the quantum spin systems.

Systematic spin-cluster approximation is applied to the quantum spin systems and the critical temperature and the critical exponent are estimated by use of the coherent anomaly method. The critical exponent γ of the three dimensional S = ½ Heisenberg ferromagnet is estimated to be 1.44.

A Kosterlitz-Thouless (KT)-type transition in the one-dimensional S= 1/2 XXZ model at the ground state is analyzed using the double-cluster approximation (DCA) and the coherent-anomaly method (CAM). It is numerically shown that the periodic boundary condition (PBC) is suitable for the construction of a systematic series of cluster approximations in quantum antiferromagnets. The transition point and the exponent of the essential singularity are estimated as g^*c ≃ 0.9969 and σ ≃ 0.527, which are in good agreement with the exact values, g^*_c = 1 and σ=0.5, respectively. Such accuracy of estimation is due to the consideration of a power-law factor in front of the exponential singularity of x̄, which had been neglected until now.

How to impose boundary conditions is crucial to finite-cluster calculations of quantum spin systems, but only open-boundary clusters have been used in cluster-effective-field approximations up to now. In the present paper, periodic-boundary clusters are also considered for the formulation of cluster-effective-field approximations in frustrated quantum spin systems. Namely, the open-boundary-condition double-cluster approximation (OBC-DCA) and the periodic-boundary-condition double-cluster approximation (PBC-DCA) are applied to the one-dimensional S = ½ frustrated XXZ model at the ground state. These two approximations are compared using the coherent-anomaly method (CAM). Within the limitation of cluster sizes in exact-diagonalization calculations, the PBC-DCA can reproduce the true phase boundary of the Neel-dimer transition. On the other hand, the OBC-DCA severely underestimates the existence of magnetic orders as quantum fluctuation is increased. These findings suggest that previous cluster-effective-field studies based on open-boundary clusters should be reconsidered.

Applying the coherent anomaly method (CAM) to site percolation problems, we estimate the percolation threshold ϱ_c and critical exponents. We obtain p_c = 0.589, Β=0.140, Γ = 2.426 on the two-dimensional square lattice. These values are in good agreement with the values already known. We also investigate higher-dimensional cases by this method.

We study the mean-field approximation in the site-percolation problem. Using the analog of the Simon-Lieb inequality, we show that the mean-field critical probability is convergent to the exact value when the size of clusters tends to infinity. Applying this approximation to the one-dimensional further-neighbor percolation problem and calculating some critical coefficients, we prove that the asymptotic scaling relations predicted by the coherent-anomaly method are satisfied.

Self-avoiding walk (SAW), being a nonequilibrium cooperative phenomenon, is investigated with a finite-order-restricted-walk (finite-ORW or FORW) coherent-anomaly method (CAM). The coefficient β_1r in the asymptotic form C_nr ≃ β_lrλⁿ_1r for the total number C_nr of r-ORW's with respect to the step number n is investigated for the first time. An asymptotic form for SAW's is thus obtained from the series of FORW approximants, C_nr ≃ br^gμ(1 + a/r)ⁿ, as the envelope curve C_n ≃ b(ae/g)^gμⁿn^g. Numerical results are given by C_n ≃ 1.424n^0.27884.1507ⁿ and C_n ≃ 1.179n^0.158710.005ⁿ for the plane triangular lattice and f.c.c. lattice, respectively. A good coincidence of the total numbers estimated from the above simple formulae with exact enumerations for finite-step SAW's implies that the essential nature of SAW (non-Markov process) can be understood from FORW (Markov process) in the CAM framework.

We analyzed the generalized diffusion limited aggregation model, the Η-model, applying the coherent anomaly method. Fractal dimensions of the growth patterns are estimated from a set of mean densities, {n(L)}, which are defined in a modified Η-model with a screened potential (Δ − 1/L²)Ø=0.

The coherent-anomaly method (CAM) is applied to the kinetic Ising model. Dynamical cluster-mean-field approximations are formulated to obtain a series of the dynamical mean-field critical coefficients. The coherent-anomaly scaling relations are derived on the basis of the scaling form of the generating function in nonequilibrium systems to estimate the exponent Δ of the critical slowing down. The dynamical critical exponent is estimated as Δ ≃ 2.15 (±0.02) for the kinetic Ising model on the two-dimensional triangular lattice.

The one-dimensional contact process (CP) is studied by a systematic series of approximations. A new decoupling procedure of correlation functions is proposed by combining the idea of Suzuki's correlation-identity-decoupling (CID) with a concept of window. Liggett's approximations are also considered. Applying Suzuki's coherent-anomaly method (CAM) to the mean-field-type solutions, the values of the critical point and the critical exponents are estimated as λ_c = 1.6490(±0.0008), β=0.280(±0.013), Δ(= Δ/δ)= 1.734(±O.OO1), β=0.627(±0.005). Finally a comparison with other estimates is shown.

The branching annihilation random walk (in short BARW) exhibiting an extinction-survival phase transition in one dimension is studied by the coherent anomaly method. This is the first theoretical evidence that the BRAW belongs to the universality class of directed percolation.

The critical behaviour of a long-range, one-dimensional non-equilibrium system where the desorption rate decreases with distance as some power ~ r^−α−d is studied by means of the coherent anomaly method. The α-dependence of the critical parameter λ_c and the critical exponent β is determined.

The following sections are included:

• SUBJECT INDEX