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Journal of Physics A Mathematical and General

Thet-expansion study of critical phenomena in quantum systems

Ladislav Šamaj, Pavol Kalinay, P. Markoš, Igor Travěnec

Published Mar 7, 1997• 14 citations
Publisher

Abstract

For quantum systems in the ground state, the t-expansion technique is used to construct a canonical sequence of classical theories. The consequent application of the coherent-anomaly method provides an estimation of the non-classical values of critical indices related to the singular behaviour of the ground-state energy. The accuracy of the results is tested on the Ising model with a transverse field, formulated on two- and three-dimensional lattices.

DOI
10.1088/0305-4470/30/5/015

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For quantum systems in the ground state, the t-expansion technique is used to construct a canonical sequence of classical theories. The consequent application of the coherent-anomaly method provides an estimation of the non-classical values of critical indices related to the singular behaviour of the ground-state energy. The accuracy of the results is tested on the Ising model with a transverse field, formulated on two- and three-dimensional lattices.
A systematic procedure of constructing a series of cluster-mean-field approximations is established to investigate the coherent anomalies of the mean-field critical coefficients \barχ( L , T c ( L )) in the two- and three-dimensional Ising models. The values of critical exponents as well as the critical temperatures are estimated by means of the new method (Coherent Anomaly Method) as T c * =2.230(0.038) J / k B , γ=1.785(0.116) for the two-dimensional square lattice and T c * =4.529(0.058) J / k B , γ=1.258(0.068) for the three-dimensional simple cubic lattice. The results suggest that the present approach to study critical phenomena is very promising.
The applicability of the coherent-anomaly method (CAM) to series expansions is checked using test functions and the critical exponents γ for the Ising models are estimated by this method. Among three canonical series studied here, the Padé series and the continued-fraction series are most effective in their convergence and the accuracy of estimation. There are some functions which are difficult to analyze with the power-series CAM. Our estimate for γ of the three-dimensional Ising model is 1.246. This value is higher than the latest results obtained by means of other methods taking into account the correction-to-scaling term.
A cluster-effective-field theory, called the double-cluster approximation, is formulated for quantum spin systems at T=0. Combining this approximation with the coherent-anomaly method (CAM), critical phenomena in such systems can be analysed. To test their theory, the authors have estimated the critical points and critical exponents of the one-dimensional S=1/2 transverse Ising model and the one-dimensional S=1/2 XZ model. The estimates are in good agreement with the exact solutions. These results show that the CAM approach is useful in treating zero-temperature phase transitions in quantum spin systems.
For the symmetric two-state vertex model on the honeycomb lattice we construct a series expansion of the free energy which, at finite order, depends on free gauge parameters. We treat these gauge parameters as variational ones, and derive a canonical series of classical approximations which possesses the property of coherent anomaly. As a test model we use the vertex formulation of the Ising antiferromagnet in a field and, within the coherent-anomaly method, determine with a high accuracy its critical frontier and exponent γ. Numerical checks on the constancy of critical exponents along the phase boundary are presented, too.
A series of spin-cluster approximation which is a mean field type approximation is derived systematically on the basis of a spin-cluster expansion formula of the free energy. Mean field critical temperature and mean field critical coefficients are calculated by the spin-cluster approximation for the two-, three- and more than four-dimensional ferromagnetic Ising systems. Analyzing them by the coherent anomaly method, the critical temperature and the critical exponents α, β, γ and δ are estimated. Accuracy of estimated values in this method are comparable to those given by other methods such as the high temperature series analysis. The present theory provides a new and useful tool to study the critical phenomena, especially of three-dimensional systems.
A new scheme of the coherent-anomaly method (CAM) is proposed to study critical phenomena in the models for which a mean-field description gives spurious first-order phase transition. A canonical series of mean-field-type approximations are constructed so that the spurious discontinuity should vanish asymptotically as the approximate critical temperature approachs the true value. The true value of the critical exponents β and γ are related to the coherent-anomaly exponents defined among the classical approximations. The formulation is demonstrated in the two-dimensional q -state Potts models for q =3 and 4. The result shows that the present method enables us to estimate the critical exponents with high accuracy by using the date of the cluster-mean-field approximations.