Journal of the Physical Society of Japan

Exact Critical Temperature by Mean-Field Approximation

Adam Lipowski, Masuo Suzuki

Published Dec 15, 1992• 28 citations

Abstract

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.

DOI: 10.1143/jpsj.61.4356

Related papers

Calculation of the Critical Temperature for 2- and 3-Dimensional Ising Models and for 2-Dimensional Potts Models Using the Transfer Matrix Method

Mehrdad Ghaemi, Gholamabbas Parsafar, Mahmud Ashrafizaadeh

A new graphical method is developed to calculate the critical temperature of 2- and 3-dimensional Ising models as well as that of the 2-dimensional Potts models. This method is based on the transfer matrix method and using the limited lattice for the calculation. The reduced internal energy per site has been accurately calculated for different 2-D Ising and Potts models using different size-limited lattices. All calculated energies intersect at a single point when plotted versus the reduced temperature. The reduced temperature at the intersection is 0.4407, 0.2746, and 0.6585 for the square, triangular, and honeycombs Ising lattices and 1.0050, 0.6309, and 1.4848 for the square, triangular, and honeycombs Potts lattices, respectively. These values are exactly the same as the critical temperatures reported in the literature, except for the honeycomb Potts lattice. For the two-dimensional Ising model, we have shown that the existence of such an intersection point is due to the duality relation. The method is then extended to the simple cubic Ising model, in which the intersection point is found to be dependent on the lattice sizes. We have found a linear relation between the lattice size and the intersection point. This relation is used to obtain the critical temperature of the unlimited simple cubic lattice. The obtained result, 0.221(2), is in a good agreement with the accurate value of 0.22165 reported by others.

arXiv:physics/0406135The Journal of Physical Chemistry BSep 27, 200115 citations

Mean-field renormalization group for the boundary magnetization of strip clusters

Alessandro Pelizzola, Attilio L. Stella

We analyse in some detail a recently proposed transfer matrix mean-field approximation which yields the exact critical point for several two-dimensional nearest-neighbour Ising models. For the square lattice model we show explicitly that this approximation yields not only the exact critical point, but also the exact boundary magnetization of a semi-infinite Ising model, independent of the size of the strips used. Then we develop a new mean-field renormalization-group strategy based on this approximation and make connections with finite size scaling. Applying our strategy to the quadratic Ising and three-state Potts models we obtain results for the critical exponents which are in very good agreement with the exact ones. In this way we also clarify some advantages and limitations of the mean field renormalization group approach.

arXiv:cond-mat/9307023Journal of Physics A Mathematical and GeneralDec 7, 19937 citations

Exact Decimation Approach with Mean-Field Approximation for Randomly Diluted Ising Models

TANG KUN-FA, HU JIA-ZHEN

A new renormalization approach developed by authors previously is generalized to disordered systems. Calculations of the critical temperature, concentration, exponents, and of limiting slope of critical curve for the random bond-diluted Ising model on the square lattice, even in the lowest approximation, are in very satisfactory agreement with all known exact or series results.

Communications in Theoretical PhysicsJun 1, 19880 citations

EXACT BOUNDARY MAGNETIZATION OF THE LAYERED ISING MODEL ON TRIANGULAR AND HONEYCOMB LATTICES

Alessandro Pelizzola

In the present paper we extend a previous exact calculation of the boundary magnetization of the square lattice layered Ising model, based on a transfer-matrix effective-field technique, to the triangular lattice. The result is then further extended to the honeycomb lattice by means of a star-triangle transformation. Finally, the (11) boundary of the square lattice is analyzed as a particular case of the triangular and honeycomb lattices.

Modern Physics Letters BFeb 28, 19962 citations

Anisotropic triangular Ising model in the extended mean-field renormalization-group approach

Christos N. Likos

The recently proposed multi-interaction mean-field renormalization technique [C. N. Likos and A. Maritan, Phys. Rev. E 53, 3303 (1996)] is applied to the Ising model on a triangular lattice with equal couplings in two directions and a different coupling in the third, a model equivalent to a square Ising model with additional second-neighbor interactions along a single diagonal direction. Three different clusters are considered and the possible mappings between them are discussed. The estimates for critical couplings and exponents are in satisfactory agreement with exact results. An explanation of the way in which the finite-size features of the method bring about a systematic overestimation of the critical temperature is also given.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsFeb 1, 19976 citations

A brief account of the Ising and Ising-like models: Mean-field, effective-field and exact results

Jozef Strečka, M. Jaščur

The article provides a tutorial review on how to treat Ising models within mean-field (MF), effective-field (EF) and exact methods. MF solutions of the spin-1 Blume-Capel (BC) model and the mixed-spin Ising model demonstrate a change of continuous phase transitions to discontinuous ones at a tricritical point. A quantum phase transition of the spin-S Ising model driven by a transverse field is explored within MF method. EF theory is elaborated within a single- and two-spin cluster approach to demonstrate an efficiency of this approximate method. The long-standing problem of this method concerned with a self-consistent determination of the free energy is addressed. EF theory is adapted for the spin-1/2 Ising model, the spin-S BC model and the transverse Ising model. The particular attention is paid to continuous and discontinuous transitions. Exact results for the spin-1/2 Ising chain, spin-1 BC chain and mixed-spin Ising chain are obtained using the transfer-matrix method, the crucial steps of which are reviewed for a spin-1/2 Ising square lattice. Critical points of the spin-1/2 Ising model on several lattices are rigorously obtained with the help of dual, star-triangle and decoration-iteration transformations. Mapping transformations are adapted to obtain exact results for the mixed-spin Ising model on planar lattices. An increase in the coordination number of the mixed-spin Ising model on decorated planar lattices gives rise to reentrant transitions, while the critical temperature of the mixed-spin Ising model on a regular honeycomb lattice is always greater than that of two semi-regular archimedean lattices. The effect of selective site dilution of the mixed-spin Ising model on a honeycomb lattice upon phase diagrams is examined. The review affords a brief account of the Ising models solved within MF, EF and exact methods along with a few comments on their future applicability.

arXiv:1511.03031Nov 10, 201545 citations