Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics C Solid State Physics

Transfer matrix and phenomenological renormalisation methods applied to a triangular Ising ferrimagnetic model

David A. Lavis, A G Quinn

Published May 20, 1987• 1 citation
Publisher

Abstract

The authors study a two-dimensional Ising model on a triangular lattice which is divided into two sublattices. One consists of a honeycomb of sites and the other of the remaining interstitial sites. The interaction energies within the honeycomb and between the honeycomb and the interstitial sites differ from each other. The method used consists of a combination of transfer matrix techniques and the phenomenological renormalisation procedure. Results are presented for the critical temperature and exponents in zero field and the phase diagram in the temperature-field plane is obtained.

DOI
10.1088/0022-3719/20/14/014

Related papers

Antiferromagnetic Ising Model on a Triangular Lattice

Yukio Saitō, Kazuhiro Igeta

An antiferromagnetic Ising system on a triangular lattice is studied by means of a transfer matrix and the phenomenological scaling method. System consists of a semi-infinite strip and helical boundary conditions are imposed at the end of each row. At external magnetic fields h =0 and 6 J , where J is a nearest neighbor coupling constant, a unite residual entropy is obtained at the zero temperature. With a small field, specific heat is found to have two peaks; a high-temperature peak corresponding to the enhanced short-range order, and a low-temperature peak to the appearance of a long-range order. A phase diagram in h - T space is obtained by means of the phenomenological scaling, and agrees fairly well with the one obtained by Monte Carlo simulation. The main deficiency is the finite transition temperature at h =0, which may be due to a crossover to the XY-like phase transition.

Journal of the Physical Society of Japan•Sep 15, 1984•13 citations

The 2×1 and 2×2 Structures on a Triangular Lattice

Yukio Saitō, Kazushi Furuta, Masato Hojou

Phase transitions of the triangular Ising model in a field with nearest and second nearest neighbor antiferromagnetic interactions with couplings, - J and - J /2, respectively, are studied by the interfacial free energy method, by the mean field approximation, by the transfer matrix and phenomenological renormalization methods and by the Monte Carlo simulation. At low temperatures and at fields weaker than the lower critical field h c1 the system takes the 2 × 1 structure, and at fields between h c1 and the upper critical field h c2 it takes the 2×2 structure. Between the 2×1 and 2×2 phases near the field h c1 , a partial disordered phase, in which a part of the sublattices are in disorder, is found. Specific heat shows no divergence, when the partial disordered phase changes to other phases.

Journal of the Physical Society of Japan•Jan 15, 1987•7 citations

Study on Phase Transitions of Antiferromagnetic Ising Models Using the Coherent-Anomaly Method

Adam Lipowski, Masuo Suzuki

The coherent-anomaly method proposed by one of the authors (M.S.) combined with a transfer-matrix version of the double-cluster approximation is applied to antiferromagnetic Ising models. We determined both the critical temperature and the critical exponent γ as functions of an external magnetic field. For a square lattice, our results are in a good agreement with the ones obtained recently by the phenomenological renormalization method. For triangle and simple cubic lattices, our results are less definite but still in a qualitative agreement with values predicted by other methods.

Journal of the Physical Society of Japan•Jul 15, 1992•6 citations

Mean-field renormalisation group transformations for the triangular Ising antiferromagnet

Per Arne Slotte

The mean-field renormalisation group of Indekeu et al. (1982) is applied to the antiferromagnetic nearest-neighbour Ising model on the triangular lattice. The resulting phase diagram in the temperature-field plane is in good agreement with other calculations, while the predicted specific-heat index, alpha , is negative and thus qualitatively wrong. The study indicates that the method may be a useful approach, at least for determining phase diagrams, for frustrated systems and systems with competing interactions where conventional mean-field theory is dubious.

Journal of Physics A Mathematical and General•Feb 1, 1984•16 citations

Kadanoff's variational renormalisation group method: the Ising model on the square and triangular lattices

B. W. Southern

The renormalisation group method of Kadanoff is applied to the spin- 1 / 2 Ising model in two dimensions using a transformation applicable to both the square and triangular lattices. Two types of cluster are studied for the triangular lattice.The first cluster yields a new fixed point but the results for the critical temperature and critical exponents are in poor agreement with the known exact values. The second cluster maps directly onto the fixed point found by Kadanoff for the square lattice yielding excellent values for both the critical exponents and the critical temperature.

Journal of Physics A Mathematical and General•Jan 1, 1978•5 citations

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/0406135•The Journal of Physical Chemistry B•Sep 27, 2001•15 citations
The authors study a two-dimensional Ising model on a triangular lattice which is divided into two sublattices. One consists of a honeycomb of sites and the other of the remaining interstitial sites. The interaction energies within the honeycomb and between the honeycomb and the interstitial sites differ from each other. The method used consists of a combination of transfer matrix techniques and the phenomenological renormalisation procedure. Results are presented for the critical temperature and exponents in zero field and the phase diagram in the temperature-field plane is obtained.
An antiferromagnetic Ising system on a triangular lattice is studied by means of a transfer matrix and the phenomenological scaling method. System consists of a semi-infinite strip and helical boundary conditions are imposed at the end of each row. At external magnetic fields h =0 and 6 J , where J is a nearest neighbor coupling constant, a unite residual entropy is obtained at the zero temperature. With a small field, specific heat is found to have two peaks; a high-temperature peak corresponding to the enhanced short-range order, and a low-temperature peak to the appearance of a long-range order. A phase diagram in h - T space is obtained by means of the phenomenological scaling, and agrees fairly well with the one obtained by Monte Carlo simulation. The main deficiency is the finite transition temperature at h =0, which may be due to a crossover to the XY-like phase transition.
Phase transitions of the triangular Ising model in a field with nearest and second nearest neighbor antiferromagnetic interactions with couplings, - J and - J /2, respectively, are studied by the interfacial free energy method, by the mean field approximation, by the transfer matrix and phenomenological renormalization methods and by the Monte Carlo simulation. At low temperatures and at fields weaker than the lower critical field h c1 the system takes the 2 × 1 structure, and at fields between h c1 and the upper critical field h c2 it takes the 2×2 structure. Between the 2×1 and 2×2 phases near the field h c1 , a partial disordered phase, in which a part of the sublattices are in disorder, is found. Specific heat shows no divergence, when the partial disordered phase changes to other phases.
The coherent-anomaly method proposed by one of the authors (M.S.) combined with a transfer-matrix version of the double-cluster approximation is applied to antiferromagnetic Ising models. We determined both the critical temperature and the critical exponent γ as functions of an external magnetic field. For a square lattice, our results are in a good agreement with the ones obtained recently by the phenomenological renormalization method. For triangle and simple cubic lattices, our results are less definite but still in a qualitative agreement with values predicted by other methods.
The mean-field renormalisation group of Indekeu et al. (1982) is applied to the antiferromagnetic nearest-neighbour Ising model on the triangular lattice. The resulting phase diagram in the temperature-field plane is in good agreement with other calculations, while the predicted specific-heat index, alpha , is negative and thus qualitatively wrong. The study indicates that the method may be a useful approach, at least for determining phase diagrams, for frustrated systems and systems with competing interactions where conventional mean-field theory is dubious.
The renormalisation group method of Kadanoff is applied to the spin- 1 / 2 Ising model in two dimensions using a transformation applicable to both the square and triangular lattices. Two types of cluster are studied for the triangular lattice.The first cluster yields a new fixed point but the results for the critical temperature and critical exponents are in poor agreement with the known exact values. The second cluster maps directly onto the fixed point found by Kadanoff for the square lattice yielding excellent values for both the critical exponents and the critical temperature.
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.