Free-fermion, checkerboard and Z -invariant lattice models in statistical mechanics

1986 ◽  
Vol 404 (1826) ◽  
pp. 1-33 ◽  
Keyword(s):  
Ising Model ◽  
Potts Model ◽  
Three Dimensional ◽  
Lattice Models ◽  
Square Lattice ◽  
Free Fermion ◽  
Local Correlations ◽  
Free Fermion Model ◽  
Zamolodchikov Model ◽  
Special Case

It is shown that the two-dimensional free fermion model is equivalent to a checkerboard Ising model, which is a special case of the general ‘ Z -invariant’ Ising model. Expressions are given for the partition function and local correlations in terms of those of the regular square lattice Ising model. Corresponding results are given for the self-dual Potts model, and the application of the methods to the three-dimensional Zamolodchikov model is discussed. The paper ends with a discussion of the critical and disorder surfaces of the checkerboard Potts model.

Local three-spin correlations in the free-fermion and planar Ising models

1989 ◽  
Vol 423 (1865) ◽  
pp. 279-300 ◽  
Keyword(s):  
Ising Model ◽  
Ising Models ◽  
Square Lattice ◽  
Free Fermion ◽  
Two Dimensional ◽  
Solvable Models ◽  
Spin Correlations ◽  
Fermion Model ◽  
Free Fermion Model

A number of local three-spin correlations are calculated exactly for various related ferromagnetic two-dimensional solvable models in statistical mechanics.They are the square lattice free-fermion model, the equivalent checkerboard Ising model, and the anisotropic triangular, honeycomb and square lattice Ising models. The different results are all applications of a single unifying formula.

ORDERING PROBLEM FOR THE ANTIFERROMAGNETIC POTTS MODEL: T=0 MONTE-CARLO STUDIES

1991 ◽  
Vol 05 (19) ◽  
pp. 3061-3071 ◽  
Author(s):  
A.V. BAKAEV ◽  
V.I. KABANOVICH ◽  
A.M. KURBATOV
Keyword(s):  
Monte Carlo ◽  
Long Range ◽  
Potts Model ◽  
Triangular Lattice ◽  
Lattice Models ◽  
Range Order ◽  
Square Lattice ◽  
Long Range Order ◽  
Monte Carlo Studies ◽  
Antiferromagnetic Potts Model

AF Potts model MC dynamics at T=0 is considered. It is shown that q=3 square lattice and q=4 triangular lattice models are frozen for local MC algorithm. The nature of the previously discussed long-range order phases is examined and entropically favored states are considered.

scholarly journals FUSION OF A-D-E LATTICE MODELS

1994 ◽  
Vol 08 (25n26) ◽  
pp. 3531-3577 ◽  
Author(s):  
YU-KUI ZHOU ◽  
PAUL A. PEARCE
Keyword(s):  
Ising Model ◽  
Potts Model ◽  
Functional Equations ◽  
Lattice Models ◽  
Transfer Matrices ◽  
The Face

Fusion hierarchies of A-D-E face models are constructed. The fused critical D, E and elliptic D models yield new solutions of the Yang-Baxter equations with bond variables on the edges of faces in addition to the spin variables on the corners. It is shown directly that the row transfer matrices of the fused models satisfy special functional equations. Intertwiners between the fused A-D-E models are constructed by fusing the cells that intertwine the elementary face weights. As an example, we calculate explicitly the fused 2×2 face weights of the 3-state Potts model associated with the D4 diagram as well as the fused intertwiner cells for the A5-D4 intertwiner. Remarkably, this 2×2 fusion yields the face weights of both the Ising model and 3-state CSOS models.

scholarly journals On anomalous vibrational spectra

1938 ◽  
Vol 164 (916) ◽  
pp. 62-79 ◽  
Keyword(s):  
Linear Chain ◽  
Three Dimensional ◽  
Low Frequency ◽  
Lattice Models ◽  
Force Constants ◽  
Square Lattice ◽  
Elastic Continuum ◽  
Simple Lattice ◽  
Lattice Distance ◽  
Limiting Case

In previous work on the vibrational spectrum of simple lattice models, it has been shown (Blackman 1937) that the spectrum becomes anomalous when special values are chosen for the force constants entering into the description of the model. As an example we may take the case of a square lattice of lattice distance d containing one particle per cell, in which the force constants α (for particles at a distance α) and γ (for particles at a distance d √2) were used. When γ / α tended to zero it could be shown that the spectrum changed from the two-dimensional to that of the linear chain. A similar result holds in the corresponding three-dimensional case. Although one would not expect any actual crystal to correspond to a limiting case (it would not be stable) it is conceivable that there should be crystals which approach the limiting case. In all cases which have hitherto been discussed, the anomaly is associated with the behaviour of the low-frequency end of the spectrum. This suggests that one could trace the effect to some property of the elastic continuum. An examination of the lattice mentioned above, for which the elastic continuum has two elastic constants c 11 and c 12 = c 44 , shows that the transition to the limiting case ( γ / α = 0) can be put in the form c 44 / c 11 → 0.

Ising model critical exponent η from a criticality equation

1989 ◽  
Vol 67 (10) ◽  
pp. 946-951 ◽  
Author(s):  
B. Frank ◽  
L. Macot ◽  
K. V. Bassias ◽  
M. Danino
Keyword(s):  
Ising Model ◽  
Critical Exponent ◽  
Pair Correlation ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Square Lattice ◽  
Specific Method ◽  
Hamiltonian Methods ◽  
Order Of Magnitude

A pair-correlation function ansatz, previously used in the derivation of the Ising model critical exponent η for the square lattice from a criticality equation within the i-δ approximation, is investigated further. Various methods are used to calculate the multispin correlation functions entering the criticality equation. The calculations are extended to the three-dimensional cubic lattices. It is found that the values obtained for η are relatively insensitive to the specific method used. In two dimensions, η takes the values 0.2506 (i-δ method), 0.2471, and 0.2490 (renormalized Hamiltonian methods), thereby remaining within 1.2% of the exact value 1/4. In three dimensions, η ranges from 0.030 to 0.075, being of the same order of magnitude as the series value 0.041 ± 0.01.

ISING MODEL ON A SELF-DUAL 3-6 LATTICE

1990 ◽  
Vol 04 (01) ◽  
pp. 113-121 ◽  
Author(s):  
N. C. CHAO ◽  
FELIX J. LEE ◽  
K. Y. LIN
Keyword(s):  
Phase Transition ◽  
Critical Temperature ◽  
Ising Model ◽  
Antiferromagnetic Interaction ◽  
Interaction Parameters ◽  
Square Lattice ◽  
Rectangular Lattice ◽  
The One ◽  
Special Case

We consider the Ising model on a self-dual 3-6 lattice with six interaction parameters which includes the rectangular lattice as a special case. Depending on the signs and magnitudes of the interactions, the model has either a unique critical temperature or no phase transition. In the special case where all six interactions are the same, the unique critical temperature is identical with (different from) the one of the square lattice Ising model with the same ferromagnetic (antiferromagnetic) interaction.

scholarly journals QUANTUM MODEL OF INTERACTING "STRINGS" ON THE SQUARE LATTICE

2000 ◽  
Vol 15 (01) ◽  
pp. 105-131
Author(s):  
H. E. BOOS
Keyword(s):  
Square Lattice ◽  
Quantum Model ◽  
Free Fermion ◽  
Two Dimensional ◽  
Fermion Model ◽  
One Dimensional ◽  
Single String ◽  
Free Fermion Model ◽  
Xy Spin Chain ◽  
The One

The model which is the generalization of the one-dimensional XY-spin chain for the case of the two-dimensional square lattice is considered. The subspace of the "string" states is studied. The solution to the eigenvalue problem is obtained for the single "string" in cases of the "string" with fixed ends and "string" of types (1, 1) and (1, 2) living on the torus. The latter case has the features of a self-interacting system and does not seem to be integrable while the previous two cases are equivalent to the free-fermion model.

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