Star-Triangle and Star-Star Relations in Statistical Mechanics

1997 ◽  
Vol 11 (01n02) ◽  
pp. 27-37 ◽  
Author(s):  
R. J. Baxter
Keyword(s):  
Statistical Mechanics ◽  
Potts Model ◽  
Original Model ◽  
State Model ◽  
Nearest Neighbour ◽  
Transfer Matrices ◽  
Zamolodchikov Model ◽  
Adequate Condition

The homogeneous three-layer Zamolodchikov model is equivalent to a four-state model on the checkerboard lattice which closely resembles the four-state critical Potts model, but with some of its Boltzmann weights negated. Here we show that it satisfies a "star-to-reverse-star" (or simply star-star) relation, even though we know of no star-triangle relation for this model. For any nearest-neighbour checkerboard model, we show that this star-star relation is sufficient to ensure that the decimated model (where half the spins have been summed over) satisfies a "twisted" Yang-Baxter relation. This ensures that the transfer matrices of the original model commute in pairs, which is an adequate condition for "solvability".

FUNCTIONAL RELATIONS FOR TRANSFER MATRICES OF THE CHIRAL POTTS MODEL

1990 ◽  
Vol 04 (05) ◽  
pp. 803-870 ◽  
Author(s):  
R.J. Baxter ◽  
V.V. Bazhanov ◽  
J.H.H. Perk
Keyword(s):  
Potts Model ◽  
Finite Size ◽  
State Model ◽  
Integrable Case ◽  
Vertex Model ◽  
Chiral Potts Model ◽  
Transfer Matrices ◽  
Functional Relations ◽  
Finite Size Corrections

It has recently been shown that the solvable N-state chiral Potts model is related to a vertex model with N-state spins on vertical edges, two-state spins on horizontal edges. Here we generalize this to a “j-state by N-state” model and establish three sets of functional relations between the various transfer matrices. The significance of the “super-integrable” case of the chiral Potts model is discussed, and results reported for its finite-size corrections at criticality.

scholarly journals Parallel family trees for transfer matrices in the Potts model

2015 ◽  
Vol 187 ◽  
pp. 55-71 ◽  
Author(s):  
Cristobal A. Navarro ◽  
Fabrizio Canfora ◽  
Nancy Hitschfeld ◽  
Gonzalo Navarro
Keyword(s):  
Potts Model ◽  
Transfer Matrices ◽  
Family Trees

scholarly journals SIMULATIONS OF FINANCIAL MARKETS IN A POTTS-LIKE MODEL

2005 ◽  
Vol 16 (08) ◽  
pp. 1311-1317 ◽  
Author(s):  
TETSUYA TAKAISHI
Keyword(s):  
Financial Markets ◽  
Financial Market ◽  
Potts Model ◽  
State Model ◽  
Exponential Distributions ◽  
Stylized Facts ◽  
Time Correlations ◽  
Long Time ◽  
Three State Model ◽  
Two State Model

A three-state model based on the Potts model is proposed to simulate financial markets. The three states are assigned to "buy", "sell" and "inactive" states. The model shows the main stylized facts observed in the financial market: fat-tailed distributions of returns and long time correlations in the absolute returns. At low inactivity rate, the model effectively reduces to the two-state model of Bornholdt and shows similar results to the Bornholdt model. As the inactivity increases, we observe the exponential distributions of returns.

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.

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