Some exact results for the Ashkin-Teller model

1979 ◽  
Vol 365 (1722) ◽  
pp. 371-380 ◽  
Keyword(s):  
Renormalization Group ◽  
Partition Function ◽  
Potts Model ◽  
Critical Line ◽  
Square Lattice ◽  
Direct Connection ◽  
Exact Results ◽  
Renormalization Group Theory ◽  
Vertex Model ◽  
Special Cases

It is shown that various cases of the Ashkin-Teller model on the square, triangular and hexagonal lattices can be transformed by the dual and star-triangle transformations and, further, that these problems can be reduced to special cases of the eight vertex model on the Kagomé lattice. In general, we can only obtain the partition function of the Ashkin-Teller model if we are on its line of fixed points, and it then turns out that it is reducible to the six vertex model. Since the partition function of the q -state Potts model at its critical point can also be related to the six vertex model, a direct connection between the Ashkin-Teller model and the Potts model can be made. It turns out that moving along the critical line of the Ashkin-Teller model corresponds to varying q for the Potts model. For the square lattice comparison is made with renormalization group calculations, and the agreement found is a satisfactory check of renormalization group theory.

General Square Lattice Vertex Models

2019 ◽  
pp. 430-453
Author(s):  
Hans-Peter Eckle
Keyword(s):  
Square Lattice ◽  
Vertex Model ◽  
Transfer Matrices ◽  
Spectral Parameters ◽  
Vertex Models ◽  
Integrable Quantum ◽  
Special Cases ◽  
Mechanical Models ◽  
The One ◽  
Ice Model

Vertex models more general than the ice model are possible and often have physical applications. The square lattice admits the general sixteen-vertex model of which the special cases, the eight- and the six-vertex model, are the most relevant and physically interesting, in particular through their connection to the one-dimensional integrable quantum mechanical models and the Bethe ansatz. This chapter introduces power- ful tools to examine vertex models, including the R- and L-matrices to encode the Boltzmann vertex weights and the monodromy and transfer matrices, which encode the integrability of the vertex models (i.e. that transfer matrices of different spectral parameters commute). This integrability is ultimately expressed in the Yang–Baxter relations.

scholarly journals PARTITION FUNCTION ZEROS OF A RESTRICTED POTTS MODEL ON SELF-DUAL STRIPS OF THE SQUARE LATTICE

2007 ◽  
Vol 21 (10) ◽  
pp. 1755-1773 ◽  
Author(s):  
SHU-CHIUAN CHANG ◽  
ROBERT SHROCK
Keyword(s):  
Partition Function ◽  
Potts Model ◽  
Square Lattice ◽  
Partition Function Zeros ◽  
Large Width ◽  
Function Zeros ◽  
Continuous Accumulation

We calculate the partition function Z(G, Q, v) of the Q-state Potts model exactly for self-dual cyclic square-lattice strips of various widths Ly and arbitrarily large lengths Lx, with Q and v restricted to satisfy the relation Q=v2. From these calculations, in the limit Lx→∞, we determine the continuous accumulation locus [Formula: see text] of the partition function zeros in the v and Q planes. A number of interesting features of this locus are discussed and a conjecture is given for properties applicable to arbitrarily large width. Relations with the loci [Formula: see text] for general Q and v are analyzed.

Partition Function Zeros of the Square Lattice Potts Model

1996 ◽  
Vol 76 (2) ◽  
pp. 169-172 ◽  
Author(s):  
Chi-Ning Chen ◽  
Chin-Kun Hu ◽  
F. Y. Wu
Keyword(s):  
Partition Function ◽  
Potts Model ◽  
Square Lattice ◽  
Partition Function Zeros ◽  
Function Zeros

c = 1 − 6(n − 1)2/n, THEORIES COUPLED TO GRAVITY: A COMMENT ON THEIR POSSIBLE LATTICE MODELS REALIZATIONS

1991 ◽  
Vol 06 (18) ◽  
pp. 1709-1719 ◽  
Author(s):  
HUBERT SALEUR
Keyword(s):  
Free Energy ◽  
Partition Function ◽  
Potts Model ◽  
Planar Graphs ◽  
Critical Line ◽  
Central Charge ◽  
Lattice Models ◽  
Genus Zero ◽  
Regular Lattices ◽  
Beraha Numbers

We discuss the recently proposed logarithmic violation of scaling for c = 1 − 6(n − 1)2/n theories in the light of lattice models. We study for this purpose the Q state Potts model in its antiferromagnetic regime eK − 1 = −Q1/2, coupled to gravity. Setting Q1/2 = 2 cos π/t, this model is known to have a generic central charge c = 1 − 6(t − 1)2/t. Summing over all possible planar graphs allows us to make connection with Kostov's solution of IRF models, and to calculate the genus zero properties along the critical line. Except for n = 1, 2 we do not get indications of logarithmic violations. The apparent regularity of the thermodynamic properties (γ str = −(n − 1) = integer ) is explained by a discontinuity of the free energy of the Potts model when Q crosses the Beraha numbers [Formula: see text], n ≥ 3 in the antiferromagnetic region. Such behavior was recently observed for some regular lattices. The logarithmic terms for n = 2, c = −2 appear simply because a derivative with respect to Q has to be taken to define a non-vanishing partition function. Only the n = 1, c = 1 logarithmic terms seem to have a non-trivial origin.

Sign in / Sign up

Export Citation Format

Share Document