scholarly journals ZEROS OF THE POTTS MODEL PARTITION FUNCTION IN THE LARGE-q LIMIT

2007 ◽  
Vol 21 (07) ◽  
pp. 979-994 ◽  
Author(s):  
SHU-CHIUAN CHANG ◽  
ROBERT SHROCK
Keyword(s):  
Partition Function ◽  
Boundary Conditions ◽  
Coordination Number ◽  
Potts Model ◽  
The Other ◽  
Square Lattice ◽  
Various Boundary Conditions ◽  
Temperature Variable

We calculate zeros of the q-state Potts model partition function Z(GΛ,q,v) for large q, where v is the temperature variable and GΛ is a section of a lattice Λ with coordination number κΛ and various boundary conditions. Lattice types studied include square, triangular, honeycomb, and kagomé. We show that for large q these zeros take on approximately circular patterns in the complex xΛ plane, where xΛ=v/q2/κΛ. This generalizes a known result for the square lattice to the other lattices considered.

Analysis of Plate Examples by Difference Methods and the Superposition Principle

1936 ◽  
Vol 3 (3) ◽  
pp. A81-A90
Author(s):  
D. L. Holl
Keyword(s):  
Boundary Conditions ◽  
Critical Stress ◽  
Plate Theory ◽  
Superposition Principle ◽  
Equivalent Stress ◽  
Lower Surface ◽  
The Other ◽  
Principle Of Superposition ◽  
Various Boundary Conditions ◽  
Actual Plate

Abstract In this paper the author applies the membrane analogs of H. Marcus to some elementary cases of thin homogeneous isotropic square plates having central-point loads and various boundary conditions. The analogy is made possible by two theorems: (a) The deflection of a membrane loaded with loads proportional to those on a given plate may be considered as the sum of the principal moments of the actual plate. (b) A second membrane may be loaded with elastic weights proportional to these moment sums and, subject to appropriate boundary conditions, the deflections of the latter membrane will be proportional to the deflections of the actual plate under the given loading system. The principle of superposition of deflection surfaces or equivalent stress systems is utilized in this paper both by difference and differential methods. The problems treated are (1) a square plate with pinned or simply supported edges, (2) two opposite edges pinned and the other two free, (3) two opposite edges pinned and the other two clamped, (4) all four edges clamped, (5) all four edges free with only corner post supports. The correct critical stress at the center of the lower surface of the plate was obtained from special thick-plate theory for a particular thickness-to-span ratio. The effect of this critical stress on the whole plate action is depicted for various boundary conditions.

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.

Determination of the stress concentration in the corner point of the wedge-shaped region reinforced by a more rigid thin coating

2020 ◽  
Vol 26 (1) ◽  
pp. 80-89
Author(s):  
AN Soloviev ◽  
BV Sobol ◽  
EV Rashidova ◽  
AI Novikova
Keyword(s):  
Boundary Conditions ◽  
Corner Point ◽  
Theory Of Elasticity ◽  
The Other ◽  
Thin Coating ◽  
Various Boundary Conditions ◽  
Singular Behaviour ◽  
Isotropic Theory ◽  
Made In

We analysed the problem of determining the exponents in the asymptotic solution of the isotropic theory of elasticity problem at the top of the wedge-shaped region where its sides (or one of them) are supported by a thin coating and lean without friction on the rigid bases. On the other side of the wedge-shaped region, it is assumed that there are various boundary conditions, including when there is a thin coating. Mathematically, the problem reduces to the problem of determining the roots of transcendental characteristic equations arising from the condition for the existence of a nontrivial solution of a system of the linear homogeneous equations. The characteristics of the stress tensor components have been determined for the various combinations of boundary conditions and physical and geometric parameters. The qualitative conclusions are made. In particular, we have established the combinations of the values of these parameters at which the singular behaviour of stresses arises.

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

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.

Estimates of the critical exponent β for the Potts model using a variational approximation

1976 ◽  
Vol 54 (15) ◽  
pp. 1621-1626 ◽  
Author(s):  
S. B. Kelland
Keyword(s):  
Variational Principle ◽  
Partition Function ◽  
Critical Exponent ◽  
Transfer Matrix ◽  
Potts Model ◽  
Order Parameter ◽  
Square Lattice ◽  
Variational Approximation ◽  
Matrix Equations ◽  
Ice Model

The q-state Potts model on a square lattice is transformed into a staggered ice model. A variational principle for the largest eigenvalue of the transfer matrix of this model is used to develop a set of matrix equations. In the limit of these matrices becoming infinite, the equations determine the partition function and order parameter of the Potts model exactly. We have solved the equations numerically for finite matrices, obtaining estimates of these quantities and, as a result, the critical exponent β.

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