scholarly journals Two-dimensional Ising Model on a 4?6?12 Lattice

1985 ◽  
Vol 38 (2) ◽  
pp. 227
Author(s):  
KY Lin ◽  
WN Huang
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Two Dimensional ◽  
Special Case

We have considered a two-dimensional Ising model on a 4-6-12 lattice. The partition function is evaluated exactly by the method of Pfaffian. The Ising model on a ruby lattice is a special case of our model.

scholarly journals Complexity of Ising Polynomials

2012 ◽  
Vol 21 (5) ◽  
pp. 743-772 ◽  
Author(s):  
TOMER KOTEK
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
External Field ◽  
Polynomial Time ◽  
Tutte Polynomial ◽  
Study Phase ◽  
Exceptional Set ◽  
Exponential Time ◽  
Low Dimension ◽  
Special Case

This paper deals with the partition function of the Ising model from statistical mechanics, which is used to study phase transitions in physical systems. A special case of interest is that of the Ising model with constant energies and external field. One may consider such an Ising system as a simple graph together with vertex and edge weights. When these weights are considered indeterminates, the partition function for the constant case is a trivariate polynomialZ(G;x,y,z). This polynomial was studied with respect to its approximability by Goldberg, Jerrum and Paterson.Z(G;x,y,z) generalizes a bivariate polynomialZ(G;t,y), which was studied in by Andrén and Markström.We consider the complexity ofZ(Gt,y) andZ(G;x,y,z) in comparison to that of the Tutte polynomial, which is well known to be closely related to the Potts model in the absence of an external field. We show thatZ(G;x,y,z) is #P-hard to evaluate at all points in3, except those in an exceptional set of low dimension, even when restricted to simple graphs which are bipartite and planar. A counting version of the Exponential Time Hypothesis, #ETH, was introduced by Dell, Husfeldt and Wahlén in order to study the complexity of the Tutte polynomial. In analogy to their results, we give under #ETHa dichotomy theorem stating that evaluations ofZ(G;t,y) either take exponential time in the number of vertices ofGto compute, or can be done in polynomial time. Finally, we give an algorithm for computingZ(G;x,y,z) in polynomial time on graphs of bounded clique-width, which is not known in the case of the Tutte polynomial.

The cooperative behaviour of a two-dimensional defect crystal

1957 ◽  
Vol 53 (4) ◽  
pp. 863-869 ◽  
Author(s):  
E. W. Elcock
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Logarithmic Singularity ◽  
Perfect Crystal ◽  
Cooperative Behaviour ◽  
Two Dimensional ◽  
Combinatorial Method ◽  
Ferromagnetic Crystal ◽  
Defect Crystal ◽  
Lower Temperature

ABSTRACTThe Ising model of a two-dimensional ferromagnetic crystal containing defects is considered. It is shown that the combinatorial method is particularly appropriate to the investigation of the two-dimensional defect crystal and how it may be readily modified to calculate the partition function for such a crystal. It is found that the cooperative behaviour of the infinite perfect crystal, as indicated by the occurrence of a logarithmic singularity in the specific heat as a function of temperature, is not destroyed in the defect crystal; the singularity persists, though at a progressively lower temperature as the density of defects is increased, until a high density of defects is reached. The value of this limiting density of defects, above which the cooperative behaviour is destroyed, is calculated.

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