Some exact results for the anisotropic triangular lattice

1969 ◽  
Vol 47 (22) ◽  
pp. 2445-2448 ◽  
Author(s):  
R. W. Gibberd
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Triangular Lattice ◽  
Correlation Functions ◽  
Spin Correlation ◽  
Exact Results ◽  
Combinatorial Approach ◽  
Ising Lattice ◽  
Simple Derivation ◽  
One Dimensional

The combinatorial approach to the triangular Ising model is shown to be considerably simplified at a given temperature TD. This enables the partition function and spin-correlation functions to be evaluated without the use of pfaffians, thus providing a simple derivation of a few of the results obtained previously by Stephenson. We confirm the results of Stephenson, that at the temperature TD, the correlation functions have the same structure as the correlations of a one-dimensional Ising lattice. The only new result is a general expression for some of the spin–spin correlation functions when the spins do not lie on the axes of the lattice.

scholarly journals The bulk, surface and corner free energies of the anisotropic triangular Ising model

2020 ◽  
Vol 476 (2234) ◽  
pp. 20190713
Author(s):  
Rodney J. Baxter
Keyword(s):  
Free Energy ◽  
Ising Model ◽  
Partition Function ◽  
Temperature Series ◽  
Isotropic Case ◽  
Series Expansions ◽  
Exact Results ◽  
Free Energies ◽  
Bulk Surface ◽  
Bulk Free Energy

We consider the anisotropic Ising model on the triangular lattice with finite boundaries, and use Kaufman’s spinor method to calculate low-temperature series expansions for the partition function to high order. From these, we can obtain 108-term series expansions for the bulk, surface and corner free energies. We extrapolate these to all terms and thereby conjecture the exact results for each. Our results agree with the exactly known bulk-free energy and with Cardy and Peschel’s conformal invariance predictions for the dominant behaviour at criticality. For the isotropic case, they also agree with Vernier and Jacobsen’s conjecture for the 60 ° corners.

Asymptotic inference for an Ising lattice

1976 ◽  
Vol 13 (3) ◽  
pp. 486-497 ◽  
Author(s):  
D. K. Pickard
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Confidence Intervals ◽  
Limit Theorems ◽  
Critical Case ◽  
Ising Lattice ◽  
Asymptotic Inference ◽  
Nearest Neighbours ◽  
Sample Correlation

Kaufmann's exact characterization of the partition function for the classical Ising model is used to obtain limit theorems for the sample correlation between nearest neighbours in the non-critical case. This provides a basis for the asymptotic testing and estimation (by confidence intervals) of the correlation between nearest neighbours.

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