In the Ising model on a helical lattice with period M (in the limit M →∞ of this lattice, the model corresponds to the two-dimensional Ising model), the exact simultaneous equations to calculate physical quantities are derived. The zero field magnetic susceptibility χ 0 , and also the specific heat C are obtained numerically for the system of M ≤11. Furthermore, in the Ising model with up to the next nearest neighbor interactions on the helical lattice, χ 0 and C are calculated for the system of M ≤9. The results obtained are an excellent approximation to the exact χ 0 and C of the two-dimensional systems.In the Ising model on a helical lattice with period M (in the limit M →∞ of this lattice, the model corresponds to the two-dimensional Ising model), the exact simultaneous equations to calculate physical quantities are derived. The zero field magnetic susceptibility χ 0 , and also the specific heat C are obtained numerically for the system of M ≤11. Furthermore, in the Ising model with up to the next nearest neighbor interactions on the helical lattice, χ 0 and C are calculated for the system of M ≤9. The results obtained are an excellent approximation to the exact χ 0 and C of the two-dimensional systems.

Journal of the Physical Society of JapanApr 15, 19933 citations

Exact Solution of the Two-Dimensional Ising Model

R. Shankar

Cambridge University Press eBooksAug 31, 201712 citations

Approximate calculations for the two-dimensional Ising model

Frank A. Blood

Journal of Statistical PhysicsDec 1, 19703 citations

An exact solution to the partition function of the finite-size Ising Model

Rong Wei

There is no an exact solution to three-dimensional (3D) finite-size Ising model (referred to as the Ising model hereafter for simplicity) and even two-dimensional (2D) Ising model with non-zero external field to our knowledge. Here by using an elementary but rigorous method, we obtain an exact solution to the partition function of the Ising model with $N$ lattice sites. It is a sum of $2^N$ exponential functions and holds for $D$ -dimensional ( $D=1,2,3,...$ ) Ising model with or without the external field. This solution provides a new insight into the problem of the Ising model and the related difficulties, and new understanding of the classic exact solutions for one-dimensional (1D) (Kramers and Wannier, 1941) or 2D Ising model (Onsager, 1944). With this solution, the specific heat and magnetisation of a simple 3D Ising model are calculated, which are consistent with the results from experiments and/or numerical simulations. Furthermore, the solution here and the related approaches, can also be available to other models like the percolation and/or the Potts model.There is no an exact solution to three-dimensional (3D) finite-size Ising model (referred to as the Ising model hereafter for simplicity) and even two-dimensional (2D) Ising model with non-zero external field to our knowledge. Here by using an elementary but rigorous method, we obtain an exact solution to the partition function of the Ising model with $N$ lattice sites. It is a sum of $2^N$ exponential functions and holds for $D$-dimensional ($D=1,2,3,...$) Ising model with or without the external field. This solution provides a new insight into the problem of the Ising model and the related difficulties, and new understanding of the classic exact solutions for one-dimensional (1D) (Kramers and Wannier, 1941) or 2D Ising model (Onsager, 1944). With this solution, the specific heat and magnetisation of a simple 3D Ising model are calculated, which are consistent with the results from experiments and/or numerical simulations. Furthermore, the solution here and the related approaches, can also be available to other models like the percolation and/or the Potts model.

arXiv:1805.01366May 2, 20181 citation

Exact Solution for Three-Dimensional Ising Model

Degang Zhang

Jan 1, 20191 citation

Some remarks on Kadanoff's variational approximations for renormalization-group transformations

Michel Droz, Sheldon L. Katz, J. D. Gunton

The recursion equations corresponding to the one-hypercube approximation of Kadanoff are studied in more detail for the Wilson-Fisher $n$ -vector model in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and the $d$ -dimensional Ising model. The free energy for the one-dimensional Ising model is solved exactly in this approximation and is shown to give the correct singular behavior near zero temperature and the exact free energy in zero field. The corrections to scaling exponents are shown to be exact to first order in $\ensuremath{\epsilon}$ for the Wilson-Fisher model and are given for the $d=2,3, \mathrm{and} 4$ dimensional Ising models. The amplitude of the specific-heat divergence is calculated for the two-dimensional Ising model and is shown to be in reasonable agreement with the exact result.The recursion equations corresponding to the one-hypercube approximation of Kadanoff are studied in more detail for the Wilson-Fisher $n$-vector model in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and the $d$-dimensional Ising model. The free energy for the one-dimensional Ising model is solved exactly in this approximation and is shown to give the correct singular behavior near zero temperature and the exact free energy in zero field. The corrections to scaling exponents are shown to be exact to first order in $\ensuremath{\epsilon}$ for the Wilson-Fisher model and are given for the $d=2,3, \mathrm{and} 4$ dimensional Ising models. The amplitude of the specific-heat divergence is calculated for the two-dimensional Ising model and is shown to be in reasonable agreement with the exact result.

Physical review. B, Solid stateJan 1, 19777 citations

N

lattice sites. It is a sum of

2^N

exponential functions and holds for

D

-dimensional (

D=1,2,3,...

) Ising model with or without the external field. This solution provides a new insight into the problem of the Ising model and the related difficulties, and new understanding of the classic exact solutions for one-dimensional (1D) (Kramers and Wannier, 1941) or 2D Ising model (Onsager, 1944). With this solution, the specific heat and magnetisation of a simple 3D Ising model are calculated, which are consistent with the results from experiments and/or numerical simulations. Furthermore, the solution here and the related approaches, can also be available to other models like the percolation and/or the Potts model.

The recursion equations corresponding to the one-hypercube approximation of Kadanoff are studied in more detail for the Wilson-Fisher

n

-vector model in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and the

d

-dimensional Ising model. The free energy for the one-dimensional Ising model is solved exactly in this approximation and is shown to give the correct singular behavior near zero temperature and the exact free energy in zero field. The corrections to scaling exponents are shown to be exact to first order in $\ensuremath{\epsilon}$ for the Wilson-Fisher model and are given for the

d=2,3, \mathrm{and} 4

dimensional Ising models. The amplitude of the specific-heat divergence is calculated for the two-dimensional Ising model and is shown to be in reasonable agreement with the exact result.