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arXiv:cond-mat/0408657 · Journal of Physics A Mathematical and General

1D action and partition function for the 2D Ising model with a boundary magnetic field

Maxime Clusel, Jean-Yves Fortin

Published Mar 15, 2005• 5 citations
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Abstract

In this paper we present an alternative method to that developed by McCoy and Wu to obtain some exact results for the 2D Ising model with a general boundary magnetic field and for a finite size system. This method is a generalization of ideas from Plechko presented for the 2D Ising model in zero field, based on the representation of the Ising model using a Grassmann algebra. A Gaussian 1D action is obtained for a general configuration of the boundary magnetic field. When the magnetic field is homogeneous, we check that our results are in agreement with McCoy and Wu's previous work. This 1D action is used to compute in an efficient way the free energy in the special case of inhomogeneous boundary magnetic field. This method is useful to obtain new exact results for interesting boundary problems, such as wetting transitions.

DOI
10.1088/0305-4470/38/13/003

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In this paper we present an alternative method to that developed by McCoy and Wu to obtain some exact results for the 2D Ising model with a general boundary magnetic field and for a finite size system. This method is a generalization of ideas from Plechko presented for the 2D Ising model in zero field, based on the representation of the Ising model using a Grassmann algebra. A Gaussian 1D action is obtained for a general configuration of the boundary magnetic field. When the magnetic field is homogeneous, we check that our results are in agreement with McCoy and Wu's previous work. This 1D action is used to compute in an efficient way the free energy in the special case of inhomogeneous boundary magnetic field. This method is useful to obtain new exact results for interesting boundary problems, such as wetting transitions.
There is no an accepted exact partition function (PF) for the two-dimensional (2D) Ising model with a non-zero external magnetic field to our knowledge. Here we infer an empirical PF for such an Ising model. We compare the PFs for two finite-size Ising lattices (4×44\times 44×4 and 4×64\times 64×6) from this empirical PF with those from Wei (2018) (Wei, R.Q., 2018. An exact solution to the partition function of the finite-size Ising Model, arXiv: General Physics: 1805.01366.), and find that they are consistent very well. Based on this empirical PF, we further analyze and calculate the thermodynamic functions (heat capacity, magnetization, susceptibility) of this 2D Ising model and discuss the model's singularity semiquantitatively. Analysis and calculations from this PF show that they are coincident with those from other related studies; Especially the 2D Ising model in an external magnetic field has spontaneous magnetization (SM) calling the phenomenon at the critical temperature a phase transition, and the SM decreases with the increasing temperature. However, the decreasing variation of the SM here is different from that obtained by Yang (1952) from the 2D Ising model in a weak magnetic field.
The exact solution of the two-dimensional (2D) Ising model at an external magnetic field is derived by a modified Clifford algebraic approach. At first, the transfer matrices are analyzed in three representations, i.e., Clifford algebraic representation, transfer tensor representation and schematic representation, to inspect nonlocal effects in this many-body interacting system. It is ensured that nontrivial topological structures exist in this system, which is analogous to (but different with) those in the three-dimensional (3D) Ising model at zero magnetic field. Therefore, the approaches developed for the 3D Ising models are modified to be appropriable for solving analytically the solution of the 2D Ising model at a magnetic field. An additional rotation, serving as a topological Lorentz transformation, is applied for dealing with the topological problems in the present system. The rotation angle for the transformation is determined by Yang-Baxter relations and a subsequent average of rotation angles treating the linear change of the topological actions. Application of a magnetic field increases the magnetization, shifting the critical point to higher temperatures. At the temperature above the critical point, the magnetization keeps zero until a critical field at which it jumps rapidly as a first-order magnetization process. The partition function and the magnetization obtained are helpful for understanding the physical properties, in particular, the magnetization processes of the 2D magnetic materials.
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 NNN lattice sites. It is a sum of 2N2^N2N exponential functions and holds for DDD-dimensional (D=1,2,3,...D=1,2,3,...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.
Abstract The partition function of the square lattice Ising model on the rectangle with open boundary conditions in both directions is calculated exactly for arbitrary system size L × M and temperature. We start with the dimer method of Kasteleyn, McCoy and Wu, construct a highly symmetric block transfer matrix and derive a factorization of the involved determinant, effectively decomposing the free energy of the system into two parts, F ( L , M ) = F s t r i p ( L , M ) + F s t r i p r e s ( L , M ) , where the residual part F s t r i p r e s ( L , M ) contains the nontrivial finite- L contributions for fixed M . It is given by the determinant of a M / 2 × M / 2 matrix and can be mapped onto an effective spin model with M Ising spins and long-range interactions. While F s t r i p r e s ( L , M )