The partition function
Z of a two-dimensional Ising lattice is evaluated by expressing it in the form $Z=\ensuremath{\Sigma}{i}^{}\ensuremath{\lambda}_{i}^{}{}_{}{}^{\mathrm{nm}}$ where ${\ensuremath{\lambda}}_{i}$ are the eigenvalues of a
2n-dimensional matrix M,
n is the number of rows and
m the number of columns of the lattice. M is a generalization of the V-shaped matrix studied by Kramers and Wannier in connection with the same problem. It is shown that M can be expressed in terms of
2n-dimensional representations of
2n-dimensional orthogonal matrices. The eigenvalues of M are determined for
n large by making use of the known relations between the eigenvalues of a
2n-dimensional orthogonal matrix and the eigenvalues of its
2n-dimensional representative.The lattice considered differs only slightly from that treated by Onsager and Kaufman. In the lattice considered here, the boundary effects are eliminated by introducing an interaction between the last lattice point of each row and the first point of the next row. The matrix V considered by Onsager and Kaufman is, except for the boundary effects, equivalent to the matrix
Mn.
Z is shown to differ from the partition function calculated by Onsager and Kaufman only in order ${n}^{\ensuremath{-}2}$.
The model is a linear chain in which each spin interacts with its 2r nearest neighbors, the interaction energy being proportional to 1/r. Using a method similar to that of Montroll, the partition function of the model in the thermodynamic limit is shown to be related to the largest eigenvalue of a certain matrix. The largest eigenvalue of the matrix is determined numerically for 3 < r < 12. Also, a correct method is demonstrated for evaluating an improper limit of the model, in which the interaction range is set to the chain length before the limit of an infinite number of spins is taken. Previously published works have performed this calculation incorrectly. Although the numerical results show some evidence of convergence to the improper limit, the results are inconclusive and furthermore raise doubts about the practicality of the numerical method in this context.
We describe an extension of an approximation scheme proposed earlier for calculating the critical properties of spin-(1/2) Ising systems in terms of the duo-diagonal, sparse matrices of Kramers and Wannier. The computational framework is refined and extended to accommodate a variety of two-, three-, and higher-dimensional lattices as well as interactions other than nearest neighbor. The relation between the underlying symmetries of the lattice and the structure of its associated phase diagram is clarified. Making use of a probabilistic interpretation for the components of the eigenvectors associated with the maximum eigenvalue of the duo-diagonal matrices, it is shown that this makes practical the computation of various thermodynamic quantities---as well as correlation functions---in terms of the expectation values of appropriate operators taken between these eigenvectors. When combined with the techniques of finite-size scaling the present scheme is confirmed to be effective for extracting the critical-point properties of the infinite lattice.
We derive a general formula giving a representation of the partition function of the one-dimensional Ising model of a system of N particles in the form of an explicitly defined functional of the spectral invariants of finite submatrices of a certain infinite Toeplitz matrix. We obtain an asymptotic representation of the partition function for large N, which can be a base for explicitly calculating some thermodynamic averages, for example, the specific free energy, in the case of a general translation-invariant spin interaction (not necessarily only between nearest neighbors). We estimate the partition function from above and below in the plane of the complex variable β (β is the inverse temperature) and consider the conditions under which these estimates are asymptotically equivalent as N → ∞
Using a method suggested by Montroll, we extend the well-known matrix formulation of the nearest-neighbor one-dimensional Ising problem to allow for interactions with an arbitrary finite range n, general spin l, and an applied magnetic field B. We exhibit the relevant matrix element explicitly and hence formally obtain the partition function via an eigenvalue problem of order (2l + 1)n. For the case B = 0, l = ½ we introduce a change of variable which simplifies the partition function while still allowing a matrix formulation. Using this approach we have computed specific-heat curves for infinite, ferromagnetic Ising chains with interactions of range n (n ≤ 7). We prove in an appendix that open and cyclic boundary conditions are equivalent for the system under consideration.
Numerical methods are used to examine the thermodynamic characteristics of the two-dimensional Ising model as a function of the number of spins N. Onsager's solution is generalized to a finite-size lattice, and experimentally validated analytical expressions for the free energy and its derivatives are computed. The heat capacity at the critical point is shown to grow logarithmically with N. Due to the finite extent of the system the critical temperature can only be determined to some accuracy.