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Physical Review

Statistics of the Two-Dimensional Ferromagnet. Part I

H. A. Kramers, Gregory H. Wannier

Published Aug 1, 1941• 1,810 citations
Publisher

Abstract

In an effort to make statistical methods available for the treatment of cooperational phenomena, the Ising model of ferromagnetism is treated by rigorous Boltzmann statistics. A method is developed which yields the partition function as the largest eigenvalue of some finite matrix, as long as the manifold is only one dimensionally infinite. The method is carried out fully for the linear chain of spins which has no ferromagnetic properties. Then a sequence of finite matrices is found whose largest eigenvalue approaches the partition function of the two-dimensional square net as the matrix order gets large. It is shown that these matrices possess a symmetry property which permits location of the Curie temperature if it exists and is unique. It lies at $\frac{J}{k{T}_{c}}=0.8814$ if we denote by $J$ the coupling energy between neighboring spins. The symmetry relation also excludes certain forms of singularities at ${T}_{c}$, as, e.g., a jump in the specific heat. However, the information thus gathered by rigorous analytic methods remains incomplete.

DOI
10.1103/physrev.60.252

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Statistics of the Two-Dimensional Ferromagnet. Part II

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The partition function of a two-dimensional "ferromagnetic" with scalar "spins" (Ising model) is computed rigorously for the case of vanishing field. The eigenwert problem involved in the corresponding computation for a long strip crystal of finite width ($n$ atoms), joined straight to itself around a cylinder, is solved by direct product decomposition; in the special case $n=\ensuremath{\infty}$ an integral replaces a sum. The choice of different interaction energies ($\ifmmode\pm\else\textpm\fi{}J,\ifmmode\pm\else\textpm\fi{}{J}^{\ensuremath{'}}$) in the (0 1) and (1 0) directions does not complicate the problem. The two-way infinite crystal has an order-disorder transition at a temperature $T={T}_{c}$ given by the condition $sinh(\frac{2J}{k{T}_{c}}) sinh(\frac{2{J}^{\ensuremath{'}}}{k{T}_{c}})=1.$ The energy is a continuous function of $T$; but the specific heat becomes infinite as $\ensuremath{-}log |T\ensuremath{-}{T}_{c}|$. For strips of finite width, the maximum of the specific heat increases linearly with $log n$. The order-converting dual transformation invented by Kramers and Wannier effects a simple automorphism of the basis of the quaternion algebra which is natural to the problem in hand. In addition to the thermodynamic properties of the massive crystal, the free energy of a (0 1) boundary between areas of opposite order is computed; on this basis the mean ordered length of a strip crystal is ${(\mathrm{exp} (\frac{2J}{\mathrm{kT}}) tanh(\frac{2{J}^{\ensuremath{'}}}{\mathrm{kT}}))}^{n}.$

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Thermodynamics Property of Magnetic Materials in a Ferromagnetic System using the Ising Model

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There is hardly any branch of physics one can approach successfully without resorting to quantum statistical mechanics.It has been proven that the quantum mechanical description of a system gives accurate results.To use quantum statistical mechanics, the system are treated at their microscopic level.Most systems consist of many particles and in order o deal with such large numbers, one has to resort to statistical method related to the partition function of statistical mechanics.In this study, we haves used the partition function which is a statistical parameter to explore the thermodynamic properties such as Helmoltz free energy, Entropy, internal energy and heat capacity of a magnetic materials in a ferromagnetic system using the first and second dimensional Ising model.The results show that in one-dimensional model, phase transition does not occur at finite temperature but susceptibility of a system can be measured from the fluctuations in the magnetization in the second dimensional Ising model.

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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 ${2}^{n}$-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 ${2}^{n}$-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 ${2}^{n}$-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 ${\mathbf{M}}^{n}$. $Z$ is shown to differ from the partition function calculated by Onsager and Kaufman only in order ${n}^{\ensuremath{-}2}$.

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A Numerical Investigation of a Certain One-Dimensional Ising Model

Jimmy Green

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.

•Jan 1, 1981•0 citations
In an effort to make statistical methods available for the treatment of cooperational phenomena, the Ising model of ferromagnetism is treated by rigorous Boltzmann statistics. A method is developed which yields the partition function as the largest eigenvalue of some finite matrix, as long as the manifold is only one dimensionally infinite. The method is carried out fully for the linear chain of spins which has no ferromagnetic properties. Then a sequence of finite matrices is found whose largest eigenvalue approaches the partition function of the two-dimensional square net as the matrix order gets large. It is shown that these matrices possess a symmetry property which permits location of the Curie temperature if it exists and is unique. It lies at JkTc=0.8814\frac{J}{k{T}_{c}}=0.8814kTc​J​=0.8814 if we denote by JJJ the coupling energy between neighboring spins. The symmetry relation also excludes certain forms of singularities at Tc{T}_{c}Tc​, as, e.g., a jump in the specific heat. However, the information thus gathered by rigorous analytic methods remains incomplete.
The study of the two-dimensional Ising model is continued. Its specific heat at the Curie point is investigated. The quantity in question is computed for six successive finite matrix problems and the conclusion is drawn that the specific heat is infinite at the Curie point. A new closed form approximation of the partition function $\ensuremath{\lambda}$ is then developed by using the matrix method in its variational form. The two power series for $\ensuremath{\lambda}$ at extreme temperatures are used as a test for this and various other approximations, and it is found that the new result is a considerable improvement over the existing solutions. Finally it is pointed out that these closed form solutions support our conclusion as to the place and nature of the Curie point transition.
This is a continuation of an effort to reduce to the solution of a characteristic value problem the rigorous calculation of thermodynamic properties of systems in which the intermolecular forces are sufficiently short ranged so that practically the entire potential energy of the system results from interactions between nearest neighbors. The partition function of such a system can be expressed in terms of the largest characteristic value of a linear operator equation and finally as the ratio of partition functions of systems with relatively few particles whose potential energy functions differ somewhat from those in the original system. A method of evaluating grand partition functions and one of introducing interactions between more distant neighbors are discussed. The general theory is applied to the calculation of magnetization, internal energy, and specific heat of two-dimensional ferromagnets on the basis of the Ising model. There seems to exist a λ-point phase transition in the change from ferromagnetic to nonferromagnetic states.
The partition function of a two-dimensional "ferromagnetic" with scalar "spins" (Ising model) is computed rigorously for the case of vanishing field. The eigenwert problem involved in the corresponding computation for a long strip crystal of finite width (nnn atoms), joined straight to itself around a cylinder, is solved by direct product decomposition; in the special case $n=\ensuremath{\infty}$ an integral replaces a sum. The choice of different interaction energies ($\ifmmode\pm\else\textpm\fi{}J,\ifmmode\pm\else\textpm\fi{}{J}^{\ensuremath{'}}$) in the (0 1) and (1 0) directions does not complicate the problem. The two-way infinite crystal has an order-disorder transition at a temperature T=TcT={T}_{c}T=Tc​ given by the condition $sinh(\frac{2J}{k{T}_{c}}) sinh(\frac{2{J}^{\ensuremath{'}}}{k{T}_{c}})=1.$ The energy is a continuous function of TTT; but the specific heat becomes infinite as $\ensuremath{-}log |T\ensuremath{-}{T}_{c}|$. For strips of finite width, the maximum of the specific heat increases linearly with lognlog nlogn. The order-converting dual transformation invented by Kramers and Wannier effects a simple automorphism of the basis of the quaternion algebra which is natural to the problem in hand. In addition to the thermodynamic properties of the massive crystal, the free energy of a (0 1) boundary between areas of opposite order is computed; on this basis the mean ordered length of a strip crystal is ${(\mathrm{exp} (\frac{2J}{\mathrm{kT}}) tanh(\frac{2{J}^{\ensuremath{'}}}{\mathrm{kT}}))}^{n}.$
There is hardly any branch of physics one can approach successfully without resorting to quantum statistical mechanics.It has been proven that the quantum mechanical description of a system gives accurate results.To use quantum statistical mechanics, the system are treated at their microscopic level.Most systems consist of many particles and in order o deal with such large numbers, one has to resort to statistical method related to the partition function of statistical mechanics.In this study, we haves used the partition function which is a statistical parameter to explore the thermodynamic properties such as Helmoltz free energy, Entropy, internal energy and heat capacity of a magnetic materials in a ferromagnetic system using the first and second dimensional Ising model.The results show that in one-dimensional model, phase transition does not occur at finite temperature but susceptibility of a system can be measured from the fluctuations in the magnetization in the second dimensional Ising model.
The partition function ZZZ 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{2}^{n}2n-dimensional matrix M, nnn is the number of rows and mmm 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{2}^{n}2n-dimensional representations of 2n2n2n-dimensional orthogonal matrices. The eigenvalues of M are determined for nnn large by making use of the known relations between the eigenvalues of a 2n2n2n-dimensional orthogonal matrix and the eigenvalues of its 2n{2}^{n}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{\mathbf{M}}^{n}Mn. ZZZ 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.