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Diffusion and defect data, solid state data. Part B, Solid state phenomena/Solid state phenomena

Partition Function and Density of States in Models of a Finite Number of Ising Spins with Direct Exchange between the Minimum and Maximum Number of Nearest Neighbors

Petr Dmitrievich Andriushchenko, Konstantin Valentinovich Nefedev

Published Mar 1, 2016• 1 citation
Publisher

Abstract

The results of studies of 1D Ising models and Curie-Weiss models partition functions structure are presented in this work. Exact calculation of the partition function using the authors combinatorial approach for such system is discussed. The distribution of the energy levels degeneracy was calculated. Analytical solution for density of states of 1D Ising chain were obtained. Generating functions for these models were obtained. It was suggested that in Curie-Weiss model the transition to a low-energy state occurs without the formation of separation boundaries

DOI
10.4028/www.scientific.net/ssp.247.142

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The results of studies of 1D Ising models and Curie-Weiss models partition functions structure are presented in this work. Exact calculation of the partition function using the authors combinatorial approach for such system is discussed. The distribution of the energy levels degeneracy was calculated. Analytical solution for density of states of 1D Ising chain were obtained. Generating functions for these models were obtained. It was suggested that in Curie-Weiss model the transition to a low-energy state occurs without the formation of separation boundaries
The approach to determination of the order parameter in the finite 1D Ising model was proposed. For such systems a root mean square magnetization and an average size of the maximal ground state cluster which is formed by neighboring spins with an energy {-2J} does not disappear at T!=0. The temperature dependence of heat capacity and its derivative with respect to temperature from the average size of the maximum quasi ground state cluster, formed by the neighboring spins with an energy {-2J,0} in the 1D Ising chain were considered. The percolation approach in Curie-Weiss does not work due the internal structure of the model.
The partition functions of the finite Ising models of spin S =1, 3/2, 2, 5/2 and 3 have been calculated by a high speed computer. The periodicity condition at the boundaries has been imposed, and only the nearest neighbor interaction has been assumed. The specific heats of these finite systems have been calculated as a function of temperature. The maxima of the specific heat increase proportionally to long N in the two-dimensional Ising model with spin S =1, N being the number of lattice points, which indicates that the specific heat for the above lattice has a singularity such as C v / k ∼ A log | T - T c |+ B ± : A ∼0.10 and B - - B + =0.24. The difference ( B + - B - ) has been estimated from the asymmetric distribution of the zeros of the partition function in the complex temperature plane.
The problem of N interacting spins on a lattice is equivalent to one of N clusters linked in a specific manner. The energy of any configuration of spins can be expressed in terms of the energy levels of this cluster. A new expression is obtained for the probability of occurrence of any configuration. A closed form expression is obtained for the partition function per spin in terms of the energy levels of this cluster with the degeneracies being a function of temperature. On physical grounds it is suggested to be the form of the exact partition function per spin. The partition functions of Ising-like systems all have a common form. For the 3D Ising model seven functions need to be determined to describe the partition function completely. The key to understanding second order phase transitions and critical phenomena lies in the temperature dependence of various degeneracies. It is necessary to develop new techniques to determine the partition function that account for this temperature dependence, as it would represent the underlying physics correctly.
To study the relations between Yang-Lee's theory of the phase transitions and the distribution of the zeros of the partition functions in the antiferromagnetic case, the partition functions of finite (4×4 and 4×6) Ising model with the nearest and the next nearest neighbor interactions are calculated. The distributions of the zeros in the fugacity z plane of the partition function for various combinations of the values of J / k T - J '/ k T or tanh ( J /2 k T )-tanh ( J '/2 k T ) are investigated. The patterns of the distribution of the zeros show several types corresponding to ferromagnetic state, antiferromagnetic state and superantiferromagnetic state. In the antiferromagnetic ordered state the locus of the zeros is found to be nearly two concentric circles and to cross the positive real axis at two points z c and 1/ z c when the next neighbor ferromagnetic interaction is introduced. This means that our system shows the phase transitions at finite critical fields H c and - H c . The locus of zeros of Fisher's model of super-exchange antiferromagnet is also shown to cross the positive real axis at z c and 1/ z c below the Néel temperature.
Two different methods of obtaining the partition function for a spin ½ open ended Ising chain with interactions between nearest and second nearest neighbour spins are presented. These are non-matrix methods, in that they do not require the diagonalization of a transfer matrix, and they provide a useful representation for studying the analytical structure of the partition function. The pair correlation functions are also derived, using the second of these methods. These correlation functions are utilized in a study of the ground state and the finite temperature spin configurations. An interesting application of these results to the problem of out-of-phase domain structure in binary alloys is discussed.
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.