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Doklady Mathematics

Features of the Spectral Density of a Spin System

Boris Kryzhanovsky

Published Mar 1, 2018• 6 citations
Publisher

Abstract

The relationship between the spectral density and the free energy of a spin system is analyzed. Analytical expressions for calculating the spectral density for exactly solvable models are derived. The approach is tested as applied to the one-dimensional Ising model. Additionally, the approach is used to analyze the spectral density of the two-dimensional Ising model, the Bethe-lattice model, and the mean-field model. It is shown that even a small change in the spectral density is able to radically change the parameters of the system.

DOI
10.1134/s1064562418020126

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Physical review. B, Condensed matter•Feb 1, 1992•84 citations
The relationship between the spectral density and the free energy of a spin system is analyzed. Analytical expressions for calculating the spectral density for exactly solvable models are derived. The approach is tested as applied to the one-dimensional Ising model. Additionally, the approach is used to analyze the spectral density of the two-dimensional Ising model, the Bethe-lattice model, and the mean-field model. It is shown that even a small change in the spectral density is able to radically change the parameters of the system.
The relationship between the spectral density and free energy of a spin system is considered. The analytical expressions allowing for the calculation of the spectral density for solvable models are determined. A linear Ising model is taken for testing the approach. Additionally, the approach is used to analyze the spectral density of a 2D Ising model, a Bethe-lattice model and a mean-field model. Even a small change of the spectral density is shown to be able to radically change the parameters of the system.
The partition function of a thermodynamic system of N spins can be calculated with the aid of its spectral density, which characterizes the degeneracy of each energy level. We compare two approximations for the spectral density, which are the approximation by polynomials [1] and the nv-approximation presented earlier by us [2]. The approximation by polynomials provides the results that contradict the fundamental physical principles, namely, the system reaches the ground state at a finite temperature. In addition, this approximation fails to predict correctly the phase transition type. The nv-approximation is free from these defects. Furthermore, it allows us to calculate the critical temperature, which deviates from the commonly accepted value by less than 3%.
An approximation for calculating the thermodynamic properties of spin-\textonehalf{} Ising models is described. The method is based on the formulation of Kramers and Wannier for the square lattice, and involves expressing the free energy in terms of the limiting maximum eigenvalue of a matrix sequence which represents the contribution from a single spin of the lattice. The application of this formulation to various two- and three-dimensional lattices shows that many of these are characterized by an identically structured matrix sequence with exactly two nonvanishing elements in each row and column. To explore the potential of using this similarity and its simplicity for analyzing Ising models we consider the two-dimensional square lattice in an external field and approximate the doubly infinite lattice by a semi-infinite one consisting of $n (\ensuremath{\le}10)$ infinitely long rows. Calculations are carried out and compared with (1) Onsager's formula for the free energy, (2) the Onsanger-Yang formula for the spontaneous magnetization, and (3) values calculated from series. Except for the region around the critical point where the correlation length exceeds nnn lattice spacings, substantial agreement---to better than 12 places of decimals in some cases---is obtained. As an additional measure of the potential of this method, a value for the critical exponent $\ensuremath{\delta}$ is computed by the extrapolation of the magnetization along the critical isotherm. The result is found to be in excellent agreement with the known value for $\ensuremath{\delta}$.
Собственные значения гамильтониана многомерной модели Изинга выражаются через константы меж-спинового взаимодействия и собственные значения одномерного гамильтониана. Для периодических граничных условий точные аналитические выражения могут быть получены практически для произвольного дальнодействия. For the Hamiltonian of multidimensional Ising system we express the eigenvalues in terms of the spin-spin interaction constants and the well-known eigenvalues of the one-dimensional spin Hamiltonian. In the case of periodic boundary conditions, we can obtain exact analytic expressions for eigenvalues of the multidimensional Hamiltonian for an arbitrary long-range interaction.
The recently introduced concept of exact mean-field theories is applied to the extended simplified Hubbard model, which is a one-band extended Hubbard model (including nearest-neighbor interaction) where only the \ensuremath{\downarrow} electrons can hop. An exact mean-field Hamiltonian is derived for the model on a Bethe lattice in the limit of a large coordination number (Z\ensuremath{\rightarrow}\ensuremath{\infty}). The mean-field Hamiltonian is used to calculate the Green functions and the free energy at half-filling. This exact solution is analyzed analytically at small and large values of the on-site (U) and nearest-neighbor (V) interaction strengths. One finds that a phase transition occurs at sufficiently low temperatures for all U,V>0. The low-temperature phase is a charge-density wave for V>1/2U and a spin-density wave for V1/2U. Special attention is paid to (i) the critical temperature as a function of U and V, (ii) the order parameter as a function of temperature, and (iii) the density of states. The density of states shows that the low-temperature phase is an insulator for all U,V provided that V/U\ensuremath{\ne}1/2, while the high-temperature phase is a paramagnetic metal for all V if U2, and a paramagnetic insulator if U>2. Many detailed results are given.