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arXiv:1704.01351

The Spectral Density of a Spin System Calculated for Solvable Models

Boris Kryzhanovsky

Published Apr 5, 2017• 2 citations
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Abstract

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.

DOI
10.48550/arxiv.1704.01351

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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 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 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 $\ensuremath{\delta}$.
We analyze the ground-state energy and correlation energy of the Heisenberg model as a function of spin, both in the ferromagnetic and the antiferromagnetic case, and in one, two and three dimensions. First, we present a comparative analysis of known expressions for the ground-state energy E0(S){E}_{0}(S)E0​(S) of homogeneous Heisenberg models. In the one-dimensional antiferromagnetic case we propose an improved expression for E0(S),{E}_{0}(S),E0​(S), which takes into account Bethe-ansatz data for S=1/2.S=1/2.S=1/2. Next, we consider inhomogeneous Heisenberg models (e.g., exposed to spatially varying external fields). We prove a Hohenberg-Kohn-like theorem stating that in this case the ground-state energy is a functional of the spin distribution and that this distribution encapsulates the entire physics of the system, regardless of the external fields. Building on this theorem, we then propose a local-density-type approximation that allows us to utilize the results obtained for homogeneous systems in inhomogeneous situations too. We conjecture a scaling law for the dependence of the correlation functional on dimensionality, which is well satisfied by existing numerical data. Finally, we investigate the importance of the spin-correlation energy by comparing results obtained with the proposed correlation functional to ones from an uncorrelated mean-field calculation, taking as our example a linear spin-density wave state.
In this work, the critical properties of the classical spin-1 Ising Hamiltonian applied to magnetic systems characterized by the first-neighbors biquadratic exchange, the anisotropy and the external magnetic field contributions are theoretically investigated. The first-neighbors bilinear exchange interaction is set equal to zero. For magnetic systems the bicubic exchange interaction must be set equal to zero as it would break the time-reversal invariance of the exchange Hamiltonian. To determine the critical behavior, the spin-1 Ising Hamiltonian is mapped onto the spin-1/2 Ising Hamiltonian by using the Griffith’s variable transformation. The critical surface of a 2D square magnetic lattice is determined in the parameter space as a function of the magnetic parameters and the phase transition occurring across it is quantitatively discussed by calculating, for each spin, the free energy and the magnetization. The free energy of the 2D square magnetic lattice, described via the three-state spin-1 Ising model, is obtained from an empirical expression of the partition function recently proposed for a spin-1/2 Ising model in an external magnetic field and applied to a 2D magnetic lattice. These results could pave the way to numerical simulations and to measurements able to confirm the analytical predictions.