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Theoretical and Mathematical Physics

Thermodynamical Averages For The Ising Model And Spectral Invariants Of Toeplitz Matrices

Виталий Маркович Каплицкий

Published Jun 1, 2020• 0 citations
Publisher

Abstract

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 → ∞

DOI
10.1134/s0040577920060069

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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 → ∞
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 partition function of a three-dimensional Ising model was calculated by a high-speed computer for a cubic array of twenty-seven points. The periodicity condition was imposed. The specific heat of the 3×3×3 lattice was calculated as a function of temperature. The critical temperature estimated from the specific heat curve seems slightly lower that obtained from the Padé approximation. The zeros of the partition function of the complex temperature plane were also obtained. The critical temperature estimated from the zeros near the positive real axis seems slightly higher than both of the above values. Similar calculations were carried out for the case of the two-dimensional Ising model. Several results are compared with those obtained from Onsager's exact solution.
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.
In this paper, we consider the one-dimensional Ising model (shortly, 1D-MSIM) having mixed spin-(s,(2t−1)/2) with nearest neighbors and the external magnetic field. We establish the partition function of the model by means of the transfer matrix. Under a null boundary condition, we compute certain thermodynamic quantities for the 1D-MSIM. We find some precise formulas to determine the model’s free energy, entropy, magnetization, and susceptibility. By examining the model’s associated iterative equations, we use the cavity approach to investigate the phase transition problem. We numerically determine the model’s periodicity.
Abstract We reconsider the problem of a one‐dimensional Ising model with an arbitrary nearest‐neighbor random exchange integral, temperature, and random magnetic field in each site. A convenient formalism is developed that reduces the partition function to a recurrence equation, which is convenient both for numerical as well as for analytical approaches. We have calculated asymptotic expressions for an ensemble averaged free energy and the averaged magnetization in the case of strong and weak couplings in external constant magnetic field. With a random magnetic field at each site in addition to nearest‐neighbor random exchange integrals we also evaluated the free energy. We show that the zeros of the partition function for the Ising model in the complex external magnetic field plane formally coincide with the singularities of the real part of electron's transmission amplitude through the chain of δ‐function potentials.
In contrast to an infinite chain, the low-temperature expansion of a one-dimensional free-field Ising model has a strong dependence on boundary conditions. I derive an explicit formula for the leading term of the expansion both under open and periodic boundary conditions and show that they are related to different distributions of the partition function zeros on the complex temperature plane. In particular, when a periodic boundary condition is imposed, the leading coefficient of the expansion grows with increasing size of the chain, due to the zeros approaching the origin.