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Journal of Experimental and Theoretical Physics

Generalized Ising Model in a Magnetic Field

E. S. Tsuvarev, Felix A. Kassan-Ogly

Published Aug 1, 2021• 3 citations
Publisher

Abstract

Abstract Results on the generalization of the Ising model to an arbitrary number of translations of a linear chain in an external magnetic field, taking into account various interactions between spins, are presented. An exact analytical expression has been derived for the largest eigenvalue of the Kramers–Wannier transfer matrix with a translation per two chain periods in an external magnetic field when the nearest and second neighbors are taken into account. Exact expressions have been established for the zero-temperature entropies and magnetizations at various magnitudes and signs of the exchange interactions and the magnetic field. Many of the zero-temperature entropies and zero-temperature magnetizations found are represented via the so-called mathematical ratios long known in the wonderful world of number mathematics (the golden ratio φ, the silver ratio δ, the supergolden ratio ψ, the plastic number ρ, and new (nameless) ones). A result whereby the frustrating entropies and magnetizations can be expressed via the limit of the ratio of certain number sequences without invoking the formalism of the Kramers–Wannier transfer matrix has been obtained.

DOI
10.1134/s1063776121080112

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arXiv:2004.02427•Apr 6, 2020•1 citation
Abstract Results on the generalization of the Ising model to an arbitrary number of translations of a linear chain in an external magnetic field, taking into account various interactions between spins, are presented. An exact analytical expression has been derived for the largest eigenvalue of the Kramers–Wannier transfer matrix with a translation per two chain periods in an external magnetic field when the nearest and second neighbors are taken into account. Exact expressions have been established for the zero-temperature entropies and magnetizations at various magnitudes and signs of the exchange interactions and the magnetic field. Many of the zero-temperature entropies and zero-temperature magnetizations found are represented via the so-called mathematical ratios long known in the wonderful world of number mathematics (the golden ratio φ, the silver ratio δ, the supergolden ratio ψ, the plastic number ρ, and new (nameless) ones). A result whereby the frustrating entropies and magnetizations can be expressed via the limit of the ratio of certain number sequences without invoking the formalism of the Kramers–Wannier transfer matrix has been obtained.
Abstract The Kramers–Wannier transfer matrix method is generalized to an arbitrary decoration number of the Ising chain. An exact analytical expression is obtained for the largest eigenvalue of the transfer matrix of a decorated Ising chain in the presence of an external magnetic field. Frustration points and the values of frustration magnetic fields are found that depend on the magnitudes and signs of exchange interactions. Exact expressions are obtained for zero-temperature entropies and zero-temperature magnetizations of the model under consideration. Magnetic phase diagrams of the ground state of the system are constructed for decoration values of d = 1 and d = 2, including those in the absence of a magnetic field. A comparison is made with a decorated square lattice not only in the absence but also in the presence of a magnetic field.
Abstract The Ising model on an one-dimensional monoatomic equidistant lattice with different exchange interactions between atomic spins at the sites of nearest neighbors and different interactions between atomic spins at the sites of second neighbors is investigated. The generalized Kramers–Wannier transfer matrix with translation on two periods of a lattice is derived. The phase diagrams representing all possible magnetic types of ordering in the ground state are constructed. Many triple points of intersection of phases and lines of pair coexistence of phases in strict accordance with the Gibbs phase rule are found and a property similar to supercooling and superheating is detected. It is shown that at triple points the phases are not individualized but significantly frustrated, which corresponds to the phenomenon of critical opalescence. Exact analytical expressions for the Gibbs free energy, as well as for such thermodynamic characteristics of the system as internal energy, heat capacity and entropy including zero-temperature entropy are derived. A variety of particular cases are analyzed and a comparison is made with all known results including on a two-dimensional lattice.
Using the Kramers–Wannier transfer matrix method, we studied several decorated Ising chains. The exact expressions for thermodynamic characteristics, including the ground state characteristics, were obtained. We considered several modeling chains with different signs and absolute values of exchange constants for the nearest- and the next-nearest neighbors. For these models, we calculated the magnetization curves. The critical values of magnetic fields and corresponding magnetization plateau parameters were obtained. Analytic expressions for the ground state entropy were obtained for the chains with different interaction constants. The dependencies of the number of states with minimum energy (the degeneration of the ground state) as the function of the number of particles were found. It was shown that these dependencies are expressed in terms of well-known numerical sequences, namely Lucas numbers and Pell numbers, which, in the limit of a large number of particles, are proportional to the powers of the golden and silver sections. Therefore, the ground state entropy (per particle) of the systems under consideration can be described in terms of these sections and, therefore, is nonzero.
Using the Kramers-Wannier transfer matrix method we studied several decorated Ising chains. The exact expressions for thermodynamic characteristics, including the ground state characteristics, were obtained. We considered a number of modeling chains with different signs and absolute values of exchange constants for the nearest- and the next-nearest neighbors. For these models we calculated the magnetization curves. The critical values of magnetic fields and corresponding magnetization plateau parameters were obtained. Analytic expressions for the ground state entropy were obtained for the chains with different interaction constants. The dependencies of the number of states with minimum energy (the degeneration of the ground state) as the function of the number of particles were found. It was shown that these dependencies are expressed in terms of well-known numerical sequences - Lucas numbers and Pell numbers, which, in the limit of a large number of particles, are proportional to the powers of the golden and silver sections. Therefore, the ground state entropy (per particle) of the systems under consideration can be described in terms of these sections and, therefore, is nonzero.
For generalized 2D Ising model in an external magnetic field with the interaction of nearest neighbors, next nearest neighbors, all kinds of triple interactions and the quadruple interaction the formulas for finding free energy per lattice site in the thermodynamic limit were derived on a certain set of exact disordered solutions depending on seven parameters. Lattice models are considered with boundary conditions with a shift (similar to helical ones), and with cyclic closure of the set of all points in natural ordering. The elementary transfer matrix with nonnegative matrix elements are constructed. On the set of disorder solutions the largest eigenvalue of the transfer matrix is constant for every size of considering planar lattice, and, in particular, in the thermodynamic limit. Free energy per lattice site in the thermodynamic limit is expressed through the natural logarithm of the largest eigenvalue of transfer matrix. This largest eigenvalue can be found for a special form of eigenvector with positive components. The numerical example show the existence of nontrivial solutions of the resulting systems of equations. The system of equations and the value of free energy in the thermodynamic limit will remain the same for 2D generalized Ising models with Hamiltonians, in which the values of two (out of four) neighboring maximal spins in the natural ordering are replaced by the values of the spins at any other two lattice points adjacent in the natural ordering, this significantly expands the set of models having disordered exact solutions. The high degree of symmetry and inductive construction of the components of the eigenvectors, which disappear when going beyond the framework of the obtained set of exact solutions, is an occasion to search for phase transitions in the vicinity of this set of disordered solutions.