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arXiv:2407.03782 · Physical review. E

Frustration in the Ising model on a decorated square lattice

Felix A. Kassan-Ogly, Alexander V. Zarubin

Published Apr 23, 2025• 2 citations
arXivHTMLPDF

Abstract

In this paper, we performed the comprehensive studies of magnetic frustration properties in the Ising model on a decorated square lattice in the framework of an exact analytical approach based on the Kramers-Wannier transfer matrix method. The assigned challenge consisted in finding rigorous formulas for residual entropies of all possible cases of degeneration of the system without exception and also elucidating the influence of frustrations on the behavior of the thermodynamic functions of the system under discussion. The accomplished results turned out to be very rich and intricate, especially in comparison with the original undecorated square lattice, in which frustration properties are completely absent. Such an abundance of properties is due to the multiparametric problem of decorated lattice with four exchange interactions J_{x}, J_{y}, J_{dx}, and J_{dy} taking on arbitrary continuous values and different signs: positive (ferromagnetic) and negative (antiferromagnetic), as well as with two discrete variables d_{x} and d_{y} (multiplicities of decorating spins in both lattice directions), corresponding to infinite natural series of numbers.

DOI
10.1103/physreve.111.044133

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In this paper, we performed the comprehensive studies of magnetic frustration properties in the Ising model on a decorated square lattice in the framework of an exact analytical approach based on the Kramers-Wannier transfer matrix method. The assigned challenge consisted in finding rigorous formulas for residual entropies of all possible cases of degeneration of the system without exception and also elucidating the influence of frustrations on the behavior of the thermodynamic functions of the system under discussion. The accomplished results turned out to be very rich and intricate, especially in comparison with the original undecorated square lattice, in which frustration properties are completely absent. Such an abundance of properties is due to the multiparametric problem of decorated lattice with four exchange interactions J_{x}, J_{y}, J_{dx}, and J_{dy} taking on arbitrary continuous values and different signs: positive (ferromagnetic) and negative (antiferromagnetic), as well as with two discrete variables d_{x} and d_{y} (multiplicities of decorating spins in both lattice directions), corresponding to infinite natural series of numbers.
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.
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.
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.
We study the spin-1/2 Ising-XXZ model on a decorated honeycomb lattice composed of five spins per unit cell, one Ising spin, and four Heisenberg spins. This model involving the Heisenberg exchange interaction is one of the few models that can be exactly solvable through the generalized star-triangle transformation. The significance of this model is its close relationship to the fully decorated quantum Heisenberg honeycomb lattice since 4/5 of the particles are Heisenberg spins. We investigate the phase diagram at zero temperature and identify a relevant quantum spin frustrated phase resulting from the contribution of quantum Heisenberg exchange interaction. We obtain an exact residual entropy for the quantum spin frustrated phase, which coincides with the residual entropy of the antiferromagnetic spin-1/2 Ising model on a triangular lattice. We also thoroughly explore its thermodynamic properties, focusing mainly on the frustrated region such as entropy, specific heat, spontaneous magnetization, and critical temperature under several conditions.
Abstract The frustration properties of the Ising model on a one-dimensional monoatomic equidistant lattice are investigated taking into account the exchange interactions of atomic spins at the sites of the first (nearest), second (next-nearest) and third neighbors. The exact solution of the model was obtained using the Kramers–Wannier transfer matrix method. The types of magnetic ordering of the ground state of the model are determined, and a magnetic phase diagram is constructed. Criteria are formulated for the occurrence of magnetic frustrations in the presence of competition between the energies of exchange interactions. Non-zero entropy values in the ground state of the frustrated system were found.
Abstract The Ising model on a one-dimensional monoatomic equidistant lattice with different nearest-neighbour and second-neighbour exchange interactions is researched. Generalized Kramers-Wannier transfer-matrix with translation on two periods of a lattice is introduced. A property similar to supercooling and superheating is detected. At the triple points phases are not individualized, but completely frustrated which corresponds to the phenomenon of critical opalescence. Exact analytical expressions for free energy, heat capacity and entropy including zero-temperature entropy are obtained. Various new special cases were analyzed and compared with all known results. All frustration fields for magnetization, frustration values for the zero-temperature entropy and magnetization are found.