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Physical Review E

Solution of the antiferromagnetic Ising model on a tetrahedron recursive lattice

E. Jurčišinová, Marian Jurčisin

Published Mar 20, 2014• 27 citations
Publisher

Abstract

We consider the antiferromagnetic spin-1/2 Ising model on the recursive tetrahedron lattice on which two elementary tetrahedrons are connected at each site. The model represents the simplest approximation of the antiferromagnetic Ising model on the real three-dimensional tetrahedron lattice which takes into account effects of frustration. An exact analytical solution of the model is found and discussed. It is shown that the model exhibits neither the first-order nor the second-order phase transitions. A detailed analysis of the magnetization of the model in the presence of the external magnetic field is performed and the existence of the magnetization plateaus for low temperatures is shown. All possible ground states of the model are found and discussed. The existence of nontrivial singular ground states is proven and exact explicit expressions for them are found.

DOI
10.1103/physreve.89.032123

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We consider the antiferromagnetic spin-1/2 Ising model on the recursive tetrahedron lattice on which two elementary tetrahedrons are connected at each site. The model represents the simplest approximation of the antiferromagnetic Ising model on the real three-dimensional tetrahedron lattice which takes into account effects of frustration. An exact analytical solution of the model is found and discussed. It is shown that the model exhibits neither the first-order nor the second-order phase transitions. A detailed analysis of the magnetization of the model in the presence of the external magnetic field is performed and the existence of the magnetization plateaus for low temperatures is shown. All possible ground states of the model are found and discussed. The existence of nontrivial singular ground states is proven and exact explicit expressions for them are found.
We consider the antiferromagnetic spin-1/2 Ising model with multisite interaction in an external magnetic field on an infinite zigzag ladder. The model is solved exactly by using the transfer matrix method. Using the exact expression for the total magnetization per site, the magnetic properties of the model are investigated in detail. The model exhibits the formation of magnetization plateaus for low temperatures, and it is shown that their properties depend strongly on the strength of the multisite interaction. All possible ground states of the model are found and discussed. The existence of nontrivial singular ground states is proven and exact explicit expressions for them are found. The macroscopic degeneracy of the ground states is investigated and discussed.
Abstract. The antiferromagnetic spin-1/2 Ising model with the presence of the multisite interaction and in the external magnetic field on the infinite tetrahedral chain is solved exactly by using the transfer matrix method. Using the exact expression for the free energy per site of the model the thermodynamic properties of the model are investigated in detail. It is shown that the total magnetisation per site forms the magnetisation plateaus at low temperatures, the properties of which depend on the strength of the multisite interaction. The set of all possible ground states is found and discussed. The existence of the so-called single-point ground states is shown and their properties are studied in detail. The entropy per site of the model is investigated and the exact expressions for the residual entropies of all ground states are found and their macroscopic degeneracies are discussed. The existence of the ground states with the same value of magnetisation but with different entropies is observed. In addition, the behaviour of the susceptibility and the specific heat is also studied.
The spin-3/2 Ising model is investigated for the case of antiferromagnetic (AFM/AFM) interactions on the two-layer Bethe lattice by using the exact recursion relations in the pairwise approach for given coordination numbers q = 3, 4 and 6 when the layers are under the influences of equal external magnetic and equal crystal fields. The ground state (GS) phase diagrams are obtained on the different planes in detail and then the temperature-dependent phase diagrams of the system are calculated accordingly. It is observed that the system presents both second- and first-order phase transitions for all q , therefore, tricritical points. It is also found that the system exhibits double-critical end points and isolated points. The model also presents two Néel temperatures, T N , and the existence of which leads to the reentrant behaviour.
Two-layer Bethe lattice whose lattice sites are occupied with spin-3/2 atoms is solved exactly by using the recursion relations in a pairwise approach for given coordination numbers q = 3, 4 and 6 with equal external magnetic fields acting on the layers.The ferromagnetic (FM) and antiferromagnetic (AFM) interactions for the spins of the upper and lower layers, respectively, and either FM or AFM type interactions between the adjacent spins of the layers are assumed.The phase diagrams of the model are studied on different planes for given system parameters by obtaining the ground state (GS) phase diagrams and the thermal variations of the order parameters and the response functions, i. e. the susceptibility and the specific heat, in detail.It was found that the model presents both second-and first-order phase transitions.The reentrant behavior is seen when the model presents two N éel temperatures for higher q values.The existence of the tricritical point and critical end points is also confirmed.
The phase transitions in two-dimensional antiferromagnetic Ising model are studied on a square lattice by taking the interactions of next-nearest neighbors into account. The model is investigated on basis of the replica Monte Carlo algorithm and the histogrammic analysis of data. The diagram of the critical temperature dependence on an interaction value of next-nearest neighbors is plotted. The studied model reveals the phase transition of second order and the frustration points depending on relative interaction values and magnetic field. A effect, heat capacity splitting near the frustration points, is found.
The phase transitions and thermodynamics of the antiferromagnetic system with the presence of the next-to-nearest-neighbor interaction on the body-centered cubic lattice are studied within an exactly solvable model using the recursive lattice approach. It is shown that, besides the N\'eel and collinear antiferromagnetic phases and the paramagnetic phase discussed in the literature, the model clearly exhibits the existence of an intermediate phase that separates the antiferromagnetic phases. This new phase is observed in a narrow interval of the frustration parameter for nonzero temperatures and is realized in the form of a discrete (single-point-like) ground state in the zero-temperature limit. Transitions between all model phases are studied and it is shown that all transitions between each antiferromagnetic phase of the model and the paramagnetic one as well as between the antiferromagnetic phases and the intermediate phase have the second-order phase transition nature. The magnetization, entropy, and specific heat capacity properties of all phases of the model are studied in detail. Three different ground states are identified and their macroscopic degeneracies are determined. The anomalous Schottky-type behavior of the specific heat capacity in the vicinity of the highly macroscopically degenerated ground state realized for exactly given value of the frustration parameter is observed and discussed. Besides, the existence of the intermediate phase leads to the nontrivial temperature behavior of the specific heat capacity where, in addition to the Schottky-type anomaly related to the frustration, three discontinuous jumps related to the three consecutive second-order phase transitions between corresponding model phases are observed.