Phase transitions in the antiferromagnetic ising model on a square lattice with next-nearest-neighbor interactions
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DISORDER, AND PHASE TRANSITION IN CONDENSED SYSTEM
Phase Transitions in the Antiferromagnetic Ising Model on a Square Lattice with NextNearestNeighbor Interactions A. K. Murtazaeva,b, M. K. Ramazanova,c, F. A. KassanOglyd, and M. K. Badieva,* a
Institute of Physics, Dagestan Scientific Center, Russian Academy of Sciences, Makhachkala, 367003 Dagestan, Russia *email: [email protected] b Dagestan State University, Makhachkala, 367025 Dagestan, Russia c Dagestan State Pedagogical University, Makhachkala, 367003 Dagestan, Russia d Institute of Metal Physics, Ural Branch, Russian Academy of Sciences, ul. S. Kovalevskoi 18, Yekaterinburg, 620990 Russia Received June 17, 2013
Abstract—The phase transitions in the twodimensional Ising model on a square lattice are studied using a replica algorithm, the Monte Carlo method, and histogram analysis with allowance for the nextnearest neighbor interactions in the range 0.1 ≤ r < 1.0. A phase diagram is constructed for the dependence of the crit ical temperature on the nextnearestneighbor interaction. A secondorder phase transition is detected in this range and the model under study. DOI: 10.1134/S1063776113140021
INTRODUCTION To describe phase transitions (PTs) and critical phenomena (CP) quantitatively, the modern con densed matter physics uses various lattice models. Using theoretical methods and simple models, researchers were able to exactly solve a small number of problems. One of these models is the twodimen sional Ising model; with nearestneighbor interac tions, this model was thoroughly studied by various methods and approaches [1–4]. This model with nearestneighbor and nextnearestneighbor ferro magnetic interactions on a square lattice was also exactly solved. However, taking into account next nearestneighbor antiferromagnetic interactions in the classical Ising model is accompanied by the degen eration of the ground state and the appearance of var ious phases and PTs; in addition, nextnearestneigh bor interactions can also affect the critical behavior of the model (in particular, various anomalies in the crit ical properties appear) [5]. Late in the 1970s, the authors of [6, 7] were the first to perform renormalization group calculations and a Monte Carlo (MC) numerical simulation for the two dimensional antiferromagnetic Ising model on a square lattice with nextnearestneighbor interac tions. They assumed that a secondorder phase transi tion takes place in this model and calculated the phasetransition temperature and the critical indices. It was also shown in [8–12] that a secondorder phase transition occurs in the antiferromagnetic Ising model on a square lattice with nextnearestneighbor interactions. This model can also have “anomalous” critical indices. Moreover, the critical indices were
found to depend on the ratio r = J2/J1, where J1 and J2 are the exchange interaction constants of nearest and nextnearest neighbors, respectively. Neverthe less, the scenario of a continuous phase transition was disputable after meanfield
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