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Physica Scripta

Uniaxial Anisotropy and Next-Nearest-Neighbour Interaction Effects in the 2D Ising Model

L.L. Gonçalves

Published Feb 1, 1987• 2 citations
Publisher

Abstract

An Ising model consisting of mixed spins of magnitudes s and 1/2, on the honeycomb lattice is considered. The spins of a given magnitude occupy one of the two equivalent sublattices of the honeycomb lattice and the next-nearest-neighbour interactions are allowed between the spins 1/2 only. The exact solution of the problem, in the presence of uniaxial anisotropy, is obtained by mapping the model onto an effective spin 1/2 Ising model on a triangular lattice. The transition temperature as a function of the parameters of the Hamiltonian is explicitly determined when the second neighbour interaction is allowed in one direction or two directions of the sublattice occupied by the spins 1/2. It is shown that in the first case, frustration effects are the dominant mechanism for suppression of the ground state long range order, irrespective of s being integer or half integer. The important feature in the second case is the appearance of three transition temperatures for some values of the parameters, and this will imply in a reentrant critical surface. Explicit results are worked out for s = 1 and s = 3/2, which present the main features of the model. By using a general result for decorated lattices the critical exponents are determined for arbitrary s .

DOI
10.1088/0031-8949/35/2/002

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An Ising model consisting of mixed spins of magnitudes s and 1/2, on the honeycomb lattice is considered. The spins of a given magnitude occupy one of the two equivalent sublattices of the honeycomb lattice and the next-nearest-neighbour interactions are allowed between the spins 1/2 only. The exact solution of the problem, in the presence of uniaxial anisotropy, is obtained by mapping the model onto an effective spin 1/2 Ising model on a triangular lattice. The transition temperature as a function of the parameters of the Hamiltonian is explicitly determined when the second neighbour interaction is allowed in one direction or two directions of the sublattice occupied by the spins 1/2. It is shown that in the first case, frustration effects are the dominant mechanism for suppression of the ground state long range order, irrespective of s being integer or half integer. The important feature in the second case is the appearance of three transition temperatures for some values of the parameters, and this will imply in a reentrant critical surface. Explicit results are worked out for s = 1 and s = 3/2, which present the main features of the model. By using a general result for decorated lattices the critical exponents are determined for arbitrary s .
The effect of the uniaxial anisotropy in an Ising model consisting of mixed spins of magnitudes s and 1/2, on the honeycomb lattice, is studied. The exact solution is determined by mapping the model onto an effective spin 1/2 Ising model on the triangular lattice. The transition temperature is obtained as a function of the parameters and it is shown that for s integer there is no long range order for the parameter values in a determined region which is independent of the value of s. The critical exponents are obtained and it is shown that they are the same as those of the two-dimensional spin 1/2 Ising model.
The Ising model on the honeycomb lattice with spins of magnitudes 1/2 and s with random exchange constant and random uniaxial anisotropy is considered. The two cases are studied separately, and in both cases binary annealed distributions of the random parameters are considered. An exact solution is obtained by mapping the model onto the uniform spin 1/2 Ising model on the triangular lattice, and the transition temperature is determined as a function of the parameters for arbitrary s. Explicit results are presented for s=1 and s=3/2, and it is shown that some critical parameters, surprisingly, are not universal.
In this work the ordering processes in classical frustrated Ising models and the ground-state behavior of frustrated quantum spin models are investigated. In the first part the phase transition from a high-temperature paramagnetic phase to a antiferromagnetic ground state is analyzed for the square-lattice Ising model with competing antiferromagnetic interactions between nearest- and next-nearest-neighbor spins. In the case of spatially isotropic interactions the phase transition to a collinear state is of special interest. Using Monte-Carlo simulations on finite lattices as well as field-theoretical methods for the continuous model it is shown that depending on the frustration three different sectors can be distinguished that show (i) Ising-universal behavior (ii) Ashkin-Teller-like non-universal behavior (varying critical exponents) and (iii) non-critical first-order behavior. For spatially anisotropic interactions the emergence of an intermediate incommensurate phase is verified by means of Monte-Carlo simulations which has been predicted by earlier mean-field calculations. The nature of this phase is studied analyzing spin-spin correlation functions and its ordering wave-vector reveals a strong temperature and size dependence. In the second part the influence of quantum fluctuations on classical antiferromagnetic ground states are investigated in anisotropic quantum spin-1/2 Heisenberg models with competing interactions. Using quantum Monte-Carlo simulations, exact diagonalization and perturbation theory the ground-state phase diagrams are calculated for a square lattice with nearest- and next-nearest-neighbor interactions and a honeycomb lattice with up to third-nearest-neighbor interactions whereby the fluctuations are chosen to be non-frustrating. For both cases the stability regions of classical magnetic states are identified and an intermediate phase is detected that shows no sign of long-range magnetic order. Further calculations also rule out valence-bond-crystal or topological order for this state, thus a quantum disordered ground state is stabilized in these models.
Magnetic behaviour of a mixed spin-1/2 and spin-1 Ising model on the diced lattice is studied by the use of an exact star-triangle mapping transformation. It is found that the uniaxial as well as biaxial single-ion anisotropy acting on the spin-1 sites may potentially cause a reentrant transition with two consecutive critical points. Contrary to this, the effect of next-nearest-neighbour interaction between the spin-1/2 sites possibly leads to a reentrant transition with three critical temperatures in addition to the one with two critical points only. The shape of the total magnetization versus temperature dependence is particularly investigated for the case of ferrimagnetically ordered system.
Abstract The static critical behavior of a 2D decorated ferromagnetic Ising model on a triangular lattice has been investigated by the methods of computational physics. For nodal spins, ferromagnetic exchange interaction between the first nearest neighbors and antiferromagnetic interaction between the next nearest neighbors have been taken into account. Exchange interaction Jd between nodal and decorated spins varies within [–3.0, 3.0]. For different values of Jd, the ground state of the model has been determined, critical temperatures have been calculated, and a phase diagram has been constructed. It has been shown that the magnitude of exchange interaction with decorated spins influences the type of phase transition to the paramagnetic state.
The critical behaviour of the square-lattice Ising model, with nearest and next-nearest neighbour interactions of either sign, has been investigated by means of high-temperature series. lhe location of the critical lines in the coupling constant plane has been accurately determined. Along the critical line which corresponds to transitions to the layered or superantiferromagnetic state a breakdown of universality is observed and explicit numerical estimates obtained for the exponent of the ordering susceptibility.