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Journal of the Physical Society of Japan

Spin-One Ising Model with Nearest-Neighbour and Next-Nearest-Neighbour Interactions

Motoyuki Tanaka, Takeshi Kawabe

Published Oct 1, 1983• 10 citations
Publisher

Abstract

Phase transitions of the spin-one Ising model with nearest-neighbour and next-nearest-neighbour interactions are investigated for a square lattice using a molecular-field theory and a Monte Carlo method. Special attention is paid to the effects of the biquadratic next-nearest-neighbour interaction on the phase transitions of the model. Phase diagrams characterizing types of phase transition are obtained for several values of the coupling constants. The existence of the multi-critical points and the successive phase transitions are also discussed in connection with these phase diagrams.

DOI
10.1143/jpsj.52.3325

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Phase transitions of the spin-one Ising model with nearest-neighbour and next-nearest-neighbour interactions are investigated for a square lattice using a molecular-field theory and a Monte Carlo method. Special attention is paid to the effects of the biquadratic next-nearest-neighbour interaction on the phase transitions of the model. Phase diagrams characterizing types of phase transition are obtained for several values of the coupling constants. The existence of the multi-critical points and the successive phase transitions are also discussed in connection with these phase diagrams.
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.
Monte Carlo simulations are used to study the location and nature of phase boundaries for Ising square lattices with antiferromagnetic coupling JNN{J}_{\mathrm{NN}}JNN​ between nearest neighbors and additional interactions JNNN{J}_{\mathrm{NNN}}JNNN​ between next-nearest neighbors and J3NN{J}_{3\mathrm{N}\mathrm{N}}J3NN​ between third-nearest neighbors. Results in zero magnetic field are obtained for a wide range of R=JNNN/JNN{J}_{\mathrm{NNN}/{J}_{\mathrm{NN}}}JNNN/JNN​​ and R'=J3NN/JNN{J}_{3\mathrm{N}\mathrm{N}/{J}_{\mathrm{NN}}}J3NN/JNN​​. In addition to the c(2\ifmmode\times\else\texttimes\fi{}2) and (2\ifmmode\times\else\texttimes\fi{}1) phases, which also occur for R'=0, we find new (4\ifmmode\times\else\texttimes\fi{}4) and (4\ifmmode\times\else\texttimes\fi{}2) ordered states, for R'\ensuremath{\ne}0, which are separated from the disordered state by lines of first-order transitions. The nonuniversal critical behavior of the (2\ifmmode\times\else\texttimes\fi{}1) phase is studied using the block-distribution method and finite-size scaling. The possible existence of incommensurate phases is also explored.
Abstract. We use improved Monte-Carlo algorithms to study the antiferromagnetic 2D-Ising model with competing interactions J1 on nearest neighbour and J2 on next-nearest neighbour bonds. The finite-temperature phase diagram is divided by a critical point at J2 = J1/2 where the groundstate is highly degenerate. To analyse the phase boundaries we look at the specific heat and the energy distribution for various ratios of J2/J1. We find a first order transition for small J2> J1/2 and the transition temperature suppressed to TC = 0 at the critical point. 1.