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Physical review. B, Condensed matter

Competing multispin interactions

K. A. Penson

Published Mar 1, 1984• 19 citations
Publisher

Abstract

A generalized ($D+1$)-dimensional anisotropic Ising model with a combination of multispin interactions in its $D$ dimensions is introduced. A (1 + 1)-dimensional case ($D=1$) with competing two- ($\ensuremath{\sim}{J}_{2}$) and four-spin ($\ensuremath{\sim}{J}_{4}$) couplings was analyzed through its one-dimensional Hamiltonian version with the use of the finite-size scaling method. Whereas for small $\frac{{J}_{4}}{{J}_{2}}$ the transition is of the usual Ising type, for sufficiently large $\frac{{J}_{4}}{{J}_{2}}$ a first-order phase transition occurs. A comparison with other models with competing interactions was made.

DOI
10.1103/physrevb.29.2404

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A generalized (D+1D+1D+1)-dimensional anisotropic Ising model with a combination of multispin interactions in its DDD dimensions is introduced. A (1 + 1)-dimensional case (D=1D=1D=1) with competing two- (\ensuremath∼J2\ensuremath{\sim}{J}_{2}\ensuremath∼J2​) and four-spin (\ensuremath∼J4\ensuremath{\sim}{J}_{4}\ensuremath∼J4​) couplings was analyzed through its one-dimensional Hamiltonian version with the use of the finite-size scaling method. Whereas for small J4J2\frac{{J}_{4}}{{J}_{2}}J2​J4​​ the transition is of the usual Ising type, for sufficiently large J4J2\frac{{J}_{4}}{{J}_{2}}J2​J4​​ a first-order phase transition occurs. A comparison with other models with competing interactions was made.
An Ising-type model including nnn-spin nearest-neighbor interactions is introduced. The ground-state properties of its Hamiltonian version are studied in one and two dimensions. The model is self-dual in one dimension. The mean-field theory indicates that for n>ncn>{n}_{c}n>nc​ the transition may become first order. Finite-size scaling suggests that nc=4{n}_{c}=4nc​=4 in one dimension. The critical exponents as functions of nnn are estimated.
The phase diagram and critical behavior of a two-dimensional system with competing multispin interactions, known as the two-plus-four model, are studied by means of simple approximate analytical methods. Using the M\"uller-Hartmann and Zittartz approximation, the interface free-energy calculation in the ferromagnetic region is reduced to a single-particle problem. We found acceptable agreement with earlier finite-size scaling results. The model is also considered in the Hamiltonian limit, where a linearized Holstein-Primakoff transformation is used to replace spin interactions by a set of independent Bose excitations. We conjecture that the phase transition is first order when the ratio of the four-spin and nearest-neighbor interactions is greater than one-half. Standard perturbative calculations for low temperatures also indicate that the model could display first-order phase transitions.
We study spin-S Ising models with p-spin interactions on the one-dimensional chain and the two-dimensional square lattice. Here, S denotes the magnitude of the spin and p represents the number of spins involved in each interaction. The analysis is performed for S=1/2,1,3/2,2, and p=3,4,5. For the one-dimensional model, we formulate transfer matrices and numerically diagonalize them to analyze the temperature dependence of the free energy and spin-spin correlations. In the case of S=1/2, the free energy does not depend on p, and the spin-spin correlations are uniformly enhanced across all temperature scales as p increases. In contrast, for S≥1, the free energy varies with p, and the spin-spin correlations are significantly enhanced at lower temperatures as p increases. For the two-dimensional model, by using multicanonical simulations, we analyze physical quantities such as an order parameter, internal energy, and specific heat. In addition, we define and examine an order parameter to distinguish ordered and disordered phases. It is found that a first-order phase transition occurs at finite temperatures for all S and p≥3, and increasing p strengthens its nature. We present S and p dependence of the transition temperature and latent heat, and discuss effects of higher-order interactions on the nature of phase transitions.
Abstract The two-dimensional anisotropic Ising model with competing interactions in the region of a transition from the ferromagnetic to paramagnetic phase has been studied by the Monte Carlo method using the Wang–Landau algorithm. State density distributions have been obtained and the order parameters corresponding to different energies have been calculated. The temperature dependences of the thermodynamic parameters including the internal energy, magnetization, free energy, entropy, specific heat at constant magnetization, and Binder cumulants have been calculated. Using the Binder cumulant method and histogram analysis, it has been shown that a single second-order phase transition from the ferromagnetic to paramagnetic phase occurs at two exchange coupling constants ratios |J1/J| = 0.1 and 0.2 in the two-dimensional anisotropic next-nearest neighbor Ising model. The phase transition temperatures have been calculated. The main critical exponents have been determined using the finite-size scaling theory.
Ordering of a one-dimensional stochastic Ising model evolving by a combination of spin flips and spin exchanges is investigated. The spin-flip rates satisfy detailed balance for the equilibrium state of the Ising model at temperature T, while the spin exchanges are random and of arbitrary range. Analytical methods and Monte Carlo simulations are used to show that, depending on the details of the spin-flip rate, finite-temperature phase transition may or may not occur in the system. When ordering occurs, it is of mean-field type and the scaling function describing the finite-size scaling of the magnetization fluctuations is found to be indistinguishable from that of an equilibrium Ising model with infinite-range interaction.