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Physical review. B, Condensed matter

Surface energy and free-boson approximation in a competing multispin model

Marcelo D. Grynberg, Horacio Ceva

Published Oct 1, 1989• 2 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.40.7265

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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.
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- ($\ensuremath{\sim}{J}_{2}$) 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 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.
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.
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.
A transfer-matrix representation in terms of Pauli spin matrices is used to perform a phenomenological renormalization and finite-size scaling analysis of two-dimensional spin models with competing interactions in one direction. This enables us to obtain the leading eigenvalues of the transfer matrix with a significative saving of computer time using the Lanczos scheme. The phase diagrams of the axial next-nearest-neighbor Ising model and a model with competing two- and four-spin couplings in one direction are studied. The order of the phase transitions was determined by examining the finite-size scaling behavior of the persistence length. Data for the normalized scaled persistence length display scaling on the ferromagnetic-paramagnetic transition line for both models.
The magnetic critical exponent of the two-dimensional axial next-nearest-neighbor Ising model is estimated by means of the finite-size scaling of the order parameter. For a finite system without ordering field, this magnitude is computed from the two leading eigenvectors of the transfer matrix. The ferromagnetic-paramagnetic transition temperatures estimated with this method exhibit a small lattice-size dependence. Moreover, they are in good agreement with those obtained using phenomenological renormalization, and also with analytical approximations. The correlation-length critical exponent is calculated by means of phenomenological renormalization. The results found with these numerical techniques indicate that the ferromagnetic-paramagnetic transition of the model seems to be continuous, in contradiction with a previous result obtained using the persistence-length criterion, which indicated that the transition is always first order. We found that the phase transition is first order only when the ratio of the next-nearest-neighbor and nearest-neighbor interactions equals one-half.