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Physical Review Letters

Low temperature expansions around the multiphase point of an hcp Ising model with competing interactions

Dan S. Zimmerman, Catherine Kallin, A. J. Berlinsky

Published May 4, 1987• 4 citations
Publisher

Abstract

A low-temperature expansion is derived for the ferromagnetic phase boundary of an hcp Ising model with competing interactions, near its multiphase point, to second order in x=exp(-2D/T), where D is the ferromagnetic interplane coupling. The coefficient of ${\mathrm{x}}^{2}$ is a slowly converging two dimensionally infinite sum. The resulting phase boundary coincides to order ${\mathrm{x}}^{2}$ with the line onto which Domany mapped a two-dimensional kinetic Ising model. This lends support to the conjecture of Domany and Gubernatis that this special line coincides with the exact ferromagnetic phase boundary.

DOI
10.1103/physrevlett.58.1881

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A low-temperature expansion is derived for the ferromagnetic phase boundary of an hcp Ising model with competing interactions, near its multiphase point, to second order in x=exp(-2D/T), where D is the ferromagnetic interplane coupling. The coefficient of x2{\mathrm{x}}^{2}x2 is a slowly converging two dimensionally infinite sum. The resulting phase boundary coincides to order x2{\mathrm{x}}^{2}x2 with the line onto which Domany mapped a two-dimensional kinetic Ising model. This lends support to the conjecture of Domany and Gubernatis that this special line coincides with the exact ferromagnetic phase boundary.
A study is made of an Ising model on the hexagonal closed-packed lattice, with ferromagnetic interactions D between nearest-neighbor spins located in adjacent layers, and antiferromagnetic interactions J between nearest-neighbor spins located in the same layer. The ground states of the model are studied for different values of the parameter κ = —J/D. For κ < 1/2 the ground states are ferromagnetic and for κ > 1/2 the ground state spin configurations consist of stacked identical layers, such that each layer is obtained by stacking rows of alternating spins. At the point (κ — 1/2,t = 0), where t = T/D, there exists a multitude of degenerate ground state spin configurations which are not stable for κ ≠ 1/2. Mean-field theory and low temperature expansions are used to study the phase diagram at low temperatures. Mean-field theory predicts that (κ= 1/2, t = 0) is a multiphase point where an infinite sequence of modulated phases coincide. In the vicinity of the multiphase point, the mean-field phase diagram is found to be similar to the mean-field phase diagram of the three-dimensional ANNNI model near its multiphase point. Low temperature expansions are performed to second order in x, where x = e [sup -2/t], around the phase boundary between the ferromagnetic and the modulated phases. In contrast to standard low temperature expansions, the complete contribution, to order x², is obtained by grouping the contributions from excitations which contribute to arbitrarily high orders in x. The phase boundary between the ferromagnetic and the modulated phases is found to coincide, to order x², with the line onto which Domany mapped a kinetic Ising model on the honeycomb lattice. This strongly suggests that the Domany line is a phase boundary in three-dimensions. Mean-field theory shows that this Ising model contains a continuous minimum-energy surface. A renormalization group method which applies to models which contain continuous minimum-energy surfaces is used to analyze the phase transition between the paramagnetic and the modulated phases. The calculation is performed using a Landau-Ginzburg-Wilson Hamiltonian whose minimum-energy surface consists of a hexagon and which contains fourth-order invariants due to the lattice. The calculation shows that the Hamiltonian does not contain a stable fixed point. This suggests that the paramagnetic-modulated phase transition of this Ising model is a fluctuation-induced first-order transition.
The interface method is used to derive the order-disorder critical temperature of the two-dimensional Ising model on a rectangular lattice with ferromagnetic interactions in the x direction and antiferromagnetic interactions in the y direction. Employing this approximation in two different ways gives different critical temperatures Tc{T}_{c}Tc​ for nonzero uniform magnetic fields H. However, both of the phase boundaries Tc{T}_{c}Tc​(H) reduce to the exact results available at H=0 and T=0. .AE
We have studied the phase diagrams of the one-dimensional spin-1 Blume-Capel model with anisotropy constant D, in which equivalent-neighbor ferromagnetic interactions of strength -J are superimposed on nearest-neighbor antiferromagnetic interactions of strength K. A rich critical behavior is found due to the competing interactions. At zero temperature two ordered phases exist in the D/J-K/J plane, namely the ferromagnetic (F) and the antiferromagnetic one (AF). For lower values of D/J(D/J<0.25) these two ordered phases are separated by the point K_{c}=0.25J. For 0.25<D/J≤0.50, the paramagnetic phase P emerges in a region separated between the lines determined by D/J=0.5-K/J and D/J=K/J. For D/J>0.5, only phases AF and F exist and are separated by a line given by D/J=K/J. At finite temperatures, we found that the ferromagnetic region of the phase diagram in the k_{B}T/J-D/J plane is enriched by another ferromagnetic phase F^{^{'}} above a first-order line for 0.195<K/J<0.250. This first-order line, which separates phases F and F^{^{'}}, begins at a coexistence point, where phases F,F^{^{'}}, and P coexist, and ends at an ordered critical point. Similarly, we found that the phase F^{^{'}} is present in the phase diagram in the k_{B}T/J-K/J plane for 0.228<D/J<0.286.
The two-dimensional axial Ising model with competing interactions up to third neighbors is studied by analytical methods. A low-temperature approximation is used to describe the system as a sequence of interacting and nonreversing domain walls for small values of the third-neighbor interaction. The boundaries of the ferromagnetic and other low-temperature modulated phases are obtained by means of the ground-state energy of the Heisenberg-Ising chain after reducing the problem to a system of spinless interacting fermions in one dimension. The density of walls or wave number obtained within this approximation results to be continuously varying and for some relations among the coupling constants it displays plateaus and re-entrant phenomena.
The square Ising ferromagnet is investigated by computer simulation in the region of competing interactions (nearest-neighbor interaction K >0 and second-neighbor one K '<0). Spontaneous magnetization M 0 and energy E are calculated from advanced Monte Carlo calculations. The estimates of M 0 and of the Curie point T c ( K '/ K ) determined by analyzing the data on E are compared with the results from analysis of low-temperature series based on the assumption β=1/8. The values of T c ( K '/ K ) are obtained from padé-approximant method for low-temperature series. The value of β does not vary from the Onsager's value 1/8.
We study the nonequilibrium phase diagram and critical properties of a two-dimensional kinetic Ising model with competing Glauber and Kawasaki dynamics suggested by Tome and de Oliveira [Phys. Rev. A 40, 6643 (1989)]. The role of the Kawasaki dynamics, chosen with probability 1-p, is to simulate a permanent energy flux into the system. The theoretical prediction for the phase diagram is improved significantly by using four- and six-point dynamical mean-field approximations. Monte Carlo simulations support that the ferromagnetic-paramagnetic phase transition changes from second to first order for sufficiently small p. The antiferromagnetic phase is found to be stable for a nonzero value of p even at T=0.