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Physical review. B, Condensed matter

Interacting-fermion approximation in the two-dimensional axial next-nearest-neighbor Ising model

Marcelo D. Grynberg, Horacio Ceva

Published Jun 1, 1991• 4 citations
Publisher

Abstract

We investigate the effect of including doman-wall interactions in the two-dimensional axial next-nearest-neighbor Ising model. At low temperatures this problem is reduced to a one-dimensional system of interacting fermions which can be treated exactly. It is found that the critical boundaries of the low-temperature phases are in good agreement with those obtained using a free-fermion approximation. In contrast with the monotonic behavior derived from the free-fermion approach, the wall density or wave number displays reentrant phenomena when the ratio of the next-nearest-neighbor and nearest-neighbor interactions is greater than 1/2.

DOI
10.1103/physrevb.43.13630

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We investigate the effect of including doman-wall interactions in the two-dimensional axial next-nearest-neighbor Ising model. At low temperatures this problem is reduced to a one-dimensional system of interacting fermions which can be treated exactly. It is found that the critical boundaries of the low-temperature phases are in good agreement with those obtained using a free-fermion approximation. In contrast with the monotonic behavior derived from the free-fermion approach, the wall density or wave number displays reentrant phenomena when the ratio of the next-nearest-neighbor and nearest-neighbor interactions is greater than 1/2.
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.
We study thermodynamic properties of an axial next-nearest-neighbor Ising model in two dimensions using a cluster heat bath Monte Carlo method. The method drastically reduces the relaxation time, and enables us to study equilibrium properties of the model. We show that the model really exhibits two phase transitions at Tc1{T}_{c1}Tc1​ and Tc2(<Tc1),{T}_{c2}(<{T}_{c1}),Tc2​(<Tc1​), as predicted by a free fermion approximation. The one at Tc1{T}_{c1}Tc1​ is shown to be of the Kosterlitz-Thouless type, and the other to be of the Pokrovsky-Talapov type. The higher transition temperature Tc1{T}_{c1}Tc1​ is considerably lower than those estimated by previous authors. Low-temperature properties as well as the lower transition temperature Tc2{T}_{c2}Tc2​ are found to be well described by the fermion approximation.
We consider an axial Ising model with competing interactions up to third neighbors within a low-temperature approximation in terms of interacting and nonreversing domain walls. Assuming small third-neighbor coupling, we find a close relation between the transfer matrix of this classical model and the Hamiltonian of an S=1/2 anisotropic Heisenberg chain in a uniform magnetic field. This connection allows us to obtain the phase boundaries of the ferromagnetic and other low-temperature modulated states. The ground-state energy for arbitrary magnetization values of the Heisenberg chain is shown to provide information about the density of walls or wave number. In contrast with the two-dimensional axial next-nearest-neighbor Ising model, this quantity displays plateaus and reentrant behavior for some relations among the coupling constants. An alternative way to evaluate correlation functions in the antiferromagnetic Heisenberg chain is presented.
The next-nearest-neighbor two-dimensional Ising model is cast into a form which resembles a one-dimensional interacting many-fermion system. An approximation, which has previously been shown to be successful in giving the critical properties of the two-dimensional ferroelectric problem, is used. From the approximate expression obtained, the critical indices are found to be α = α′ = 0, β = ⅛, which agrees with the results obtained from series expansions and plausible physical arguments. The critical temperature obtained agrees to within 6% of the series expansion results.
We investigate the dynamics of a two-dimensional axial next-nearest-neighbor Ising model following a quench to zero temperature. The Hamiltonian is given by H= -J_(0) summation operator(L)_(i,j=1)S_(i,j)S_(i+1,j)-J_(1)summation operator_(i,j=1)(S_{i,j}S_{i,j+1}-kappaS_{i,j}S_{i,j+2}) . For kappa<1 , the system does not reach the equilibrium ground state but slowly evolves to a metastable state. For kappa>1 , the system shows a behavior similar to that of the two-dimensional ferromagnetic Ising model in the sense that it freezes to a striped state with a finite probability. The persistence probability shows algebraic decay here with an exponent theta=0.235+/-0.001 while the dynamical exponent of growth z=2.08+/-0.01 . For kappa=1 , the system belongs to a completely different dynamical class; it always evolves to the true ground state with the persistence and dynamical exponent having unique values. Much of the dynamical phenomena can be understood by studying the dynamics and distribution of the number of domain walls. We also compare the dynamical behavior to that of a Ising model in which both the nearest and next-nearest-neighbor interactions are ferromagnetic.