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Physical review. B, Condensed matter

Equilibrium properties of an axial next-nearest-neighbor Ising model in two dimensions

Akihiro Sato, Fumitaka Matsubara

Published Oct 1, 1999• 21 citations
Publisher

Abstract

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 ${T}_{c1}$ and ${T}_{c2}(<{T}_{c1}),$ as predicted by a free fermion approximation. The one at ${T}_{c1}$ is shown to be of the Kosterlitz-Thouless type, and the other to be of the Pokrovsky-Talapov type. The higher transition temperature ${T}_{c1}$ is considerably lower than those estimated by previous authors. Low-temperature properties as well as the lower transition temperature ${T}_{c2}$ are found to be well described by the fermion approximation.

DOI
10.1103/physrevb.60.10316

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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.
The spin structure of an axial next-nearest-neighbor Ising (ANNNI) model in two dimensions (2D) is a renewed problem because different Monte Carlo (MC) simulation methods predicted different spin orderings. The usual equilibrium simulation predicts the occurrence of a floating incommensurate (IC) Kosterlitz-Thouless (KT) type phase, which never emerges in non-equilibrium relaxation (NER) simulations. In this paper, we first examine previously published results of both methods, and then investigate a higher transition temperature Tc1{T}_{c1}Tc1​ between the IC and paramagnetic phases. In the usual equilibrium simulation, we calculate the chain magnetization on larger lattices (up to $512\ifmmode\times\else\texttimes\fi{}512$ sites) and estimate ${T}_{c1}\ensuremath{\approx}1.16J$ with frustration ratio $\ensuremath{\kappa}(\ensuremath{\equiv}\ensuremath{-}{J}_{2}/{J}_{1})=0.6$. We examine the nature of the phase transition in terms of the Binder ratio gL{g}_{L}gL​ of spin overlap functions and the correlation-length ratio $\ensuremath{\xi}/L$. In the NER simulation, we observe the spin dynamics in equilibrium states by means of an autocorrelation function and also observe the chain magnetization relaxations from the ground and disordered states. These quantities exhibit an algebraic decay at $T\ensuremath{\lesssim}1.17J$. We conclude that the two-dimensional ANNNI model actually admits an IC phase transition of the KT type.
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.
We examine the ground state of a dilute two-dimensional axial-next-nearest-neighbor Ising model with the interaction ratio κ (≡ −J2/J1) > 1/2. We calculate the spin overlap function Q together with the antiphase (〈2〉 phase) magnetization M〈2〉 on medium lattices of L × L (L ≤ 24) using a cluster heat bath method. The 〈2〉 phase is destroyed by a small quantity of non-magnetic atoms due to a random impurity effect, and a critical phase is realized, which is composed of segments of four types of 〈2〉 domains arranged in order. As the spin concentration c decreases, a critical phase with different ferromagnetic clusters appears near above the percolation threshold cp of the lattice. A phase diagram is proposed in the (κ,c)(\kappa ,c)(κ,c) plain at low temperatures.
Some features of the phase diagram of the two-dimensional axial next-nearest-neighbor Ising model have long been debated. The extended structural correlations and long relaxation times associated with its Kosterlitz-Thouless phase indeed result in analytical and numerical treatments making contradictory predictions. Here, we introduce a numerical transfer matrix approach that bypasses these problems and thus clears up various ambiguities. In particular, we confirm the transition temperatures and the order of the transition to the floating incommensurate phase. Our approach motivates considering transfer matrices for solving long-standing problems in related models.
An axial next-nearest-neighbor Ising model is studied by using the nonequilibrium relaxation method. We find that the incommensurate stripe phase between the ordered phase and the paramagnetic phase is negligibly narrow or may vanish in the thermodynamic limit. The phase transition is the second-order transition if approached from the ordered phase, and it is of the Kosterlitz-Thouless type if approached from the paramagnetic phase. Both transition temperatures coincide with each other within the numerical errors. The incommensurate phase which has been observed previously is a paramagnetic phase with a very long correlation length (typically \ensuremath{\xi}\ensuremath{\geqslant}500). We could resolve this phase by treating very large systems (\ensuremath{\sim}6400\ifmmode\times\else\texttimes\fi{}6400), which is first made possible by employing the present method.