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Journal of the Physical Society of Japan

Low Temperature Phase of a Dilute Axial-next-nearest-neighbor Ising Model in Two Dimensions

Fumitaka Matsubara, Nobuo Suzuki, Takayuki Shirakura

Published Nov 5, 2020• 0 citations
Publisher

Abstract

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 \((\kappa ,c)\) plain at low temperatures.

DOI
10.7566/jpsj.89.124001

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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 ((\kappa ,c)) plain at low temperatures.
The phase diagrams of the ferromagnetic two-dimensional mixed spin-1 and spin-1/2 Ising model with four-spin interaction J4 and next-nearest couplings J’ are investigated by the use of the finite cluster approximation based on single-site cluster theory. The state equations are derived for the two-dimensional square lattice. The phase diagram is qualitatively and quantitatively different from that obtained when the interaction J’ is ignored. In fact, the system exhibits a variety of interesting features resulting from the introduction of the next-nearest neighbor interaction (NNN). It undergoes two kinds of behavior according to the negative and the positive value of J’. In particular, for J4/J2=-4, the system keeps the coexistence of the two ground state up to J’-dependent finite temperature.
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 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 use Monte Carlo and transfer matrix methods in combination with extrapolation schemes to determine the phase diagram of the two-dimensional superantiferromagnetic (SAF) Ising model with next-nearest-neighbor (NNN) interactions in a magnetic field. The interactions between nearest-neighbor (NN) spins are ferromagnetic along x, and antiferromagnetic along Y. We find that for sufficiently low temperatures and fields, there exists a region limited by a critical line of second-order transitions separating a SAF phase from a magnetically induced paramagnetic phase. We did not find any region with either first-order transition or with reentrant behavior. The NNN couplings produce either an expansion or a contraction of the SAF phase. Expansion occurs when the interactions are antiferromagnetic, and contraction when they are ferromagnetic. There is a critical ratio R(c)=1/2 between NNN and NN couplings, beyond which the SAF phase no longer exists.
The ground state of an Ising-type spin-1/2 chain with ferromagnetic bond-alternating nearest-neighbor and anti-ferromagnetic uniform next-nearest-neighbor interactions is studied by using the exact-diagonalization method and the density-matrix renormalization-group method. The Hamiltonian describing the system is expressed as H = – Σi h2i–1,2i – J1 Σi h2i,2i+1 + J2 Σi hi,i+2 with hi,i' = γ(Six Si'x + Siy Si'y) + Siz Si'z, where J1 0, J2 0, and 1 > γ 0. Special attention is paid to the ground-state phase diagram on the J1 versus J2 plane for a given value of γ. The phase diagram is composed of the ferromagnetic, intermediate, and up-up-down-down phases, the intermediate phase being characterized by its magnetization, which takes finite but unsaturated values. The phase diagram obtained for γ = 0.5 shows that the region of the intermediate phase for a given value of J1 is widest when J1 = 1.0 and becomes narrower rather rapidly as J1 decreases or increases from 1.0. The J2-dependence of the ground-state magnetization for γ = 0.5 and J1 = 0.85 is also discussed. PACS Nos.: 75.10Jm, 75.40Mg