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Physical review. B, Condensed matter

Comparison of models with competing interactions: Axial next-nearest-neighbor Ising versus two-plus-four spin interactions

M. Kolb, K. A. Penson

Published Mar 1, 1985• 9 citations
Publisher

Abstract

The phase diagrams of the one-dimensional quantum versions of the two-dimensional axial next-nearest-neighbor Ising (ANNNI) and the competing two-plus-four spin-interaction models are compared. The analysis of the zero-temperature wave-vector-dependent spectrum indicates a qualitative difference between the two models: while our finite-chain results support the existence of an incommensurate phase for the ANNNI case, they suggest that there is no such phase in the two-plus-four spin case.

DOI
10.1103/physrevb.31.3147

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Wave-vector-dependent spectra of quantum systems with competing interactions

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The analysis of phase diagrams of one-dimensional quantum versions of axial next-nearest-neighbor Ising (ANNNI) and competing two- and four-spin interaction models in two dimensions was made. The zero-temperature wave-vector-dependent spectra for finite chains with up to L≤16 spins indicate a qualitative difference between the two models: while the existence of an incommensurate phase is confirmed for the ANNNI case, our results do not suggest that such a phase exists in the two- and four-spin model.

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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 ${T}_{c1}$ 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 ${g}_{L}$ 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.

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Ising models with competing axial interactions in transverse fields

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The mean-field phase diagram of a three-dimensional axial-next-nearest-neighbor Ising (ANNNI) model in a transverse field has been studied here. The dynamical modes are found by using the random-phase approximation and the phase diagram is also obtained from the softening condition of these modes. For a one-dimensional ANNNI model in a transverse field, these dynamical modes are obtained here more accurately using a Jordan-Wigner-type transformation (with some approximations). Softening of these modes gives the zero-temperature phase diagram in one dimension.

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Journal of Physics A Mathematical and General•Jul 1, 1981•121 citations
The phase diagrams of the one-dimensional quantum versions of the two-dimensional axial next-nearest-neighbor Ising (ANNNI) and the competing two-plus-four spin-interaction models are compared. The analysis of the zero-temperature wave-vector-dependent spectrum indicates a qualitative difference between the two models: while our finite-chain results support the existence of an incommensurate phase for the ANNNI case, they suggest that there is no such phase in the two-plus-four spin case.
The analysis of phase diagrams of one-dimensional quantum versions of axial next-nearest-neighbor Ising (ANNNI) and competing two- and four-spin interaction models in two dimensions was made. The zero-temperature wave-vector-dependent spectra for finite chains with up to L≤16 spins indicate a qualitative difference between the two models: while the existence of an incommensurate phase is confirmed for the ANNNI case, our results do not suggest that such a phase exists in the two- and four-spin model.
A finite-size-scaling analysis is applied to two axial next-nearest-neighbor Ising (ANNNI) models. By using the scaling behavior of both the correlation length and the modulation wave vector we are able to clearly identify and distinguish between the ordered commensurate phases, the lock-in fluid phases, and in particular, the two types of incommensurate (floating) phases occurring in the models. As expected, the brickwork ANNNI model does not have a floating-solid phase, but instead has a q=(1/4) lock-in fluid phase above its 〈2〉 phase. On the basis of our results, we conclude that, in contrast to a recent suggestion, transfer-matrix methods can be a powerful tool in the study of two-dimensional models which exhibit incommensurate phases.
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.
The mean-field phase diagram of a three-dimensional axial-next-nearest-neighbor Ising (ANNNI) model in a transverse field has been studied here. The dynamical modes are found by using the random-phase approximation and the phase diagram is also obtained from the softening condition of these modes. For a one-dimensional ANNNI model in a transverse field, these dynamical modes are obtained here more accurately using a Jordan-Wigner-type transformation (with some approximations). Softening of these modes gives the zero-temperature phase diagram in one dimension.
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.