Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0108414 · Physical review. B, Condensed matter

Infinitesimal incommensurate stripe phase in an axial next-nearest-neighbor Ising model in two dimensions

Takashi Shirahata, Tota Nakamura

Published Dec 6, 2001• 28 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.65.024402

Related papers

Kosterlitz-Thouless phase transition of the axial next-nearest-neighbor Ising model in two dimensions

Takayuki Shirakura, Fumitaka Matsubara, Naoki Suzuki

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.

arXiv:1406.6464•Physical Review B•Oct 8, 2014•8 citations

Resolving the two-dimensional axial next-nearest-neighbor Ising model using transfer matrices

Yi Tao Hu, Patrick Charbonneau

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.

arXiv:2012.09747•Physical review. B./Physical review. B•Mar 25, 2021•8 citations

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

Akihiro Sato, Fumitaka Matsubara

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.

Physical review. B, Condensed matter•Oct 1, 1999•21 citations

Finite-size-scaling estimation of critical exponents in the two-dimensional axial next-nearest-neighbor Ising model

Marcelo D. Grynberg, Horacio Ceva

The magnetic critical exponent of the two-dimensional axial next-nearest-neighbor Ising model is estimated by means of the finite-size scaling of the order parameter. For a finite system without ordering field, this magnitude is computed from the two leading eigenvectors of the transfer matrix. The ferromagnetic-paramagnetic transition temperatures estimated with this method exhibit a small lattice-size dependence. Moreover, they are in good agreement with those obtained using phenomenological renormalization, and also with analytical approximations. The correlation-length critical exponent is calculated by means of phenomenological renormalization. The results found with these numerical techniques indicate that the ferromagnetic-paramagnetic transition of the model seems to be continuous, in contradiction with a previous result obtained using the persistence-length criterion, which indicated that the transition is always first order. We found that the phase transition is first order only when the ratio of the next-nearest-neighbor and nearest-neighbor interactions equals one-half.

Physical review. B, Condensed matter•Nov 1, 1988•6 citations

Finite-size scaling of two-dimensional axial next-nearest-neighbor Ising models

Paul D. Beale, Phillip M. Duxbury, Julia M. Yeomans

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.

Physical review. B, Condensed matter•Jun 1, 1985•44 citations

Nearest-neighbor Ising model with a uniaxial incommensurate phase and a Lifshitz point

Eytan Domany, Bernd Schaub

The anisotropic triangular nearest-neighbor Ising model, with antiferromagnetic interactions, is studied. The phase diagram, as a function of temperature and field, has a complex structure. Two commensurate ordered phases are found. The transition to one is Ising-type. The second is reached from the disordered phase in one of two ways: either via an intermediate uniaxial incommensurate phase---which is reached at high temperatures by a Kosterlitz-Thouless transition and at low temperatures undergoes an incommensurate-commensurate transition---or by a single continuous transition. The latter transition is in the same universality class as the chiral three-state Potts model, with thermal exponent within 4% of that of the three-state Potts model. These two transition regimes are found to be separated by a Lifshitz point. The phase diagram and critical behavior were determined by analytical (symmetry analysis and free-fermion approximation) as well as numerical (phenomenological renormalization-group and Monte Carlo) methods.

Physical review. B, Condensed matter•Apr 1, 1984•18 citations
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.
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.
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.
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 magnetic critical exponent of the two-dimensional axial next-nearest-neighbor Ising model is estimated by means of the finite-size scaling of the order parameter. For a finite system without ordering field, this magnitude is computed from the two leading eigenvectors of the transfer matrix. The ferromagnetic-paramagnetic transition temperatures estimated with this method exhibit a small lattice-size dependence. Moreover, they are in good agreement with those obtained using phenomenological renormalization, and also with analytical approximations. The correlation-length critical exponent is calculated by means of phenomenological renormalization. The results found with these numerical techniques indicate that the ferromagnetic-paramagnetic transition of the model seems to be continuous, in contradiction with a previous result obtained using the persistence-length criterion, which indicated that the transition is always first order. We found that the phase transition is first order only when the ratio of the next-nearest-neighbor and nearest-neighbor interactions equals one-half.
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 anisotropic triangular nearest-neighbor Ising model, with antiferromagnetic interactions, is studied. The phase diagram, as a function of temperature and field, has a complex structure. Two commensurate ordered phases are found. The transition to one is Ising-type. The second is reached from the disordered phase in one of two ways: either via an intermediate uniaxial incommensurate phase---which is reached at high temperatures by a Kosterlitz-Thouless transition and at low temperatures undergoes an incommensurate-commensurate transition---or by a single continuous transition. The latter transition is in the same universality class as the chiral three-state Potts model, with thermal exponent within 4% of that of the three-state Potts model. These two transition regimes are found to be separated by a Lifshitz point. The phase diagram and critical behavior were determined by analytical (symmetry analysis and free-fermion approximation) as well as numerical (phenomenological renormalization-group and Monte Carlo) methods.