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

Communications in Theoretical Physics

Ising Model on an Infinite Ladder Lattice

Gao Xing-Ru, Yang Zhan-Ru

Published Sep 1, 2007• 1 citation
Publisher

Abstract

In this paper we propose an Ising model on an infinite ladder lattice, which is made of two infinite Ising spin chains with interactions. It is essentially a quasi-one-dimessional Ising model because the length of the ladder lattice is infinite, while its width is finite. We investigate the phase transition and dynamic behavior of Ising model on this quasi-one-dimessional system. We use the generalized transfer matrix method to investigate the phase transition of the system. It is found that there is no nonzero temperature phase transition in this system. At the same time, we are interested in Glauber dynamics. Based on that, we obtain the time evolution of the local spin magnetization by exactly solving a set of master equations.

DOI
10.1088/0253-6102/48/3/032

Related papers

Glauber Dynamics for One-dimensional Spin Models with Random Fields

Gabor Forgács, David Mukamel, Robert A. Pelcovits

We present an exact solution of the long-time relaxational behavior of the magnetization in the Ising and $\mathrm{XY}$ chains in a quenched random field. The random field is assumed to be of infinite strength but present at only a fraction of the spin sites. We use Glauber dynamics for the Ising case, and a suitably generalized master equation for the $\mathrm{XY}$ case. In both models we find that for $T=0$, as time $t\ensuremath{\rightarrow}\ensuremath{\infty}$ the magnetization decays as exp (${\ensuremath{-}Ct}^{\frac{1}{3}}$), where $C$ is a constant. At finite temperatures the ultimate asymptotic behavior is purely exponential.

Physical review. B, Condensed matter•Jul 1, 1984•24 citations

Spontaneous Magnetization in the Finite-Size Two-Leg Ising Ladder in the External Magnetic Field

Vardan Apinyan

This paper considers a finite-sized two-leg ladder of spin-1/2 particles with local inter-chain Ising interaction. We include the effects of an external magnetic field by incorporating a Zeeman-type coupling between the magnetic field and the spins localized on the lattice sites. To analyze the system, we utilize the transfer-matrix formalism to calculate the partition function. We compute the magnetization for different configurations by varying the system's parameters. We observe the presence 3of spontaneous magnetization within the system. As the inter-chain Ising interaction changes its sign, the magnetization value changes from negative to positive values. Furthermore, we identify a phase transition from diamagnetic to ferromagnetic states when examining the temperature dependence of the magnetization at zero magnetic fields. We observe a step-like behavior in the magnetization curve in the low temperature limit.

Recent Progress in Materials•Jun 24, 2023•0 citations

Dynamics of the spin-1/2 Ising two-leg ladder with four-spin plaquette interaction and transverse field

Wescley Luiz de Souza, Érica de Mello Silva, P. H. L. Martins

Multiple-spin coupling is a key tool for investigating magnetic quantum systems. In particular, models with four-spin interactions became of great interest since they have been applied to describe material properties such as superconductivity. In this framework, the dynamics of the spin-1/2 Ising two-leg ladder with four-spin plaquette interaction in a transverse field is investigated. By means of the recurrence relations method, the first four recurrants are determined exactly for the general model, while five exact recurrants are calculated for two particular models. In all cases, higher-order recurrants are computed through extrapolation to obtain time-dependent autocorrelation functions and spectral densities in the long-time regime and infinite-temperature limit. It is found that the relaxation functions decay slower than the expected behavior of the one-dimensional transverse Ising model. The spectral lines, characterized by a Lorentzian behavior in the central body and a Gaussian shape in the tails, display exchange narrowing as the coupling intensities increase.

Physical review. E•Apr 3, 2020•19 citations

Effect of antiferromagnetic exchange interactions on the Glauber dynamics of one-dimensional Ising models

M. G. Pini, A. Rettori

We study the effect of antiferromagnetic interactions on the single spin-flip Glauber dynamics of two different one-dimensional Ising models with spin $\ifmmode\pm\else\textpm\fi{}1$. The first model is an Ising chain with antiferromagnetic exchange interaction limited to nearest neighbors and subject to an oscillating magnetic field. The system of master equations describing the time evolution of sublattice magnetizations can easily be solved within a linear field approximation and a long time limit. Resonant behavior of the magnetization as a function of temperature (stochastic resonance) is found, at low frequency, only when spins on opposite sublattices are uncompensated owing to different gyromagnetic factors (i.e., in the presence of a ferrimagnetic short range order). The second model is the axial next-nearest-neighbor Ising (ANNNI) chain, where an antiferromagnetic exchange between next-nearest neighbors is assumed to compete with a nearest-neighbor exchange interaction of either sign. The long time response of the model to a weak, oscillating magnetic field is investigated in the framework of a decoupling approximation for three-spin correlation functions, which is required to close the system of master equations. The calculation, within such an approximate theoretical scheme, of the dynamic critical exponent $z$, defined as $1∕\ensuremath{\tau}\ensuremath{\approx}{(1∕\ensuremath{\xi})}^{z}$ (where $\ensuremath{\tau}$ is the longest relaxation time and $\ensuremath{\xi}$ is the correlation length of the chain), suggests that the $T=0$ single spin-flip Glauber dynamics of the ANNNI chain is in a different universality class than that of the unfrustrated Ising chain.

arXiv:0707.1953•Physical Review B•Aug 6, 2007•14 citations

On the dynamics of the Ising model (master equation/spin correlation functions/hierarchy of correlation e

Elliott W. Montroll

A two-dimensional (and to some degree three-di- mensional) version of Glauber's one-dimensional relaxation model is described. The model is constructed to yield the Ising model of cooperative phenomena at equilibrium. A complete hi- erarchy of differential equations for multispin correlation func- tions is constructed. Some remarks are made concerning the solu- tion of them for the initial value problem of determining the relaxation of an initial set of distributions. Investigations of the equilibrium thermodynamic properties of the Ising model have expanded our intuition on the general na- ture of phase transitions. They have also provided an accurate representation of the behavior of certain real systems well char- acterized by the model. Dynamical models of cooperative phe- nomena have proven harder to develop. Glauber (1) introduced an interesting master equation for the dynamics of the Ising spin variables on a one-dimensional chain. This equation is constructed so that the spin-flipping process at a given time is determined by two competing influences: (i) temperature ef- fects which at higher temperatures tend to randomize the sys- tem, and (ii) correlation effects of a given with its neighbors which tend to align a given with its neighbors. Although some literature has been generated for the investigation of two- dimensional and three-dimensional generalizations of the Glauber model (cf. Kawasaki (2) and Matsudaira (3, 4)), a de- tailed analysis of the appropriate hierarchy of rate equations for the dynamics of correlation functions has not yet been given. The purpose of this report is to make a contribution to such a program.

•Jan 1, 2017•0 citations

Low-Temperature Properties of Infinite-Spin Ising Model in One-Dimensional Lattice

Tsuyoshi Horiguchi

We investigate the low-temperature behaviors of the spin-pair correlation functions at large distances, the correlation length and the zero-field susceptibility for the infinite-spin Ising model in a one-dimensional lattice. By solving an integral equation, we find that the divergent behavior of the correlation length is weaker than the one for the finite-spin Ising models.

Journal of the Physical Society of Japan•Sep 15, 1990•11 citations
In this paper we propose an Ising model on an infinite ladder lattice, which is made of two infinite Ising spin chains with interactions. It is essentially a quasi-one-dimessional Ising model because the length of the ladder lattice is infinite, while its width is finite. We investigate the phase transition and dynamic behavior of Ising model on this quasi-one-dimessional system. We use the generalized transfer matrix method to investigate the phase transition of the system. It is found that there is no nonzero temperature phase transition in this system. At the same time, we are interested in Glauber dynamics. Based on that, we obtain the time evolution of the local spin magnetization by exactly solving a set of master equations.
We present an exact solution of the long-time relaxational behavior of the magnetization in the Ising and XY\mathrm{XY}XY chains in a quenched random field. The random field is assumed to be of infinite strength but present at only a fraction of the spin sites. We use Glauber dynamics for the Ising case, and a suitably generalized master equation for the XY\mathrm{XY}XY case. In both models we find that for T=0T=0T=0, as time $t\ensuremath{\rightarrow}\ensuremath{\infty}$ the magnetization decays as exp (${\ensuremath{-}Ct}^{\frac{1}{3}}$), where CCC is a constant. At finite temperatures the ultimate asymptotic behavior is purely exponential.
This paper considers a finite-sized two-leg ladder of spin-1/2 particles with local inter-chain Ising interaction. We include the effects of an external magnetic field by incorporating a Zeeman-type coupling between the magnetic field and the spins localized on the lattice sites. To analyze the system, we utilize the transfer-matrix formalism to calculate the partition function. We compute the magnetization for different configurations by varying the system's parameters. We observe the presence 3of spontaneous magnetization within the system. As the inter-chain Ising interaction changes its sign, the magnetization value changes from negative to positive values. Furthermore, we identify a phase transition from diamagnetic to ferromagnetic states when examining the temperature dependence of the magnetization at zero magnetic fields. We observe a step-like behavior in the magnetization curve in the low temperature limit.
Multiple-spin coupling is a key tool for investigating magnetic quantum systems. In particular, models with four-spin interactions became of great interest since they have been applied to describe material properties such as superconductivity. In this framework, the dynamics of the spin-1/2 Ising two-leg ladder with four-spin plaquette interaction in a transverse field is investigated. By means of the recurrence relations method, the first four recurrants are determined exactly for the general model, while five exact recurrants are calculated for two particular models. In all cases, higher-order recurrants are computed through extrapolation to obtain time-dependent autocorrelation functions and spectral densities in the long-time regime and infinite-temperature limit. It is found that the relaxation functions decay slower than the expected behavior of the one-dimensional transverse Ising model. The spectral lines, characterized by a Lorentzian behavior in the central body and a Gaussian shape in the tails, display exchange narrowing as the coupling intensities increase.
We study the effect of antiferromagnetic interactions on the single spin-flip Glauber dynamics of two different one-dimensional Ising models with spin $\ifmmode\pm\else\textpm\fi{}1$. The first model is an Ising chain with antiferromagnetic exchange interaction limited to nearest neighbors and subject to an oscillating magnetic field. The system of master equations describing the time evolution of sublattice magnetizations can easily be solved within a linear field approximation and a long time limit. Resonant behavior of the magnetization as a function of temperature (stochastic resonance) is found, at low frequency, only when spins on opposite sublattices are uncompensated owing to different gyromagnetic factors (i.e., in the presence of a ferrimagnetic short range order). The second model is the axial next-nearest-neighbor Ising (ANNNI) chain, where an antiferromagnetic exchange between next-nearest neighbors is assumed to compete with a nearest-neighbor exchange interaction of either sign. The long time response of the model to a weak, oscillating magnetic field is investigated in the framework of a decoupling approximation for three-spin correlation functions, which is required to close the system of master equations. The calculation, within such an approximate theoretical scheme, of the dynamic critical exponent zzz, defined as $1∕\ensuremath{\tau}\ensuremath{\approx}{(1∕\ensuremath{\xi})}^{z}$ (where $\ensuremath{\tau}$ is the longest relaxation time and $\ensuremath{\xi}$ is the correlation length of the chain), suggests that the T=0T=0T=0 single spin-flip Glauber dynamics of the ANNNI chain is in a different universality class than that of the unfrustrated Ising chain.
A two-dimensional (and to some degree three-di- mensional) version of Glauber's one-dimensional relaxation model is described. The model is constructed to yield the Ising model of cooperative phenomena at equilibrium. A complete hi- erarchy of differential equations for multispin correlation func- tions is constructed. Some remarks are made concerning the solu- tion of them for the initial value problem of determining the relaxation of an initial set of distributions. Investigations of the equilibrium thermodynamic properties of the Ising model have expanded our intuition on the general na- ture of phase transitions. They have also provided an accurate representation of the behavior of certain real systems well char- acterized by the model. Dynamical models of cooperative phe- nomena have proven harder to develop. Glauber (1) introduced an interesting master equation for the dynamics of the Ising spin variables on a one-dimensional chain. This equation is constructed so that the spin-flipping process at a given time is determined by two competing influences: (i) temperature ef- fects which at higher temperatures tend to randomize the sys- tem, and (ii) correlation effects of a given with its neighbors which tend to align a given with its neighbors. Although some literature has been generated for the investigation of two- dimensional and three-dimensional generalizations of the Glauber model (cf. Kawasaki (2) and Matsudaira (3, 4)), a de- tailed analysis of the appropriate hierarchy of rate equations for the dynamics of correlation functions has not yet been given. The purpose of this report is to make a contribution to such a program.