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

One-dimensional Ising model with long-range and random short-range interactions

A. P. Vieira, L. L. Gonçalves

Published Jan 1, 2008• 4 citations
OpenAlex

Abstract

The one-dimensional Ising model in an external magnetic field with uniform long-range interactions and random short-range interactions satisfying bimodal annealed distributions is studied. This generalizes the random model discussed by Paladin et al. (J. Phys. I France 4, 1994, p. 1597). Exact results are obtained for the thermodynamic functions at arbitrary temperatures, and special attention is given to the induced and spontaneous magnetization. At low temperatures the system can exist in a “ferrimagnetic” phase with magnetization 0 < σ < 1, in addition to the usual paramagnetic, ferromagnetic and antiferromagnetic phases. For a fixed distribution of the random variables the system presents up to three tricritical points for different intensities of the long-range interactions. Field-temperature diagrams can present up to four critical points. 1

Related papers

Ferrimagnetism in a disordered Ising model

Giovanni Paladin, Michele Pasquini, Maurizio Serva

We introduce a one dimensional Ising model with two competing interactions: nearest neighbor random couplings +/-J with equal probability and a positive infinite range coupling Lambda. At low temperature T the model exhibits a first order phase transition between a ferromagnetic state (with magnetization m(1) = 1 at T = 0) and a << ferrimagnetic >> state (with m(2) = 2/3 at T = 0), when the disorder strength J/Lambda is increased. For 5/12 < J/Lambda < 1, a whole spectrum of ferrimagnetic ground states with magnetization m(n) = 2/(n + 1) (n = 2, ..., infinity) is present while for J/lambda > 1 the ground state is given by a trivial one dimensional spin glass with m = 0. The main qualitative features of the model can be described by a simplified annealed model where the random couplings can arrange themselves to minimize free energy with the constraint that the number of positive couplings is fixed by the law of large numbers in the thermodynamic limit. This model is exactly solved at all temperatures and the diagram of phase is calculated.

Journal de Physique I•Nov 1, 1994•7 citations

First-order phase transition in a 2D random-field Ising model with conflicting dynamics

Nuno Crokidakis

The effects of locally random magnetic fields are considered in a nonequilibrium Ising model defined on a square lattice with nearest-neighbor interactions. In order to generate the random magnetic fields, we have considered random variables { h } that change randomly with time according to a double-Gaussian probability distribution, which consists of two single-Gaussian distributions, centered at + h o and − h o , with the same width σ. This distribution is very general and can recover in appropriate limits the bimodal distribution ( ) and the single-Gaussian one ( h o = 0). We performed Monte Carlo simulations in lattices with linear sizes in the range L = 32–512. The system exhibits ferromagnetic and paramagnetic steady states. Our results suggest the occurrence of first-order phase transitions between the above-mentioned phases at low temperatures and large random-field intensities h o for some small values of the width σ. By means of finite size scaling, we estimate the critical exponents in the low-field region, where we have continuous phase transitions. In addition, we show a sketch of the phase diagram of the model for some values of σ.

arXiv:0901.2659•Journal of Statistical Mechanics Theory and Experiment•Feb 24, 2009•7 citations

Crossover between short- and long-range interactions in the one-dimensional Ising model

V.B. Kislinsky, Vyacheslav I. Yukalov

The authors consider the one-dimensional Ising model which besides the interaction of nearest neighbours includes a long-range term. The latter is ferromagnetic while the nearest-neighbour interaction can have any sign. The crossover between short- and long-range interactions when varying the Hamiltonian parameters is analysed for such a model. For some values of the antiferromagnetic short-range interaction the transition from a ferro- to paramagnetic state is of first order. At the tricritical point where the transition order changes from second to first they obtain classical tricritical exponents. For a ferromagnetic short-range interaction the Curie temperature surpasses the transition temperature of the long-range model.

Journal of Physics A Mathematical and General•Jan 7, 1988•8 citations

Exact ground states of one-dimensional long-range random-field Ising magnets

Timo Dewenter, Alexander K. Hartmann

We investigate the one-dimensional long-range random-field Ising magnet with Gaussian distribution of the random fields. In this model, a ferromagnetic bond between two spins is placed with a probability $p\ensuremath{\sim}{r}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\sigma}}$, where $r$ is the distance between these spins and $\ensuremath{\sigma}$ is a parameter to control the effective dimension of the model. Exact ground states at zero temperature are calculated for system sizes up to $L={2}^{19}$ via graph theoretical algorithms for four different values of $\ensuremath{\sigma}\ensuremath{\in}{0.25,0.4,0.5,1.0}$ while varying the strength $h$ of the random fields. For each of these values several independent physical observables are calculated, i.e., magnetization, Binder parameter, susceptibility, and a specific-heat-like quantity. The ferromagnet-paramagnet transitions at critical values ${h}_{c}(\ensuremath{\sigma})$ as well as the corresponding critical exponents are obtained. The results agree well with theory, and interestingly we find for $\ensuremath{\sigma}=1/2$ the data is compatible with a critical random-field strength ${h}_{c}>0$.

arXiv:1307.3987•Physical Review B•Jul 31, 2014•10 citations

Phase transitions in the one-dimensional Ising model with ferromagnetic and antiferromagnetic interactions

Dong Huang, Barbara J. Gabryś

A special case of the one-dimensional Ising model with short-range antiferromagnetic and long-range ferromagnetic interactions is presented. Conditions of the occurrence of phase transitions as well as the behavior of magnetization and free energy are discussed.

Physical review. B, Condensed matter•Apr 1, 1990•0 citations

First-order phase transition in a 2D random-field Ising model with conflicting dynamics

Nuno Crokidakis

The effects of locally random magnetic fields are considered in a nonequilibrium Ising model defined on a square lattice with nearest-neighbors interactions. In order to generate the random magnetic fields, we have considered random variables {h} that change randomly with time according to a double-gaussian probability distribution, which consists of two single gaussian distributions, centered at +ho and −ho, with the same width σ. This distribution is very general, and can recover in appropriate limits the bimodal distribution (σ → 0) and the single gaussian one (ho = 0). We performed Monte Carlo simulations in lattices with linear sizes in the range L = 32 − 512. The system exhibits ferromagnetic and paramagnetic steady states. Our results suggest the occurence of first-order phase transitions between the above-mentioned phases at low temperatures and large random-field intensities ho, for some small values of the width σ. By means of finite size scaling, we estimate the critical exponents in the low-field region, where we have continuous phase transitions. In addition, we show a sketch of the phase diagram of the model for some values of σ. PACS numbers: 05.10.Ln, 05.50.+q, 64.60.De, 75.10Hk, 75.40.MgFirst-order phase transition in a 2D random-field Ising model with conflicting dynamics2 1.

•Nov 20, 2012•10 citations
The one-dimensional Ising model in an external magnetic field with uniform long-range interactions and random short-range interactions satisfying bimodal annealed distributions is studied. This generalizes the random model discussed by Paladin et al. (J. Phys. I France 4, 1994, p. 1597). Exact results are obtained for the thermodynamic functions at arbitrary temperatures, and special attention is given to the induced and spontaneous magnetization. At low temperatures the system can exist in a “ferrimagnetic” phase with magnetization 0 < σ < 1, in addition to the usual paramagnetic, ferromagnetic and antiferromagnetic phases. For a fixed distribution of the random variables the system presents up to three tricritical points for different intensities of the long-range interactions. Field-temperature diagrams can present up to four critical points. 1
We introduce a one dimensional Ising model with two competing interactions: nearest neighbor random couplings +/-J with equal probability and a positive infinite range coupling Lambda. At low temperature T the model exhibits a first order phase transition between a ferromagnetic state (with magnetization m(1) = 1 at T = 0) and a << ferrimagnetic >> state (with m(2) = 2/3 at T = 0), when the disorder strength J/Lambda is increased. For 5/12 < J/Lambda < 1, a whole spectrum of ferrimagnetic ground states with magnetization m(n) = 2/(n + 1) (n = 2, ..., infinity) is present while for J/lambda > 1 the ground state is given by a trivial one dimensional spin glass with m = 0. The main qualitative features of the model can be described by a simplified annealed model where the random couplings can arrange themselves to minimize free energy with the constraint that the number of positive couplings is fixed by the law of large numbers in the thermodynamic limit. This model is exactly solved at all temperatures and the diagram of phase is calculated.
The effects of locally random magnetic fields are considered in a nonequilibrium Ising model defined on a square lattice with nearest-neighbor interactions. In order to generate the random magnetic fields, we have considered random variables { h } that change randomly with time according to a double-Gaussian probability distribution, which consists of two single-Gaussian distributions, centered at + h o and − h o , with the same width σ. This distribution is very general and can recover in appropriate limits the bimodal distribution ( ) and the single-Gaussian one ( h o = 0). We performed Monte Carlo simulations in lattices with linear sizes in the range L = 32–512. The system exhibits ferromagnetic and paramagnetic steady states. Our results suggest the occurrence of first-order phase transitions between the above-mentioned phases at low temperatures and large random-field intensities h o for some small values of the width σ. By means of finite size scaling, we estimate the critical exponents in the low-field region, where we have continuous phase transitions. In addition, we show a sketch of the phase diagram of the model for some values of σ.
The authors consider the one-dimensional Ising model which besides the interaction of nearest neighbours includes a long-range term. The latter is ferromagnetic while the nearest-neighbour interaction can have any sign. The crossover between short- and long-range interactions when varying the Hamiltonian parameters is analysed for such a model. For some values of the antiferromagnetic short-range interaction the transition from a ferro- to paramagnetic state is of first order. At the tricritical point where the transition order changes from second to first they obtain classical tricritical exponents. For a ferromagnetic short-range interaction the Curie temperature surpasses the transition temperature of the long-range model.
We investigate the one-dimensional long-range random-field Ising magnet with Gaussian distribution of the random fields. In this model, a ferromagnetic bond between two spins is placed with a probability $p\ensuremath{\sim}{r}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\sigma}}$, where rrr is the distance between these spins and $\ensuremath{\sigma}$ is a parameter to control the effective dimension of the model. Exact ground states at zero temperature are calculated for system sizes up to L=219L={2}^{19}L=219 via graph theoretical algorithms for four different values of $\ensuremath{\sigma}\ensuremath{\in}{0.25,0.4,0.5,1.0}$ while varying the strength hhh of the random fields. For each of these values several independent physical observables are calculated, i.e., magnetization, Binder parameter, susceptibility, and a specific-heat-like quantity. The ferromagnet-paramagnet transitions at critical values ${h}_{c}(\ensuremath{\sigma})$ as well as the corresponding critical exponents are obtained. The results agree well with theory, and interestingly we find for $\ensuremath{\sigma}=1/2$ the data is compatible with a critical random-field strength hc>0{h}_{c}>0hc​>0.
The effects of locally random magnetic fields are considered in a nonequilibrium Ising model defined on a square lattice with nearest-neighbors interactions. In order to generate the random magnetic fields, we have considered random variables {h} that change randomly with time according to a double-gaussian probability distribution, which consists of two single gaussian distributions, centered at +ho and −ho, with the same width σ. This distribution is very general, and can recover in appropriate limits the bimodal distribution (σ → 0) and the single gaussian one (ho = 0). We performed Monte Carlo simulations in lattices with linear sizes in the range L = 32 − 512. The system exhibits ferromagnetic and paramagnetic steady states. Our results suggest the occurence of first-order phase transitions between the above-mentioned phases at low temperatures and large random-field intensities ho, for some small values of the width σ. By means of finite size scaling, we estimate the critical exponents in the low-field region, where we have continuous phase transitions. In addition, we show a sketch of the phase diagram of the model for some values of σ. PACS numbers: 05.10.Ln, 05.50.+q, 64.60.De, 75.10Hk, 75.40.MgFirst-order phase transition in a 2D random-field Ising model with conflicting dynamics2 1.