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

Physical review. B, Condensed matter

Numerical results for the random-field Ising model and the dilute antiferromagnetic Ising model in a uniform field

Julio F. Fernández, Erling Pytte

Published Mar 1, 1985• 8 citations
Publisher

Abstract

We have studied the equilibrium properties of the random-field (H) ferromagnetic Ising model (RFIM) and the dilute antiferromagnetic Ising model (DAF) in a uniform field. For RFIM we have used the transfer-matrix technique, thus bypassing serious equilibration problems which would arise in a Monte Carlo calculation for random systems. We have calculated the structure factor S versus H. Our results are consistent with S\ensuremath{\sim}exp(c/${H}^{2}$) as predicted by theories which yield a lower critical dimension ${d}_{c}$=2, whereas the expected behavior for ${d}_{c}$=3, S\ensuremath{\sim}${H}^{\mathrm{\ensuremath{-}}4}$ is inconsistent with our results. We have carried out analogous calculations for DAF. Our results for DAF suffer seriously from finite-size effects. To circumvent this difficulty, we have also performed calculations on interface models. Together, these results indicate that DAF crosses over to RFIM behavior at H\ensuremath{\simeq}J.

DOI
10.1103/physrevb.31.2886

Related papers

The Random Field Ising Model

C. De Dominicis, Irene Giardina

The Random Field Ising Model (RFIM) represents one of the simplest models of cooperative behaviour with quenched disorder, and it is, in a way, complementary to the Ising Spin Glass which will be extensively treated later in this book. It accounts for the presence of a random external magnetic field which antagonizes the ordering induced by the ferromagnetic spin–spin interactions. From an experimental point of view, on the other hand, as shown by Fishman and Aharony (1979) and Cardy (1984), it is equivalent to a dilute anti-ferromagnet in a uniform field (see Belanger, 1998 for a recent review on experimental results). Despite twenty-five years of active and continuous research the RFIM is not yet completely understood. The problem seems related to the presence of bound states in the ferromagnetic phase, which make the standard theoretical approaches not adequate to analyze the critical behaviour. Here we discuss the RFIM in the context of perturbative field theory. The chapter is organized as follows: in Section 2.1 we define the model and outline the main expectations for its qualitative behaviour. In Section 2.2 we introduce an effective replicated ϕ 4 field model where the disorder has been integrated out. Then we perform a perturbative analysis on this model (Section 2.3) and illustrate how the so-called dimensional reduction arises (Section 2.4). Finally, in Section 2.5 we introduce some generalized couplings which need to be taken into account to properly describe the system; we perform a perturbative Renormalization Group (RG) close to the upper critical dimension (Sections 2.6 and 2.7) and discuss the occurrence of a vitrous transition (Section 2.8).

Cambridge University Press eBooks•Oct 26, 2006•1 citation

Exact ground-state properties of disordered Ising systems

Joachim Esser, U. Nowak, Klaus D. Usadel

Exact ground states are calculated with an integer optimization algorithm for two- and three-dimensional site-diluted Ising antiferromagnets in a field (DAFF) and random field Ising ferromagnets (RFIM), the latter with Gaussian- and bimodal-distributed random fields. We investigate the structure and the size distribution of the domains of the ground state and compare it to earlier results from Monte Carlo (MC) simulations for finite temperature. Although DAFF and RFIM are thought to be in the same universality class we found differences between these systems as far as the distribution of domain sizes is concerned. In the limit of strong disorder for the DAFF in two and three dimensions the ground states consist of domains with a broad size distribution that can be described by a power law with exponential cutoff. For the RFIM this is only true in two dimensions while in three dimensions above the critical field where long-range order breaks down the system consists of two infinite interpenetrating domains of up and down spins---the system is in a two-domain state. For DAFF and RFIM the structure of the domains of finite size is fractal and the fractal dimensions for the DAFF and the RFIM agree within our numerical accuracy supporting that DAFF and RFIM are in the same universality class. Also, the DAFF ground-state properties agree with earlier results from MC simulations in the whole whereas there are essential differences between our exact ground-state calculations and earlier MC simulations for the RFIM which suggested that there are differences between the fractality of domains in RFIM and DAFF. Additionally, we show that for the case of higher disorder there are strong deviations from Imry-Ma-type arguments for RFIM and DAFF in two and three dimensions.

arXiv:cond-mat/9612022•Physical review. B, Condensed matter•Mar 1, 1997•70 citations

The dilute Ising antiferromagnet in a uniform field on a square lattice

S. L. A. de Queiroz, Raimundo R. dos Santos

The bond-diluted Ising antiferromagnet in a uniform field (DAFF) on a square lattice is studied and comparison with the random-field problem is made. Position-space renormalisation group calculations for the DAFF give rise to a staircase structure in the zero-temperature phase boundary H (critical field) versus p (bond concentration), separating ordered and disordered phases. Since there is no stable long-range order in the equivalent random-field system, the authors discuss ways in which a consistent picture can be obtained, incorporating both our results and the correspondence between the two problems.

Journal of Physics C Solid State Physics•Apr 10, 1988•6 citations

Dipolar interaction in random-field systems

Thomas Nattermann

It is shown that the dipolar interaction breaks the universality of the low-temperature and critical behaviour between the random-field Ising model (RFIM) and the diluted antiferromagnet in an external field (DAFF). For the RFIM, the dipolar interaction changes the roughness and the critical exponents; in particular, the upper critical dimensionalities are diminished by one. The dipolar interaction exerts no influence on these exponents in the DAFF, nor on the lower critical dimensionality d l =2 of both systems.

Journal of Physics A Mathematical and General•Jun 21, 1988•20 citations

Diluted antiferromagnets in a field seem to be in a different universality class than the random-field Ising model

Björn Ahrens, Jianping Xiao, Alexander K. Hartmann +1 more

We perform large-scale Monte Carlo simulations using the Machta-Newman-Chayes algorithms to study the critical behavior of both the diluted antiferromagnet in a field with 30% dilution and the random-field Ising model with Gaussian random fields for different field strengths. Analytical calculations by Cardy [Phys. Rev. B 29, 505 (1984)] predict that both models map onto each other and share the same universality class in the limit of vanishing fields. However, a detailed finite-size scaling analysis of the Binder cumulant, the two-point finite-size correlation length, and the susceptibility suggests that even in the limit of small fields, where the mapping is expected to work, both models are not in the same universality class. Based on our numerical data, we present analytical expressions for the phase boundaries of both models.

arXiv:1302.2480•Physical Review B•Nov 13, 2013•15 citations

Critical behavior of the random-field Ising magnet with long-range correlated disorder

Björn Ahrens, Alexander K. Hartmann

We study the correlated-disorder-driven zero-temperature equilibrium phase transition of the random-field Ising magnet (RFIM) using exact numerical ground-state calculations for cubic lattices. We consider correlations of the quenched disorder decaying proportionally to ${r}^{a}$, where $r$ is the distance between two lattice sites and $a<0$. To obtain exact ground states, we use a well-established mapping to the graph-theoretical maximum-flow problem, which allows us to study large system sizes of more than $2\ifmmode\times\else\texttimes\fi{}{10}^{6}$ spins. We use finite-size-scaling analyses for values $a={\ensuremath{-}1,\ensuremath{-}2,\ensuremath{-}3,\ensuremath{-}7}$ to calculate the critical point and the critical exponents characterizing the behavior of the specific heat, magnetization, susceptibility, and the correlation length close to the critical point. We find basically the same critical behavior as for the RFIM with $\ensuremath{\delta}$-correlated disorder, except for the finite-size exponent of the susceptibility and for the case $a\ensuremath{\ge}\ensuremath{-}2$, where the results are also compatible with a phase transition at infinitesimal disorder strength. We numerically confirm earlier predictions.

arXiv:1106.6244•Physical Review B•Oct 7, 2011•19 citations
We have studied the equilibrium properties of the random-field (H) ferromagnetic Ising model (RFIM) and the dilute antiferromagnetic Ising model (DAF) in a uniform field. For RFIM we have used the transfer-matrix technique, thus bypassing serious equilibration problems which would arise in a Monte Carlo calculation for random systems. We have calculated the structure factor S versus H. Our results are consistent with S\ensuremath{\sim}exp(c/H2{H}^{2}H2) as predicted by theories which yield a lower critical dimension dc{d}_{c}dc​=2, whereas the expected behavior for dc{d}_{c}dc​=3, S\ensuremath{\sim}H\ensuremath−4{H}^{\mathrm{\ensuremath{-}}4}H\ensuremath−4 is inconsistent with our results. We have carried out analogous calculations for DAF. Our results for DAF suffer seriously from finite-size effects. To circumvent this difficulty, we have also performed calculations on interface models. Together, these results indicate that DAF crosses over to RFIM behavior at H\ensuremath{\simeq}J.
The Random Field Ising Model (RFIM) represents one of the simplest models of cooperative behaviour with quenched disorder, and it is, in a way, complementary to the Ising Spin Glass which will be extensively treated later in this book. It accounts for the presence of a random external magnetic field which antagonizes the ordering induced by the ferromagnetic spin–spin interactions. From an experimental point of view, on the other hand, as shown by Fishman and Aharony (1979) and Cardy (1984), it is equivalent to a dilute anti-ferromagnet in a uniform field (see Belanger, 1998 for a recent review on experimental results). Despite twenty-five years of active and continuous research the RFIM is not yet completely understood. The problem seems related to the presence of bound states in the ferromagnetic phase, which make the standard theoretical approaches not adequate to analyze the critical behaviour. Here we discuss the RFIM in the context of perturbative field theory. The chapter is organized as follows: in Section 2.1 we define the model and outline the main expectations for its qualitative behaviour. In Section 2.2 we introduce an effective replicated ϕ 4 field model where the disorder has been integrated out. Then we perform a perturbative analysis on this model (Section 2.3) and illustrate how the so-called dimensional reduction arises (Section 2.4). Finally, in Section 2.5 we introduce some generalized couplings which need to be taken into account to properly describe the system; we perform a perturbative Renormalization Group (RG) close to the upper critical dimension (Sections 2.6 and 2.7) and discuss the occurrence of a vitrous transition (Section 2.8).
Exact ground states are calculated with an integer optimization algorithm for two- and three-dimensional site-diluted Ising antiferromagnets in a field (DAFF) and random field Ising ferromagnets (RFIM), the latter with Gaussian- and bimodal-distributed random fields. We investigate the structure and the size distribution of the domains of the ground state and compare it to earlier results from Monte Carlo (MC) simulations for finite temperature. Although DAFF and RFIM are thought to be in the same universality class we found differences between these systems as far as the distribution of domain sizes is concerned. In the limit of strong disorder for the DAFF in two and three dimensions the ground states consist of domains with a broad size distribution that can be described by a power law with exponential cutoff. For the RFIM this is only true in two dimensions while in three dimensions above the critical field where long-range order breaks down the system consists of two infinite interpenetrating domains of up and down spins---the system is in a two-domain state. For DAFF and RFIM the structure of the domains of finite size is fractal and the fractal dimensions for the DAFF and the RFIM agree within our numerical accuracy supporting that DAFF and RFIM are in the same universality class. Also, the DAFF ground-state properties agree with earlier results from MC simulations in the whole whereas there are essential differences between our exact ground-state calculations and earlier MC simulations for the RFIM which suggested that there are differences between the fractality of domains in RFIM and DAFF. Additionally, we show that for the case of higher disorder there are strong deviations from Imry-Ma-type arguments for RFIM and DAFF in two and three dimensions.
The bond-diluted Ising antiferromagnet in a uniform field (DAFF) on a square lattice is studied and comparison with the random-field problem is made. Position-space renormalisation group calculations for the DAFF give rise to a staircase structure in the zero-temperature phase boundary H (critical field) versus p (bond concentration), separating ordered and disordered phases. Since there is no stable long-range order in the equivalent random-field system, the authors discuss ways in which a consistent picture can be obtained, incorporating both our results and the correspondence between the two problems.
It is shown that the dipolar interaction breaks the universality of the low-temperature and critical behaviour between the random-field Ising model (RFIM) and the diluted antiferromagnet in an external field (DAFF). For the RFIM, the dipolar interaction changes the roughness and the critical exponents; in particular, the upper critical dimensionalities are diminished by one. The dipolar interaction exerts no influence on these exponents in the DAFF, nor on the lower critical dimensionality d l =2 of both systems.
We perform large-scale Monte Carlo simulations using the Machta-Newman-Chayes algorithms to study the critical behavior of both the diluted antiferromagnet in a field with 30% dilution and the random-field Ising model with Gaussian random fields for different field strengths. Analytical calculations by Cardy [Phys. Rev. B 29, 505 (1984)] predict that both models map onto each other and share the same universality class in the limit of vanishing fields. However, a detailed finite-size scaling analysis of the Binder cumulant, the two-point finite-size correlation length, and the susceptibility suggests that even in the limit of small fields, where the mapping is expected to work, both models are not in the same universality class. Based on our numerical data, we present analytical expressions for the phase boundaries of both models.
We study the correlated-disorder-driven zero-temperature equilibrium phase transition of the random-field Ising magnet (RFIM) using exact numerical ground-state calculations for cubic lattices. We consider correlations of the quenched disorder decaying proportionally to ra{r}^{a}ra, where rrr is the distance between two lattice sites and a<0a<0a<0. To obtain exact ground states, we use a well-established mapping to the graph-theoretical maximum-flow problem, which allows us to study large system sizes of more than $2\ifmmode\times\else\texttimes\fi{}{10}^{6}$ spins. We use finite-size-scaling analyses for values $a={\ensuremath{-}1,\ensuremath{-}2,\ensuremath{-}3,\ensuremath{-}7}$ to calculate the critical point and the critical exponents characterizing the behavior of the specific heat, magnetization, susceptibility, and the correlation length close to the critical point. We find basically the same critical behavior as for the RFIM with $\ensuremath{\delta}$-correlated disorder, except for the finite-size exponent of the susceptibility and for the case $a\ensuremath{\ge}\ensuremath{-}2$, where the results are also compatible with a phase transition at infinitesimal disorder strength. We numerically confirm earlier predictions.