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

Journal of Physics A Mathematical and General

Random Ising model with long-range interactions

Alba Theumann

Published Oct 1, 1981• 12 citations
Publisher

Abstract

The critical properties of a random Ising model with long-range isotropic interactions decaying as R -(d+ sigma ) are analysed by using renormalisation group methods in an expansion in epsilon '=2 sigma -d, where d is dimensionality. For epsilon '>O the critical behaviour is described by a stable fixed point O( epsilon 1 2/') in the physical region of parameter space. The crossover to short-range behaviour is analysed and takes place when sigma =2+ epsilon /106, epsilon =4-d>O. There is then a small region O< sigma -2<O( epsilon ) where the system is still dominated by long-range behaviour. The results for d=1 are compared with those obtained previously for the random hierarchical model and it is found that the two models do not show the same critical behaviour in the epsilon ' expansion.

DOI
10.1088/0305-4470/14/10/027

Related papers

Short-time dynamics of a random Ising model with long-range interaction

Yuan Chen

Short-time critical dynamics of a random Ising model (model A) with long-range interaction decaying as r(-(d+sigma)) (where sigma is the parameter controlling the range of the interaction), is studied by the theoretic renormalization-group approach. In dimensions d<2sigma, the initial slip exponents theta(') describing the initial increase of the order parameter, and theta for the growth of the response function, which govern the short-time scaling behaviors, are calculated to the second order in sqrt[epsilon] with epsilon=2sigma-d. The crossover between the long-range interaction and the short-range interaction, which occurs at some sigma>2, is also discussed.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Sep 25, 2002•14 citations

Dynamics of the Random Ising Model with Long-Range Interaction

Chen, Yuan, Li +7 more

Critical dynamics of the random Ising model with long-range interaction decaying as r-(d+σ) where d is the dimensionality) is studied by the theoretic renormalization-group approach. The system is released to an evolution within a model A dynamics. Asymptotic scaling laws are studied in a frame of the expansion in = 2σ - d. In dimensions d ＜ 2σ. the dynamic exponent z is calculated to the second order in at the random fixed point.``

理论物理通讯：英文版•Jan 1, 2001•0 citations

Critical behaviour of the random-field Ising model with long-range interactions in one dimension

Ivan Balog, Gilles Tarjus, Matthieu Tissier

We study the critical behaviour of the 1D random field Ising model (RFIM) with long-range interactions (∝ r −( d + σ ) ) by the nonperturbative functional renormalization group. We find two distinct regimes of critical behaviour as a function of σ , separated by a critical value σ c . What distinguishes these two regimes is the presence or not of a cusp-like nonanalyticity in the functional dependence of the renormalized cumulants of the random field at the fixed point. This change of behaviour can be associated with the characteristics of the large-scale avalanches present in the system at zero temperature. We propose some ways to check these predictions through lattice simulations. We also discuss the differences with the RFIM on the Dyson hierarchical lattice.

arXiv:1407.4891•Journal of Statistical Mechanics Theory and Experiment•Oct 9, 2014•16 citations

Critical exponents of the Ising model with quenched structural disorder and long-range interactions at spatial dimension d = 3

D.V. Shapoval, M. Dudka

We analyse the critical properties of a weakly diluted (random) Ising model with the long-range interaction decaying with distance x as ∼ x - d - σ in a d-dimensional space. It is known to belong to a new long-range random universality class for certain values of the decay parameter σ. Exploiting the field-theoretic renormalization group approach within the minimal subtraction scheme, we compute the three-loop renormalization group functions. On their basis, with the help of asymptotic series resummation methods, we estimate the correlation length critical exponent ν characterising the new universality class for d = 3 and for those values of σ for which long-range interactions are relevant for the critical behaviour.

arXiv:2511.15160•Condensed Matter Physics•Dec 22, 2025•0 citations

Dynamics of the Random Ising Model with Long-Range Interaction

Yuan Chen, Zhibing Li, Fang Hai +2 more

Critical dynamics of the random Ising model with long-range interaction decaying as (where is the dimensionality) is studied by the theoretic renormalization-group approach. The system is released to an evolution within a model dynamics. Asymptotic scaling laws are studied in a frame of the expansion in . In dimensions , the dynamic exponent is calculated to the second order in at the random fixed point.

Communications in Theoretical Physics•Oct 15, 2001•1 citation

Correlated percolation with long-range interactions

Bosiljka Tadić, R. Pirc

We study the critical behavior of the q-state Potts model with long-range interactions decaying asymptotically as \ensuremath{\sim}${R}^{\mathrm{\ensuremath{-}}(d+\ensuremath{\sigma})}$ in the presence of random-${T}_{c}$ impurities correlated over large distances such that the correlations fall off as \ensuremath{\sim}${R}^{\mathrm{\ensuremath{-}}(d\mathrm{\ensuremath{-}}a)}$, a<d. We find that the renormalization-group scaling equations have a new fixed point in the appropriate double \ensuremath{\epsilon}, x expansion, where \ensuremath{\epsilon}=3\ensuremath{\sigma}-d and x=a-\ensuremath{\sigma}. This fixed point, however, is never both stable and physical.

Physical review. B, Condensed matter•May 1, 1989•0 citations
The critical properties of a random Ising model with long-range isotropic interactions decaying as R -(d+ sigma ) are analysed by using renormalisation group methods in an expansion in epsilon '=2 sigma -d, where d is dimensionality. For epsilon '>O the critical behaviour is described by a stable fixed point O( epsilon 1 2/') in the physical region of parameter space. The crossover to short-range behaviour is analysed and takes place when sigma =2+ epsilon /106, epsilon =4-d>O. There is then a small region O< sigma -2<O( epsilon ) where the system is still dominated by long-range behaviour. The results for d=1 are compared with those obtained previously for the random hierarchical model and it is found that the two models do not show the same critical behaviour in the epsilon ' expansion.
Short-time critical dynamics of a random Ising model (model A) with long-range interaction decaying as r(-(d+sigma)) (where sigma is the parameter controlling the range of the interaction), is studied by the theoretic renormalization-group approach. In dimensions d<2sigma, the initial slip exponents theta(') describing the initial increase of the order parameter, and theta for the growth of the response function, which govern the short-time scaling behaviors, are calculated to the second order in sqrt[epsilon] with epsilon=2sigma-d. The crossover between the long-range interaction and the short-range interaction, which occurs at some sigma>2, is also discussed.
We study the critical behaviour of the 1D random field Ising model (RFIM) with long-range interactions (∝ r −( d + σ ) ) by the nonperturbative functional renormalization group. We find two distinct regimes of critical behaviour as a function of σ , separated by a critical value σ c . What distinguishes these two regimes is the presence or not of a cusp-like nonanalyticity in the functional dependence of the renormalized cumulants of the random field at the fixed point. This change of behaviour can be associated with the characteristics of the large-scale avalanches present in the system at zero temperature. We propose some ways to check these predictions through lattice simulations. We also discuss the differences with the RFIM on the Dyson hierarchical lattice.
We analyse the critical properties of a weakly diluted (random) Ising model with the long-range interaction decaying with distance x as ∼ x - d - σ in a d-dimensional space. It is known to belong to a new long-range random universality class for certain values of the decay parameter σ. Exploiting the field-theoretic renormalization group approach within the minimal subtraction scheme, we compute the three-loop renormalization group functions. On their basis, with the help of asymptotic series resummation methods, we estimate the correlation length critical exponent ν characterising the new universality class for d = 3 and for those values of σ for which long-range interactions are relevant for the critical behaviour.
We study the critical behavior of the q-state Potts model with long-range interactions decaying asymptotically as \ensuremath{\sim}${R}^{\mathrm{\ensuremath{-}}(d+\ensuremath{\sigma})}$ in the presence of random-Tc{T}_{c}Tc​ impurities correlated over large distances such that the correlations fall off as \ensuremath{\sim}${R}^{\mathrm{\ensuremath{-}}(d\mathrm{\ensuremath{-}}a)}$, a<d. We find that the renormalization-group scaling equations have a new fixed point in the appropriate double \ensuremath{\epsilon}, x expansion, where \ensuremath{\epsilon}=3\ensuremath{\sigma}-d and x=a-\ensuremath{\sigma}. This fixed point, however, is never both stable and physical.