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/0005413 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Nontrivial extension of the two-dimensional Ising model: Thed-dimensional “molecular” model

Fabio Siringo

Published Nov 1, 2000• 1 citation
arXivHTMLPDF

Abstract

A recently proposed molecular model is discussed as a nontrivial extension of the Ising model. For d=2 the two models are shown to be equivalent, while for d>2 the molecular model describes a peculiar second order transition from an isotropic high-temperature phase to a low-dimensional anisotropic low-temperature state. The general mean-field analysis is compared with the results achieved by a variational Migdal-Kadanoff real space renormalization group method and by standard Monte Carlo sampling for d=3. By finite size scaling the critical exponent has been found to be nu=0.44+/-0.02, thus establishing that the molecular model does not belong to the universality class of the Ising model for d>2.

DOI
10.1103/physreve.62.6026

Related papers

Variational Migdal–Kadanoff Approach to Molecular Ordering on a Lattice

Fabio Siringo

A variational extension of the Migdal–Kadanoff real space renormalization group method is applied to a recently proposed d-dimensional lattice model describing some aspects of molecular orientation in solids. As expected the method yields the exact fixed point for d=2, given the formal equivalence to the two-dimensional Ising model. For d=3 such equivalence breaks down and an approximate estimate of the fixed point is explicitly recovered and compared to the mean-field prediction. The possible existence of an intermediate symmetric phase is discussed.

International Journal of Modern Physics B•Jul 20, 1997•2 citations

Universal finite-size scaling analysis of Ising models with long-range interactions at the upper critical dimensionality: Isotropic case

Daniel Grüneberg, Alfred Hucht

We investigate a two-dimensional Ising model with long-range interactions that emerge from a generalization of the magnetic dipolar interaction in spin systems with in-plane spin orientation. This interaction is, in general, anisotropic whereby in the present work we focus on the isotropic case for which the model is found to be at its upper critical dimensionality. To investigate the critical behavior the temperature and field dependence of several quantities are studied by means of Monte Carlo simulations. On the basis of the Privman-Fisher hypothesis and results of the renormalization group the numerical data are analyzed in the framework of a finite-size scaling analysis and compared to finite-size scaling functions derived from a Ginzburg-Landau-Wilson model in zero mode (mean-field) approximation. The obtained excellent agreement suggests that at least in the present case the concept of universal finite-size scaling functions can be extended to the upper critical dimensionality.

arXiv:cond-mat/0310252•Physical Review E•Mar 9, 2004•16 citations

Finite-size scaling analysis of the anisotropic critical behavior of the two-dimensional Ising model under shear

D. Winter, Peter Virnau, Jürgen Horbach +1 more

The critical behavior of the two-dimensional Ising Model with non-conserved order parameter in steady-state shear is studied by large-scale Monte Carlo simulations. Studying the structure factor S ( q x , q y ) in the disordered phase, the ratio of correlation length exponents ν x /ν y in the two lattice directions ( x , y ) is estimated, and the critical temperature is determined as a function of the shear rate as T c ( ) − T c (0) ∝ ϕ with ϕ≈0.45. Critical exponents β≈0.37, γ≈1.1, ; ν ⊥ ≈0.46, ν ∥ ≈1.38 are roughly compatible with anisotropic hyperscaling.

Europhysics Letters (EPL)•Sep 1, 2010•14 citations

On universality between two-dimensional Ising-like systems

Joan Adler

Questions concerning the universality between different two-dimensional Ising-like systems have recently been raised, with evidence for the existence of several different subclasses of Ising-like systems being presented. Some new facts concerning similarity between border model and low-temperature spin-1 Ising model series analyses are noted. For the spin-1 case an alternative determination of the critical temperature determines unequivocally that this model is in the Ising universality class and it is suggested that a similar independent result could lead to a similar conclusion in the border model case. The possibility of an analytic correction to scaling in the spin-1 model is proposed.

Journal of Physics A Mathematical and General•Aug 1, 1987•3 citations

Critical Properties of a Spin-3/2 Ising Model on a Square Lattice

Norihiro Tsushima, Yasushi Honda, Tsuyoshi Horiguchi

We investigate a spin-3/2 Ising model on a square lattice by using Monte Carlo (MC) simulations and a density matrix renormalization group (DMRG) method. We obtain phase diagrams of the ground state. We carried out the MC simulations in order to calculate the critical temperature and the critical exponent η. We estimate the critical exponents ν, α and γ by using a finite size scaling analysis. We also calculate the critical temperature, the central charge c and the critical exponent ν by using the DMRG method. From the results obtained by the MC simulations and by the DMRG method, we discuss a universality class of the system. It is found that the values of critical exponents are determined by means of the degeneracy of the ground state. This result suggests that the universality does not hold for the spin-3/2 Ising model. Furthermore we find a possibility of a breakdown of the hyperscaling law. We also confirm that the ferrimagnetic phase exists in a parameter region where the ground state has a fourfold degeneracy, and that a transition between the ferrimagnetic phase and the ferromagnetic phase exists in addition to that between the ferromagnetic phase and the paramagnetic phase.

Journal of the Physical Society of Japan•Oct 15, 1997•26 citations

Two-dimensional Ising model with multispin interactions

Jean‐Marc Debierre, Loïc Turban

The two-dimensional Ising model in the rectangular lattice is generalised to include m-spin interactions in one direction and two-spin interactions in the other. This model is self-dual and the critical line is the same as in the conventional Ising model with m=2. The m=3 model is solved in a generalised mean-field approximation where the transition is first order. Using a phenomenological renormalisation group approach, approximate values of the thermal and magnetic exponents y t and y h are obtained for m=3 and m=4. The m=3 results together with symmetry arguments suggest that this model belongs to the same class of universality as the Baxter-Wu and four-state ferromagnetic Potts models. When m=4 the transition is probably first order.

Journal of Physics A Mathematical and General•Oct 21, 1983•39 citations
A recently proposed molecular model is discussed as a nontrivial extension of the Ising model. For d=2 the two models are shown to be equivalent, while for d>2 the molecular model describes a peculiar second order transition from an isotropic high-temperature phase to a low-dimensional anisotropic low-temperature state. The general mean-field analysis is compared with the results achieved by a variational Migdal-Kadanoff real space renormalization group method and by standard Monte Carlo sampling for d=3. By finite size scaling the critical exponent has been found to be nu=0.44+/-0.02, thus establishing that the molecular model does not belong to the universality class of the Ising model for d>2.
A variational extension of the Migdal–Kadanoff real space renormalization group method is applied to a recently proposed d-dimensional lattice model describing some aspects of molecular orientation in solids. As expected the method yields the exact fixed point for d=2, given the formal equivalence to the two-dimensional Ising model. For d=3 such equivalence breaks down and an approximate estimate of the fixed point is explicitly recovered and compared to the mean-field prediction. The possible existence of an intermediate symmetric phase is discussed.
We investigate a two-dimensional Ising model with long-range interactions that emerge from a generalization of the magnetic dipolar interaction in spin systems with in-plane spin orientation. This interaction is, in general, anisotropic whereby in the present work we focus on the isotropic case for which the model is found to be at its upper critical dimensionality. To investigate the critical behavior the temperature and field dependence of several quantities are studied by means of Monte Carlo simulations. On the basis of the Privman-Fisher hypothesis and results of the renormalization group the numerical data are analyzed in the framework of a finite-size scaling analysis and compared to finite-size scaling functions derived from a Ginzburg-Landau-Wilson model in zero mode (mean-field) approximation. The obtained excellent agreement suggests that at least in the present case the concept of universal finite-size scaling functions can be extended to the upper critical dimensionality.
The critical behavior of the two-dimensional Ising Model with non-conserved order parameter in steady-state shear is studied by large-scale Monte Carlo simulations. Studying the structure factor S ( q x , q y ) in the disordered phase, the ratio of correlation length exponents ν x /ν y in the two lattice directions ( x , y ) is estimated, and the critical temperature is determined as a function of the shear rate as T c ( ) − T c (0) ∝ ϕ with ϕ≈0.45. Critical exponents β≈0.37, γ≈1.1, ; ν ⊥ ≈0.46, ν ∥ ≈1.38 are roughly compatible with anisotropic hyperscaling.
Questions concerning the universality between different two-dimensional Ising-like systems have recently been raised, with evidence for the existence of several different subclasses of Ising-like systems being presented. Some new facts concerning similarity between border model and low-temperature spin-1 Ising model series analyses are noted. For the spin-1 case an alternative determination of the critical temperature determines unequivocally that this model is in the Ising universality class and it is suggested that a similar independent result could lead to a similar conclusion in the border model case. The possibility of an analytic correction to scaling in the spin-1 model is proposed.
We investigate a spin-3/2 Ising model on a square lattice by using Monte Carlo (MC) simulations and a density matrix renormalization group (DMRG) method. We obtain phase diagrams of the ground state. We carried out the MC simulations in order to calculate the critical temperature and the critical exponent η. We estimate the critical exponents ν, α and γ by using a finite size scaling analysis. We also calculate the critical temperature, the central charge c and the critical exponent ν by using the DMRG method. From the results obtained by the MC simulations and by the DMRG method, we discuss a universality class of the system. It is found that the values of critical exponents are determined by means of the degeneracy of the ground state. This result suggests that the universality does not hold for the spin-3/2 Ising model. Furthermore we find a possibility of a breakdown of the hyperscaling law. We also confirm that the ferrimagnetic phase exists in a parameter region where the ground state has a fourfold degeneracy, and that a transition between the ferrimagnetic phase and the ferromagnetic phase exists in addition to that between the ferromagnetic phase and the paramagnetic phase.
The two-dimensional Ising model in the rectangular lattice is generalised to include m-spin interactions in one direction and two-spin interactions in the other. This model is self-dual and the critical line is the same as in the conventional Ising model with m=2. The m=3 model is solved in a generalised mean-field approximation where the transition is first order. Using a phenomenological renormalisation group approach, approximate values of the thermal and magnetic exponents y t and y h are obtained for m=3 and m=4. The m=3 results together with symmetry arguments suggest that this model belongs to the same class of universality as the Baxter-Wu and four-state ferromagnetic Potts models. When m=4 the transition is probably first order.