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 the Physical Society of Japan

Ising Ferromagnets with Random Impurities

Noboru Matsudaira

Published Dec 1, 1973• 99 citations
Publisher

Abstract

The Ising of spin 1/2 with the nearest-neighbor interaction, in which nonmagnetic impurity atoms are dissolved, is investigated. Based on the identity < s i >=<tanh ( K Σ s j )>, decoupling approximation is introduced for treating the many-spin correlation functions. The dependences of the critical temperature and of the spontaneous magnetization on the impurity concentration are calculated for two- and three-dimensional lattices. The results for the critical temperature and the critical concentration are better than the results of the results of the extended molecular-field approximations of Mamada and Takano, or of Oguchi and Obokata. Especially for the three-dimensional lattices the critical obtained agrees with that of Sykes and Essam, estimated from the series expansion.

DOI
10.1143/jpsj.35.1593

Related papers

Approximate Treatment of the Two-Dimensional Ising Ferromagnet with Second-Neighbor Interaction

Sei-ichi Takase

The two-dimensional square Ising ferromagnet with the nearest-neighbor interaction - J and the second-neighbor interaction - J ' is investigated starting from the identity < s i >=<tanh (ΣK i j s j )>. The Hartree approximation is used for treating the many-spin correlation functions. The critical point thus determined has two values for J '/ J <0, and is corrected by the study of the spontaneous magnetization and the free energy to give a resonable function T c ( J '/ J ). Our approximation may be valid at least qualitatively as a whole.

Journal of the Physical Society of Japan•Mar 1, 1974•8 citations

Critical exponents associated with isolated impurities in ferromagnets

T. K. Bergstresser

We have investigated the spin‐1/2 simple cubic Ising model with isolated impurities. This work does not make use of a mean‐field approximation, but does use exponents and correlation functions deduced by series exapnsions methods. Denote by σ the bulk magnetization, and denote by σi either the statistically averaged spin on the impurity or on a neighbor to the impurity. We have obtained detailed numerical results of σi/σ and σi for 0?T?Tc, and of (∂σi/∂H)/(∂σ/∂H) for T?Tc. The quantity σi has the same critical exponents β, γ, γ′ and δ as does the bulk magnetization σ. The leading singular correction terms introducted by the impurity have also been found. This work supports the hypothesis that hyperine fields on probe atoms which are dilute impurities in the magnetic lattice do indeed measure the bulk exponents for T sufficiently close to Tc. We provide quantitative estimates of the distortion in β and γ produced by an impurity when T is not close to Tc.

AIP conference proceedings•Jan 1, 1976•2 citations

Low Temperature Phase of a Dilute Axial-next-nearest-neighbor Ising Model in Two Dimensions

Fumitaka Matsubara, Nobuo Suzuki, Takayuki Shirakura

We examine the ground state of a dilute two-dimensional axial-next-nearest-neighbor Ising model with the interaction ratio κ (≡ −J2/J1) > 1/2. We calculate the spin overlap function Q together with the antiphase (〈2〉 phase) magnetization M〈2〉 on medium lattices of L × L (L ≤ 24) using a cluster heat bath method. The 〈2〉 phase is destroyed by a small quantity of non-magnetic atoms due to a random impurity effect, and a critical phase is realized, which is composed of segments of four types of 〈2〉 domains arranged in order. As the spin concentration c decreases, a critical phase with different ferromagnetic clusters appears near above the percolation threshold cp of the lattice. A phase diagram is proposed in the \((\kappa ,c)\) plain at low temperatures.

Journal of the Physical Society of Japan•Nov 5, 2020•0 citations

Effective-field model with correlations for Ising systems

G. Bruce Taggart, I. P. Fittipaldi

Using the techniques introduced by Honmura and Kaneyoshi, we show that two sets of identities for the two-state Ising model with $z$ nearest neighbors can be used as a basis for various approximation schemes which can systematically include correlation effects. In one dimension the results in zero field are exact for the magnetization, nearest-neighbor correlation function, and critical temperature. By neglecting correlations of more than three sites we introduce a simple approximation in two and three dimensions for the critical temperature. This simple approximation yields results which improve on those of the Bethe approximation. We discuss how the approximation may be improved and extended to other Ising problems.

Physical review. B, Condensed matter•Jun 1, 1982•44 citations

Computer simulation of the critical behavior of 3D disordered ising model

V. V. Prudnikov, Pavel Vladimirovich Prudnikov, A. N. Vakilov +1 more

The critical behavior of the disordered ferromagnetic Ising model is studied numerically by the Monte Carlo method in a wide range of variation of concentration of nonmagnetic impurity atoms. The temperature dependences of correlation length and magnetic susceptibility are determined for samples with various spin concentrations and various linear sizes. The finite-size scaling technique is used for obtaining scaling functions for these quantities, which exhibit a universal behavior in the critical region; the critical temperatures and static critical exponents are also determined using scaling corrections. On the basis of variation of the scaling functions and values of critical exponents upon a change in the concentration, the conclusion is drawn concerning the existence of two universal classes of the critical behavior of the diluted Ising model with different characteristics for weakly and strongly disordered systems.

arXiv:0709.1450•Journal of Experimental and Theoretical Physics•Aug 1, 2007•35 citations

Two-Dimensional Ising Ferromagnet with Second-Neighbor Interaction. II. Monte Carlo Method

Sei-ichi Takase

The static properties are investigated by computer simulation for the square Ising ferromagnet in which the nearest-neighbor interactions are ferromagnetic and the second interactions antiferromagnetic. Energy and magnetization at various temperatures are obtained by Monte Carlo calculations performed on the lattice with 20×20 spins. The Curie point determined as the inflection point of the energy-vs-temperature curve agrees with the result by Fan and Wu rather than other analytical approximations. The behavior of the spontaneous magnetization M 0 ( T ) for T → T c -0 as characterized by the critical exponent β is shown by comparing the data with an analytical equation for the magnetization. The exponent β remains probably unchanged from the original value 1/8.

Journal of the Physical Society of Japan•May 15, 1976•5 citations
The Ising of spin 1/2 with the nearest-neighbor interaction, in which nonmagnetic impurity atoms are dissolved, is investigated. Based on the identity < s i >=<tanh ( K Σ s j )>, decoupling approximation is introduced for treating the many-spin correlation functions. The dependences of the critical temperature and of the spontaneous magnetization on the impurity concentration are calculated for two- and three-dimensional lattices. The results for the critical temperature and the critical concentration are better than the results of the results of the extended molecular-field approximations of Mamada and Takano, or of Oguchi and Obokata. Especially for the three-dimensional lattices the critical obtained agrees with that of Sykes and Essam, estimated from the series expansion.
The two-dimensional square Ising ferromagnet with the nearest-neighbor interaction - J and the second-neighbor interaction - J ' is investigated starting from the identity < s i >=<tanh (ΣK i j s j )>. The Hartree approximation is used for treating the many-spin correlation functions. The critical point thus determined has two values for J '/ J <0, and is corrected by the study of the spontaneous magnetization and the free energy to give a resonable function T c ( J '/ J ). Our approximation may be valid at least qualitatively as a whole.
We have investigated the spin‐1/2 simple cubic Ising model with isolated impurities. This work does not make use of a mean‐field approximation, but does use exponents and correlation functions deduced by series exapnsions methods. Denote by σ the bulk magnetization, and denote by σi either the statistically averaged spin on the impurity or on a neighbor to the impurity. We have obtained detailed numerical results of σi/σ and σi for 0?T?Tc, and of (∂σi/∂H)/(∂σ/∂H) for T?Tc. The quantity σi has the same critical exponents β, γ, γ′ and δ as does the bulk magnetization σ. The leading singular correction terms introducted by the impurity have also been found. This work supports the hypothesis that hyperine fields on probe atoms which are dilute impurities in the magnetic lattice do indeed measure the bulk exponents for T sufficiently close to Tc. We provide quantitative estimates of the distortion in β and γ produced by an impurity when T is not close to Tc.
We examine the ground state of a dilute two-dimensional axial-next-nearest-neighbor Ising model with the interaction ratio κ (≡ −J2/J1) > 1/2. We calculate the spin overlap function Q together with the antiphase (〈2〉 phase) magnetization M〈2〉 on medium lattices of L × L (L ≤ 24) using a cluster heat bath method. The 〈2〉 phase is destroyed by a small quantity of non-magnetic atoms due to a random impurity effect, and a critical phase is realized, which is composed of segments of four types of 〈2〉 domains arranged in order. As the spin concentration c decreases, a critical phase with different ferromagnetic clusters appears near above the percolation threshold cp of the lattice. A phase diagram is proposed in the (κ,c)(\kappa ,c)(κ,c) plain at low temperatures.
Using the techniques introduced by Honmura and Kaneyoshi, we show that two sets of identities for the two-state Ising model with zzz nearest neighbors can be used as a basis for various approximation schemes which can systematically include correlation effects. In one dimension the results in zero field are exact for the magnetization, nearest-neighbor correlation function, and critical temperature. By neglecting correlations of more than three sites we introduce a simple approximation in two and three dimensions for the critical temperature. This simple approximation yields results which improve on those of the Bethe approximation. We discuss how the approximation may be improved and extended to other Ising problems.
The critical behavior of the disordered ferromagnetic Ising model is studied numerically by the Monte Carlo method in a wide range of variation of concentration of nonmagnetic impurity atoms. The temperature dependences of correlation length and magnetic susceptibility are determined for samples with various spin concentrations and various linear sizes. The finite-size scaling technique is used for obtaining scaling functions for these quantities, which exhibit a universal behavior in the critical region; the critical temperatures and static critical exponents are also determined using scaling corrections. On the basis of variation of the scaling functions and values of critical exponents upon a change in the concentration, the conclusion is drawn concerning the existence of two universal classes of the critical behavior of the diluted Ising model with different characteristics for weakly and strongly disordered systems.
The static properties are investigated by computer simulation for the square Ising ferromagnet in which the nearest-neighbor interactions are ferromagnetic and the second interactions antiferromagnetic. Energy and magnetization at various temperatures are obtained by Monte Carlo calculations performed on the lattice with 20×20 spins. The Curie point determined as the inflection point of the energy-vs-temperature curve agrees with the result by Fan and Wu rather than other analytical approximations. The behavior of the spontaneous magnetization M 0 ( T ) for T → T c -0 as characterized by the critical exponent β is shown by comparing the data with an analytical equation for the magnetization. The exponent β remains probably unchanged from the original value 1/8.