arXiv:cond-mat/9306027

Critical Point Correlation Function for the 2D Random Coupling Ising Model

Andrei Talapov, Vladimir S. Dotsenko

Published Jun 9, 1993• 0 citations

Abstract

High accuracy Monte Carlo simulation results for 1024*1024 Ising system with ferromagnetic impurity bonds are presented. Spin-spin correlation function at a critical point is found to be numerically very close to that of a pure system. This is not trivial since a critical temperature for the system with impurities is almost two times lower than pure Ising $T_c$. Small deviations from the pure behaviour contradict to some of the competing theories.

DOI: 10.48550/arxiv.cond-mat/9306027

Related papers

Critical-Point Correlation Function for the 2D Random Bond Ising Model

A. L Talapov, Щур Лев Николаевич

High-accuracy Monte Carlo simulation results for a 1024 * 1024 Ising system with ferromagnetic impurity bonds are presented. The spin-spin correlation function at a critical point is found to be numerically very close to that of a pure system. This is not trivial since the critical temperature for the system with impurities is almost two times lower than pure Ising T c . Finite corrections to the correlation function due to combined action of impurities and finite lattice size are described.

arXiv:hep-lat/9404002Europhysics Letters (EPL)Jul 20, 199430 citations

Critical point correlation function for the 2d random bond Ising model

Talapov, A L, Щур Лев Николаевич

High accuracy Monte Carlo simulation results for 1024*1024 Ising system with ferromagnetic impurity bonds are presented. Spin-spin correlation function at a critical point is found to be numerically very close to that of a pure system. This is not trivial since a critical temperature for the system with impurities is almost two times lower than pure Ising T.sub(c). Finite corrections to the correlation function due to combined action of impurities and finite lattice size are described.

Apr 6, 199429 citations

Two-point correlation function of the 2D Ising model with impurity lattice bonds

Виктор Степанович Доценко, Vl.S. Dotsenko

The two-point correlation function of the 2D Ising model with small concentration of impurity bonds ( nu imp <<1) is found to change its critical behaviour from ( sigma r sigma 0 ) approximately 1/R 1/4 for r<<R nu exp(c 1 / nu imp ) to ( sigma R delta 0 ) approximately exp<or=-(c 2 / nu imp )(ln ln R) 2 ) for R>>R, which implies the change of the critical index from 1 / 4 in the pure Ising model to 0 in the model with impurities.

Journal of Physics C Solid State PhysicsJun 20, 198220 citations

Correlation functions in the two-dimensional random-bond Ising model

S. L. A. de Queiroz, Robin B. Stinchcombe

arXiv:cond-mat/9604053Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsJul 1, 199618 citations

Critical region of the random bond Ising model

Jan 1, 19940 citations

Critical behaviour of the phase transition in the 2D Ising Model with impurities

Виктор Степанович Доценко, Victor S Dotsenko

Advances In PhysicsJan 1, 1983285 citations

arXiv:cond-mat/9306027

Critical Point Correlation Function for the 2D Random Coupling Ising Model

Andrei Talapov, Vladimir S. Dotsenko

Published Jun 9, 1993• 0 citations

arXiv HTML PDF

Abstract

T_c

. Small deviations from the pure behaviour contradict to some of the competing theories.

DOI: 10.48550/arxiv.cond-mat/9306027

Related papers

Critical-Point Correlation Function for the 2D Random Bond Ising Model

A. L Talapov, Щур Лев Николаевич

arXiv:hep-lat/9404002Europhysics Letters (EPL)Jul 20, 199430 citations

Critical point correlation function for the 2d random bond Ising model

Talapov, A L, Щур Лев Николаевич

High accuracy Monte Carlo simulation results for 1024*1024 Ising system with ferromagnetic impurity bonds are presented. Spin-spin correlation function at a critical point is found to be numerically very close to that of a pure system. This is not trivial since a critical temperature for the system with impurities is almost two times lower than pure Ising T.sub(c). Finite corrections to the correlation function due to combined action of impurities and finite lattice size are described.

Apr 6, 199429 citations

Two-point correlation function of the 2D Ising model with impurity lattice bonds

Виктор Степанович Доценко, Vl.S. Dotsenko

Journal of Physics C Solid State PhysicsJun 20, 198220 citations

Correlation functions in the two-dimensional random-bond Ising model

S. L. A. de Queiroz, Robin B. Stinchcombe

We consider long strips of finite width L\ensuremath{\le}13 sites of ferromagnetic Ising spins with random couplings distributed according to the binary distribution P( ${\mathit{J}}_{\mathit{ij}}$ )=1/2[\ensuremath{\delta}( ${\mathit{J}}_{\mathit{ij}}$ - ${\mathit{J}}_{0}$ )+\ensuremath{\delta}( ${\mathit{J}}_{\mathit{ij}}$ - ${\mathit{rJ}}_{0}$ )], 0r1. Spin-spin correlation functions 〈${\mathrm{\ensuremath{\sigma}}}_{0}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{R}}$〉 along the ``infinite'' direction are computed by transfer-matrix methods, at the critical temperature of the corresponding two-dimensional system, and their probability distribution is investigated. We show that, although in-sample fluctuations do not die out as strip length is increased, averaged values converge satisfactorily. These latter are very close to the critical correlation functions of the pure Ising model, in agreement with recent Monte Carlo simulations. A scaling approach is formulated, which provides the essential aspects of the R and L dependence of the probability distribution of ln〈${\mathrm{\ensuremath{\sigma}}}_{0}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{R}}$〉, including the result that the appropriate scaling variable is R/L. Predictions from scaling theory are borne out by numerical data, which show the probability distribution of ln〈${\mathrm{\ensuremath{\sigma}}}_{0}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{R}}$〉 to be remarkably skewed at short distances, approaching a Gaussian only as R/L\ensuremath{\gg}1. \textcopyright{} 1996 The American Physical Society.We consider long strips of finite width L\ensuremath{\le}13 sites of ferromagnetic Ising spins with random couplings distributed according to the binary distribution P(${\mathit{J}}_{\mathit{ij}}$)=1/2[\ensuremath{\delta}(${\mathit{J}}_{\mathit{ij}}$-${\mathit{J}}_{0}$)+\ensuremath{\delta}(${\mathit{J}}_{\mathit{ij}}$- ${\mathit{rJ}}_{0}$)], 0r1. Spin-spin correlation functions 〈${\mathrm{\ensuremath{\sigma}}}_{0}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{R}}$〉 along the ``infinite'' direction are computed by transfer-matrix methods, at the critical temperature of the corresponding two-dimensional system, and their probability distribution is investigated. We show that, although in-sample fluctuations do not die out as strip length is increased, averaged values converge satisfactorily. These latter are very close to the critical correlation functions of the pure Ising model, in agreement with recent Monte Carlo simulations. A scaling approach is formulated, which provides the essential aspects of the R and L dependence of the probability distribution of ln〈${\mathrm{\ensuremath{\sigma}}}_{0}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{R}}$〉, including the result that the appropriate scaling variable is R/L. Predictions from scaling theory are borne out by numerical data, which show the probability distribution of ln〈${\mathrm{\ensuremath{\sigma}}}_{0}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{R}}$〉 to be remarkably skewed at short distances, approaching a Gaussian only as R/L\ensuremath{\gg}1. \textcopyright{} 1996 The American Physical Society.

arXiv:cond-mat/9604053Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsJul 1, 199618 citations

Critical region of the random bond Ising model

PACS. 05.50 – Lattice theory and statistics; Ising problems PACS. 75.10 – General theory and models of magnetic ordering We describe results of the cluster algorithm Special Purpose Processor simulations of the 2D Ising model with impurity bonds. Use of large lattices, with the number of spins up to 10 6, permitted to define critical region of temperatures, where both finite size corrections and corrections to scaling are small. High accuracy data unambiguously show increase of magnetization and magnetic susceptibility effective exponents β and γ, caused by impurities. The M and χ singularities became more sharp, while the specific heat singularity is smoothed. The specific heat is found to be in a good agreement with Dotsenko-Dotsenko theoretical predictions in the The problem of inhomogenities influence on phase transitions has a long history. The first model with some kind of impurities to be studied, is, due to its simplicity, the Ising model. Effect of randomness on the critical behaviour of different IsingPACS. 05.50 – Lattice theory and statistics; Ising problems PACS. 75.10 – General theory and models of magnetic ordering We describe results of the cluster algorithm Special Purpose Processor simulations of the 2D Ising model with impurity bonds. Use of large lattices, with the number of spins up to 10 6, permitted to define critical region of temperatures, where both finite size corrections and corrections to scaling are small. High accuracy data unambiguously show increase of magnetization and magnetic susceptibility effective exponents β and γ, caused by impurities. The M and χ singularities became more sharp, while the specific heat singularity is smoothed. The specific heat is found to be in a good agreement with Dotsenko-Dotsenko theoretical predictions in the The problem of inhomogenities influence on phase transitions has a long history. The first model with some kind of impurities to be studied, is, due to its simplicity, the Ising model. Effect of randomness on the critical behaviour of different Ising

Jan 1, 19940 citations

Critical behaviour of the phase transition in the 2D Ising Model with impurities

Виктор Степанович Доценко, Victor S Dotsenko

The problem of the effect of inhomogeneous impurities in a lattice structure on the critical behaviour of systems which undergo a second-order phase transition is studied on a model system, the 2D Ising Model with impurity bonds randomly distributed over the lattice. It is known that in the critical region the 2D Ising Model is equivalent to the model of free fermions. We show that the effect of impurities is to add a four-fermion interaction with the corresponding charge proportional to the concentration of impurities. The resulting fermion model is simple enough and can be studied exactly by renormalization group methods. We show that for any small concentration of impurity bonds a new critical regime is established if we go sufficiently close to the phase-transition point. In particular we find that the specific heat singularity changes from C ˜ ln 1/|τ|(τ=(Τ − Τc/Τc to C ˜ ln ln 1/|τ| for τ≪τi ˜ exp(−const/c i), c i being the concentration of impurities, while the spin-spin correlation function in the critical point changes more seriously: from <σ R σΟ> ˜ 1/R1/4 to <σ R σΟ> ˜ exp { − (const/c i)(ln ln R2 )} for R≫R i˜ exp (const/c i), which corresponds to a change of the critical exponent η from 1/4 to 0.The problem of the effect of inhomogeneous impurities in a lattice structure on the critical behaviour of systems which undergo a second-order phase transition is studied on a model system, the 2D Ising Model with impurity bonds randomly distributed over the lattice. It is known that in the critical region the 2D Ising Model is equivalent to the model of free fermions. We show that the effect of impurities is to add a four-fermion interaction with the corresponding charge proportional to the concentration of impurities. The resulting fermion model is simple enough and can be studied exactly by renormalization group methods. We show that for any small concentration of impurity bonds a new critical regime is established if we go sufficiently close to the phase-transition point. In particular we find that the specific heat singularity changes from C ˜ ln 1/|τ|(τ=(Τ − Τc/Τc to C ˜ ln ln 1/|τ| for τ≪τi ˜ exp(−const/c i), c i being the concentration of impurities, while the spin-spin correlation function in the critical point changes more seriously: from <σ R σΟ> ˜ 1/R1/4 to <σ R σΟ> ˜ exp { − (const/c i)(ln ln R2 )} for R≫R i˜ exp (const/c i), which corresponds to a change of the critical exponent η from 1/4 to 0.

Advances In PhysicsJan 1, 1983285 citations

T_c

. Small deviations from the pure behaviour contradict to some of the competing theories.

We consider long strips of finite width L\ensuremath{\le}13 sites of ferromagnetic Ising spins with random couplings distributed according to the binary distribution P(

{\mathit{J}}_{\mathit{ij}}

)=1/2[\ensuremath{\delta}(

{\mathit{J}}_{\mathit{ij}}

{\mathit{J}}_{0}

)+\ensuremath{\delta}(

{\mathit{J}}_{\mathit{ij}}

{\mathit{rJ}}_{0}

)], 0r1. Spin-spin correlation functions 〈${\mathrm{\ensuremath{\sigma}}}_{0}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{R}}$〉 along the ``infinite'' direction are computed by transfer-matrix methods, at the critical temperature of the corresponding two-dimensional system, and their probability distribution is investigated. We show that, although in-sample fluctuations do not die out as strip length is increased, averaged values converge satisfactorily. These latter are very close to the critical correlation functions of the pure Ising model, in agreement with recent Monte Carlo simulations. A scaling approach is formulated, which provides the essential aspects of the R and L dependence of the probability distribution of ln〈${\mathrm{\ensuremath{\sigma}}}_{0}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{R}}$〉, including the result that the appropriate scaling variable is R/L. Predictions from scaling theory are borne out by numerical data, which show the probability distribution of ln〈${\mathrm{\ensuremath{\sigma}}}_{0}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{R}}$〉 to be remarkably skewed at short distances, approaching a Gaussian only as R/L\ensuremath{\gg}1. \textcopyright{} 1996 The American Physical Society.