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

Scaling behaviour of the correlation length for the two-point correlation function in the random field Ising chain

Adrian Lange, Robin B. Stinchcombe

Published Jul 7, 1996• 0 citations
Publisher

Abstract

We study the general behaviour of the correlation length for the two-point correlation function of the local fields in an Ising chain with binary distributed fields. At zero field it is shown that is the same as the zero-field correlation length for the spin - spin correlation function. For the field-dominated behaviour of we find an exponent for the power-law divergence which is smaller than the exponent for the spin - spin correlation length. The entire behaviour of the correlation length can be described by a single crossover scaling function involving the new critical exponent.

DOI
10.1088/0305-4470/29/13/019

Related papers

Exact spin–spin correlation function for the zero-temperature random-field Ising model

T P Handford, F-J Perez-Reche, S N Taraskin

An exact expression for the spin-spin correlation function is derived for the zero-temperature random-field Ising model defined on a Bethe lattice of arbitrary coordination number.The correlation length describing dynamic spinspin correlations and separated from the intrinsic topological length scale of the Bethe lattice is shown to diverge as a power law at the critical point.The critical exponents governing the behaviour of the correlation length are consistent with the mean-field values found for a hypercubic lattice with dimension greater than the upper critical dimension.

arXiv:1106.3424•Journal of Statistical Mechanics Theory and Experiment•Jan 10, 2012•7 citations

Correlation function of random-field Ising chains: is it Lorentzian or not?

J. M. Luck, Theo M. Nieuwenhuizen

The two-point connected correlation function, or wavevector-dependent susceptibility, of the ferromagnetic Ising chain in a random field is calculated exactly at any temperature, for a two-parameter family of diluted symmetric exponential distributions of the magnetic fields. Thermodynamic properties of this model have been derived in a previous work by the authors. Besides the correlation function itself, the solution provides exact results for the (usual) susceptibility, the correlation length and the Edwards-Anderson parameter. The low-temperature regime is examined in full detail: the authors obtain in closed form the limit value at zero temperature of various quantities, and the first correction to this limit, which behaves linearly with temperature. The correlation length is discontinuous at zero temperature. They also derive the scaling form of the correlation function for small p, where the dilution p is the probability for a spin to be subjected to a non-zero random field. Even in this limit, the correlation function is more complicated than the simple Lorentzian predicted, e.g., by mean-field theory.

Journal of Physics A Mathematical and General•Jun 21, 1989•18 citations

Crossover relationship between the magnetic susceptibility and the correlation length

Jeffrey F. Nicoll, T. S. Chang

In the asymptotic critical region of a ferromagnetic spin system, the magnetic susceptibility χ and the correlation length ξ are related by the power law: χ∼ξ2−η, where η is the correlation function exponent. Somewhat farther from the critical point, but still within the critical region, we expect the behavior to be mean-field in character: χ∼ξ2. We give an exact relationship between χ and ξ in terms of a differential renormalization group approach. The leading behavior in the disordered phase is explicitly calculated for the n-component Wilson-Fisher model to O(ε2), ε≡4−d. The result includes both limiting power laws in a smooth crossover function. For the Ising case, n=1, the relationship is extended to the entire critical region, including the ordered phase. In this case, even in the asymptotic limit, the susceptibility differs from ξ2−η by a factor which depends on the magnetization M. This function is calculated to leading order and correctly gives the associated amplitude ratios to O(ε). Applications to the general p-point vertex function are discussed.

Journal of Applied Physics•Mar 1, 1979•1 citation

The Energy-Energy Correlation Function of the Random Bond Ising Model in Two Dimensions

K. Ziegler

The energy-energy correlation function of the two-dimensional Ising model with weakly fluctuating random bonds is evaluated in the large scale limit. Two correlation lengths exist in contrast to one correlation length in the pure 2D Ising model: one is finite whereas the other is divergent at the critical points. The corresponding exponent of the divergent correlation length is $ν_e=1/2$ in contrast to the pure system where $ν_e=1$. The calculation is based on a previously developed effective field theory for the energy density fluctuations.

arXiv:cond-mat/9312017•Dec 3, 1993•0 citations

Universality of the critical behaviour of the weakly disordered Baxter model

Виктор Степанович Доценко

The two-point spin-spin correlation function of the weakly disordered Baxter model in the phase transition point is calculated. The corresponding critical exponent appears to be zero, i.e. the decay of the correlation function is slower than the power law: R - eta . Together with the previous result that the specific heat critical exponent of the weakly disordered Baxter model is also universal: alpha -0, it gives the complete set of universal critical exponents which describe the phase transition. The critical exponents of the weakly disordered Baxter model coincide with those of the weakly disordered Ising model.

Journal of Physics A Mathematical and General•Apr 1, 1985•14 citations

Correlation functions, free energies, and magnetizations in the two-dimensional random-field Ising model

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

Transfer-matrix methods are used to calculate spin-spin correlation functions (G), Helmholtz free energies (f) and magnetizations (m) in the two-dimensional random-field Ising model close to the zero-field bulk critical temperature T(c 0), on long strips of width L=3-18 sites, for binary field distributions. Analysis of the probability distributions of G for varying spin-spin distances R shows that describing the decay of their averaged values by effective correlation lengths is a valid procedure only for not very large R. Connections between field and correlation function distributions at high temperatures are established, yielding approximate analytical expressions for the latter, which are used for computation of the corresponding structure factor. It is shown that, for fixed R/L, the fractional widths of correlation-function distributions saturate asymptotically with L-2.2. Considering an added uniform applied field h, a connection between f(h), m(h), the Gibbs free energy g(m) and the distribution function for the uniform magnetization in a zero uniform field, P0(m), is derived and first illustrated for pure systems, and then applied for nonzero random field. From finite-size scaling and crossover arguments, coupled with numerical data, it is found that the width of P0(m) varies against (nonvanishing, but small) random-field intensity H0 as H(-3/7)(0).

arXiv:cond-mat/0105326•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Aug 28, 2001•5 citations
We study the general behaviour of the correlation length for the two-point correlation function of the local fields in an Ising chain with binary distributed fields. At zero field it is shown that is the same as the zero-field correlation length for the spin - spin correlation function. For the field-dominated behaviour of we find an exponent for the power-law divergence which is smaller than the exponent for the spin - spin correlation length. The entire behaviour of the correlation length can be described by a single crossover scaling function involving the new critical exponent.
An exact expression for the spin-spin correlation function is derived for the zero-temperature random-field Ising model defined on a Bethe lattice of arbitrary coordination number.The correlation length describing dynamic spinspin correlations and separated from the intrinsic topological length scale of the Bethe lattice is shown to diverge as a power law at the critical point.The critical exponents governing the behaviour of the correlation length are consistent with the mean-field values found for a hypercubic lattice with dimension greater than the upper critical dimension.
The two-point connected correlation function, or wavevector-dependent susceptibility, of the ferromagnetic Ising chain in a random field is calculated exactly at any temperature, for a two-parameter family of diluted symmetric exponential distributions of the magnetic fields. Thermodynamic properties of this model have been derived in a previous work by the authors. Besides the correlation function itself, the solution provides exact results for the (usual) susceptibility, the correlation length and the Edwards-Anderson parameter. The low-temperature regime is examined in full detail: the authors obtain in closed form the limit value at zero temperature of various quantities, and the first correction to this limit, which behaves linearly with temperature. The correlation length is discontinuous at zero temperature. They also derive the scaling form of the correlation function for small p, where the dilution p is the probability for a spin to be subjected to a non-zero random field. Even in this limit, the correlation function is more complicated than the simple Lorentzian predicted, e.g., by mean-field theory.
In the asymptotic critical region of a ferromagnetic spin system, the magnetic susceptibility χ and the correlation length ξ are related by the power law: χ∼ξ2−η, where η is the correlation function exponent. Somewhat farther from the critical point, but still within the critical region, we expect the behavior to be mean-field in character: χ∼ξ2. We give an exact relationship between χ and ξ in terms of a differential renormalization group approach. The leading behavior in the disordered phase is explicitly calculated for the n-component Wilson-Fisher model to O(ε2), ε≡4−d. The result includes both limiting power laws in a smooth crossover function. For the Ising case, n=1, the relationship is extended to the entire critical region, including the ordered phase. In this case, even in the asymptotic limit, the susceptibility differs from ξ2−η by a factor which depends on the magnetization M. This function is calculated to leading order and correctly gives the associated amplitude ratios to O(ε). Applications to the general p-point vertex function are discussed.
The energy-energy correlation function of the two-dimensional Ising model with weakly fluctuating random bonds is evaluated in the large scale limit. Two correlation lengths exist in contrast to one correlation length in the pure 2D Ising model: one is finite whereas the other is divergent at the critical points. The corresponding exponent of the divergent correlation length is νe=1/2ν_e=1/2νe​=1/2 in contrast to the pure system where νe=1ν_e=1νe​=1. The calculation is based on a previously developed effective field theory for the energy density fluctuations.
The two-point spin-spin correlation function of the weakly disordered Baxter model in the phase transition point is calculated. The corresponding critical exponent appears to be zero, i.e. the decay of the correlation function is slower than the power law: R - eta . Together with the previous result that the specific heat critical exponent of the weakly disordered Baxter model is also universal: alpha -0, it gives the complete set of universal critical exponents which describe the phase transition. The critical exponents of the weakly disordered Baxter model coincide with those of the weakly disordered Ising model.
Transfer-matrix methods are used to calculate spin-spin correlation functions (G), Helmholtz free energies (f) and magnetizations (m) in the two-dimensional random-field Ising model close to the zero-field bulk critical temperature T(c 0), on long strips of width L=3-18 sites, for binary field distributions. Analysis of the probability distributions of G for varying spin-spin distances R shows that describing the decay of their averaged values by effective correlation lengths is a valid procedure only for not very large R. Connections between field and correlation function distributions at high temperatures are established, yielding approximate analytical expressions for the latter, which are used for computation of the corresponding structure factor. It is shown that, for fixed R/L, the fractional widths of correlation-function distributions saturate asymptotically with L-2.2. Considering an added uniform applied field h, a connection between f(h), m(h), the Gibbs free energy g(m) and the distribution function for the uniform magnetization in a zero uniform field, P0(m), is derived and first illustrated for pure systems, and then applied for nonzero random field. From finite-size scaling and crossover arguments, coupled with numerical data, it is found that the width of P0(m) varies against (nonvanishing, but small) random-field intensity H0 as H(-3/7)(0).