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 Mathematical Physics

On the Two-Dimensional Ising Model with Random Impurities

Eytan Barouch

Published Aug 1, 1971• 14 citations
Publisher

Abstract

The McCoy-Wu treatment of the two-dimensional Ising model with random impurities is generalized to allow both vertical and horizontal interactions to vary randomly from row to row (while still keeping them constant in any row). If the random impurities are narrowly distributed, the order-one term in the specific heat at Tc is infinitely differentiable but nonanalytic.

DOI
10.1063/1.1665775

Related papers

Theory of a Two-Dimensional Ising Model with Random Impurities. II. Spin Correlation Functions

Barry M. McCoy, Tai Tsun Wu

We continue our investigation of an Ising model with immobile random impurities by studying the spin-spin correlation functions. These correlations are not probability-1 objects and have a probability distribution. When the random bonds have the particular distribution function studied in the first paper of this series, we demonstrate that the average value and the second moment of the temperature derivatives of these correlations are infinitely differentiable but fail to be analytic at ${T}_{c}$, the temperature at which the observable specific heat fails to be analytic. When $T<{T}_{c}$, we consider ${S}_{\ensuremath{\infty}}(l)=limit of\text{}〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{l,m}〉\text{as}m\ensuremath{\rightarrow}\ensuremath{\infty}$. This limit is not independent of $l$. In the special case that the random bonds are symmetrically distributed about the $l$th row, the geometric mean of ${S}_{\ensuremath{\infty}}(l)$ is computed and shown to vanish exponentially rapidly when $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$. We contrast this with a lower bound that shows that the spontaneous magnetization can vanish no more rapidly than ${T}_{c}\ensuremath{-}T$, and present a description of how the local magnetization ${S}_{\ensuremath{\infty}}{(l)}^{\frac{1}{2}}$ behaves as $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$.

Physical Review•Dec 10, 1969•125 citations

Saddle point solution of the Landau-Ginzburg-Wilson Hamiltonian for the McCoy-Wu Ising model

Xintian Wu

The Landau-Ginzburg-Wilson Hamiltonian for the McCoy-Wu strip-random Ising model on a two-dimensional lattice is studied. In the Landau-Ginzburg-Wilson Hamiltonian, the random temperature only depends on one of the coordinates ($x$), the horizontal coordinate. The difference (rather than differential) saddle point equation is solved numerically. In the critical regime, there exist locally ordered regions, where the saddle point solutions are not zero. Due to the translational invariance in the vertical direction, the locally ordered regions are striplike in the vertical direction. In the excited state solutions, there exist horizontal domain walls across the locally ordered regions. It is found that these horizontal domain walls are stabilized by free-energy barriers due to the discreteness. The stability conditions for such domain walls are studied. Usually, the domain wall across multiple locally ordered regions can be stable. These multiple locally ordered regions form an elementary cluster. The criterion for multiple locally ordered regions forming an elementary cluster is obtained. At the saddle point level, we get an effective Hamiltonian in which the elementary clusters behave as Ising spins and are coupled with their neighbors. For the random temperatures with narrow uniform probability distributions, the distribution of elementary cluster size, the horizontal and vertical bonds between elementary clusters are calculated numerically. The vertical bonds are extremely strong and the horizontal bonds are extremely weak. The distribution of vertical bonds is exponential. There is an exponentially small probability of elementary clusters with extremely large size. These elementary clusters with extremely large size will cause the susceptibility divergent in a range of temperature. The random temperature with binary probability distributions is also studied. The qualitative properties are the same. The Griffiths-McCoy singularity in the saddle point solutions does not depend on the distribution of random temperature.

Physical review. B./Physical review. B•Jun 20, 2018•1 citation

Theory of the two-dimensional Ising model with random impurities

Vytautas Victor Palciauskas

The effects of random impurities on the various thermodynamic functions are studied with the aid of the two-dimensional Ising model. Of special interest are the limiting forms of these functions near the critical point. This problem is approached by a Green's function formulation as developed by Kadanoff. Since experiments only measure the average effects of impurities, we consider the average of the Green's function over all distributions of impurities and calculate it by perturbation theory. From this function we find the location of the new critical temperature and the temperature dependence of the coherence length. Thermodynamic and correlation functions are calculated by means of perturbation expansions and their functional dependence on the coherence length investigated. It is found that the magnetic properties such as the magnetization, susceptibility, and spin-spin correlations retain the same functional dependence on the coherence length. The specific heat is found to have a finite value at the critical point, and the T = Tc form of the energy-energy correlation c function is changed. The thermodynamic functions exhibit new critical indices very close to Tc. These renormalized exponents have been predicted c in literature and are found to satisfy the scaling laws predictions.

•Jan 1, 1969•4 citations

Thermodynamics of Ising models with layered randomness: Exact solutions on square and triangular lattices

Theo M. Nieuwenhuizen, Henri Orland

Two-dimensional Ising models are studied on square and triangular lattices with layered disordered bonds (McCoy-Wu-type models). It is shown that exact solutions exist for diluted exponential distributions of couplings. In the ferromagnetic case the free energy exhibits an essential singularity (slightly different from the result of McCoy and Wu) at the transition point. If frustration is present the transitions occurs at T=0, and the specific heat is linear.

Physical review. B, Condensed matter•Sep 1, 1989•7 citations

Theory of a Two-Dimensional Ising Model with Random Impurities. III. Boundary Effects

Barry M. McCoy

We study the effects which immobile random impurities may have on magnetic properties and spin-spin correlation functions of a ferromagnetic system properties the Curie temperature. This is done within the context of the model introduced in the first paper of this series by computing the spin correlation functions on the boundary of a half-plane of random Ising spins, where the boundary row is allowed to intereact with a magnetic field $\mathfrak{H}$. We find that as $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$ the average boundary spontaneous magnetization vanishes as ${T}_{c}\ensuremath{-}T$. However, the average boundary magnetic susceptibility is shown not to exist for a finite range of temperature about ${T}_{c}$. Furthermore, at ${T}_{c}$ the average boundary magnetization behaves as $\ensuremath{-}{N}^{\ensuremath{-}1}$ sign $(\mathfrak{H}){[\mathrm{ln} N{\ensuremath{\beta}}_{c}|\mathfrak{H}|]}^{\ensuremath{-}1}$, where ${N}^{\ensuremath{-}1}$ is a measure of the width of the distribution of random bonds. Whenever $T\ensuremath{-}{T}_{c}=O({N}^{\ensuremath{-}2})$, the average spin-spin correlation function for two spins on the boundary is shown to approach its limit at infinite separation as some inverse power of the separation instead of as an exponential. At ${T}_{c}$ this average correlation function behaves asympotically as ${N}^{\ensuremath{-}2} {(\mathrm{ln}m{N}^{\ensuremath{-}2})}^{\ensuremath{-}1}$ when $m$ the separation between boundary spins, is large. Finally, we make the probabilistic nature of the boundary spontaneous magnetization more precise by computing its probability distribution function.

Physical Review•Dec 10, 1969•118 citations

Ising ferromagnet with quenched impurities

E. Stoll, T. Schneider

Using the Monte Carlo technique the authors have studied a two-dimensional Ising model with quenched non-magnetic impurities. In particular they calculated the temperature dependence of the order parameter, of the zero-field susceptibility and of the specific heat. These results indicate that a second-order phase transition is preserved for p>or=0.9 where 1-p denotes the concentration of randomly distributed non-magnetic sites. Moreover, they yield beta approximately=1/8 and gamma approximately=7/4, indicating that these exponents equal the corresponding ones for the pure Ising model.

Journal of Physics A Mathematical and General•Jul 1, 1976•26 citations
The McCoy-Wu treatment of the two-dimensional Ising model with random impurities is generalized to allow both vertical and horizontal interactions to vary randomly from row to row (while still keeping them constant in any row). If the random impurities are narrowly distributed, the order-one term in the specific heat at Tc is infinitely differentiable but nonanalytic.
We continue our investigation of an Ising model with immobile random impurities by studying the spin-spin correlation functions. These correlations are not probability-1 objects and have a probability distribution. When the random bonds have the particular distribution function studied in the first paper of this series, we demonstrate that the average value and the second moment of the temperature derivatives of these correlations are infinitely differentiable but fail to be analytic at Tc{T}_{c}Tc​, the temperature at which the observable specific heat fails to be analytic. When T<TcT<{T}_{c}T<Tc​, we consider ${S}_{\ensuremath{\infty}}(l)=limit of\text{}〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{l,m}〉\text{as}m\ensuremath{\rightarrow}\ensuremath{\infty}$. This limit is not independent of lll. In the special case that the random bonds are symmetrically distributed about the lllth row, the geometric mean of ${S}_{\ensuremath{\infty}}(l)$ is computed and shown to vanish exponentially rapidly when $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$. We contrast this with a lower bound that shows that the spontaneous magnetization can vanish no more rapidly than ${T}_{c}\ensuremath{-}T$, and present a description of how the local magnetization ${S}_{\ensuremath{\infty}}{(l)}^{\frac{1}{2}}$ behaves as $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$.
The Landau-Ginzburg-Wilson Hamiltonian for the McCoy-Wu strip-random Ising model on a two-dimensional lattice is studied. In the Landau-Ginzburg-Wilson Hamiltonian, the random temperature only depends on one of the coordinates (xxx), the horizontal coordinate. The difference (rather than differential) saddle point equation is solved numerically. In the critical regime, there exist locally ordered regions, where the saddle point solutions are not zero. Due to the translational invariance in the vertical direction, the locally ordered regions are striplike in the vertical direction. In the excited state solutions, there exist horizontal domain walls across the locally ordered regions. It is found that these horizontal domain walls are stabilized by free-energy barriers due to the discreteness. The stability conditions for such domain walls are studied. Usually, the domain wall across multiple locally ordered regions can be stable. These multiple locally ordered regions form an elementary cluster. The criterion for multiple locally ordered regions forming an elementary cluster is obtained. At the saddle point level, we get an effective Hamiltonian in which the elementary clusters behave as Ising spins and are coupled with their neighbors. For the random temperatures with narrow uniform probability distributions, the distribution of elementary cluster size, the horizontal and vertical bonds between elementary clusters are calculated numerically. The vertical bonds are extremely strong and the horizontal bonds are extremely weak. The distribution of vertical bonds is exponential. There is an exponentially small probability of elementary clusters with extremely large size. These elementary clusters with extremely large size will cause the susceptibility divergent in a range of temperature. The random temperature with binary probability distributions is also studied. The qualitative properties are the same. The Griffiths-McCoy singularity in the saddle point solutions does not depend on the distribution of random temperature.
The effects of random impurities on the various thermodynamic functions are studied with the aid of the two-dimensional Ising model. Of special interest are the limiting forms of these functions near the critical point. This problem is approached by a Green's function formulation as developed by Kadanoff. Since experiments only measure the average effects of impurities, we consider the average of the Green's function over all distributions of impurities and calculate it by perturbation theory. From this function we find the location of the new critical temperature and the temperature dependence of the coherence length. Thermodynamic and correlation functions are calculated by means of perturbation expansions and their functional dependence on the coherence length investigated. It is found that the magnetic properties such as the magnetization, susceptibility, and spin-spin correlations retain the same functional dependence on the coherence length. The specific heat is found to have a finite value at the critical point, and the T = Tc form of the energy-energy correlation c function is changed. The thermodynamic functions exhibit new critical indices very close to Tc. These renormalized exponents have been predicted c in literature and are found to satisfy the scaling laws predictions.
We study the effects which immobile random impurities may have on magnetic properties and spin-spin correlation functions of a ferromagnetic system properties the Curie temperature. This is done within the context of the model introduced in the first paper of this series by computing the spin correlation functions on the boundary of a half-plane of random Ising spins, where the boundary row is allowed to intereact with a magnetic field H\mathfrak{H}H. We find that as $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$ the average boundary spontaneous magnetization vanishes as ${T}_{c}\ensuremath{-}T$. However, the average boundary magnetic susceptibility is shown not to exist for a finite range of temperature about Tc{T}_{c}Tc​. Furthermore, at Tc{T}_{c}Tc​ the average boundary magnetization behaves as $\ensuremath{-}{N}^{\ensuremath{-}1}$ sign $(\mathfrak{H}){[\mathrm{ln} N{\ensuremath{\beta}}_{c}|\mathfrak{H}|]}^{\ensuremath{-}1}$, where ${N}^{\ensuremath{-}1}$ is a measure of the width of the distribution of random bonds. Whenever $T\ensuremath{-}{T}_{c}=O({N}^{\ensuremath{-}2})$, the average spin-spin correlation function for two spins on the boundary is shown to approach its limit at infinite separation as some inverse power of the separation instead of as an exponential. At Tc{T}_{c}Tc​ this average correlation function behaves asympotically as ${N}^{\ensuremath{-}2} {(\mathrm{ln}m{N}^{\ensuremath{-}2})}^{\ensuremath{-}1}$ when mmm the separation between boundary spins, is large. Finally, we make the probabilistic nature of the boundary spontaneous magnetization more precise by computing its probability distribution function.
Using the Monte Carlo technique the authors have studied a two-dimensional Ising model with quenched non-magnetic impurities. In particular they calculated the temperature dependence of the order parameter, of the zero-field susceptibility and of the specific heat. These results indicate that a second-order phase transition is preserved for p>or=0.9 where 1-p denotes the concentration of randomly distributed non-magnetic sites. Moreover, they yield beta approximately=1/8 and gamma approximately=7/4, indicating that these exponents equal the corresponding ones for the pure Ising model.