Physical review. B, Solid state

Operators along a main diagonal of the Ising model

Horacio Ceva

Published Sep 1, 1975• 1 citation

Abstract

We study products of spin and disorder variables placed along a diagonal of the Ising model, using the path formulation of correlation functions. A number $\ensuremath{\Gamma}$ used in a previous publication, associated with their ordering on the line, characterizes the correlation, in the sense that only those arrangements having $\ensuremath{\Gamma}=0$, \ifmmode\pm\else\textpm\fi{} 1/2 can have nonvanishing average for all temperatures $T\ensuremath{\ne}{T}_{c}$; at ${T}_{c}$, only those with $\ensuremath{\Gamma}=0$ are different from zero. A formal, exact expression for the $2n$-spins correlation function is presented.

DOI: 10.1103/physrevb.12.1971

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Exact expressions for diagonal correlation functions in the d=2 Ising model

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Edward Farhi, Sam Gutmann

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Determination of an Operator Algebra for the Two-Dimensional Ising Model

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Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region

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Physical review. B, Solid stateJan 1, 1976694 citations

Correlations along a Line in the Two-Dimensional Ising Model

Leo P. Kadanoff

Physical ReviewDec 10, 196951 citations

On the low temperature series expansion for the diagonal correlation functions in the two-dimensional Ising model

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A previously tested differential equation method for generating low temperature series expansion for diagonal spin-spin correlation functions in the d=2 Ising model is extended to generate the non-universal terms for arbitrary separation of the spins. This extends the earlier calculations of these correlation functions.

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Physical review. B, Solid state

Operators along a main diagonal of the Ising model

Horacio Ceva

Published Sep 1, 1975• 1 citation

Publisher

Abstract

We study products of spin and disorder variables placed along a diagonal of the Ising model, using the path formulation of correlation functions. A number

\ensuremath{\Gamma}

used in a previous publication, associated with their ordering on the line, characterizes the correlation, in the sense that only those arrangements having

\ensuremath{\Gamma}=0

, \ifmmode\pm\else\textpm\fi{} 1/2 can have nonvanishing average for all temperatures

T\ensuremath{\ne}{T}_{c}

; at

{T}_{c}

, only those with

\ensuremath{\Gamma}=0

are different from zero. A formal, exact expression for the

2n

-spins correlation function is presented.

DOI: 10.1103/physrevb.12.1971

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Exact expressions for diagonal correlation functions in the d=2 Ising model

Ranjan K. Ghosh, Robert E. Shrock

We present exact explicit expressions for the diagonal spin-spin correlation functions $〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{nn}}〉$ in the $d=2$ Ising model, in terms of elliptic integrals, for $n$ up to 6. We also give a general structural formula for $〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{nn}}〉$ and discuss its properties.We present exact explicit expressions for the diagonal spin-spin correlation functions $〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{nn}}〉$ in the $d=2$ Ising model, in terms of elliptic integrals, for $n$ up to 6. We also give a general structural formula for $〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{nn}}〉$ and discuss its properties.

Physical review. B, Condensed matterOct 1, 198424 citations

Correlation functions of the one-dimensional random-field Ising model at zero temperature

Edward Farhi, Sam Gutmann

We consider the one-dimensional random-field Ising model, where the spin-spin coupling J is ferromagnetic and the external field is chosen to be +h with probability p and -h with probability 1-p. At zero temperature, we calculate an exact expression for the correlation length of the quenched average of the correlation function 〈 ${\mathit{s}}_{0}$ ${\mathit{s}}_{\mathit{n}}$ 〉-〈 ${\mathit{s}}_{0}$ 〉〈 ${\mathit{s}}_{\mathit{n}}$ 〉 in the case that 2J/h is not an integer. The result is a discontinuous function of 2J/h. When p=1/2, we also place a bound on the correlation length of the quenched average of the correlation function 〈 ${\mathit{s}}_{0}$ ${\mathit{s}}_{\mathit{n}}$ 〉.We consider the one-dimensional random-field Ising model, where the spin-spin coupling J is ferromagnetic and the external field is chosen to be +h with probability p and -h with probability 1-p. At zero temperature, we calculate an exact expression for the correlation length of the quenched average of the correlation function 〈${\mathit{s}}_{0}$${\mathit{s}}_{\mathit{n}}$〉-〈${\mathit{s}}_{0}$〉〈${\mathit{s}}_{\mathit{n}}$〉 in the case that 2J/h is not an integer. The result is a discontinuous function of 2J/h. When p=1/2, we also place a bound on the correlation length of the quenched average of the correlation function 〈${\mathit{s}}_{0}$${\mathit{s}}_{\mathit{n}}$〉.

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Determination of an Operator Algebra for the Two-Dimensional Ising Model

Leo P. Kadanoff, Horacio Ceva

A previous publication showed how the critical indices for the two-dimensional Ising model could be derived from an assumed form of an operator algebra which describes how the product of two fluctuating variables may be reduced to a linear combination of the basic fluctuating variables. In this paper, the previously used algebra is derived from the Onsager solution of the two-dimensional Ising model. The calculation makes use of a "disorder" variable which is mathematically the result of applying the Kramers-Wannier transformation to the Ising-model spin variable. The average of products of spin and disorder variables are evaluated at the critical point for the special case in which all the variables lie on a single straight line. The ordering of these variables on the line determines a "quantum number" $\ensuremath{\Gamma}$ such that the average is nonzero only for $\ensuremath{\Gamma}=0$. Composition rules for this quantum number are derived and used to develop an algebra for the multiplication of complex variables at the critical point. Arguments are given to suggest the identifications of elements of the algebra as the spin, the energy density, the Kaufman spinors, and a stress density. The result of this calculation is the operator algebra which formed the starting point of the previous paper.A previous publication showed how the critical indices for the two-dimensional Ising model could be derived from an assumed form of an operator algebra which describes how the product of two fluctuating variables may be reduced to a linear combination of the basic fluctuating variables. In this paper, the previously used algebra is derived from the Onsager solution of the two-dimensional Ising model. The calculation makes use of a "disorder" variable which is mathematically the result of applying the Kramers-Wannier transformation to the Ising-model spin variable. The average of products of spin and disorder variables are evaluated at the critical point for the special case in which all the variables lie on a single straight line. The ordering of these variables on the line determines a "quantum number" $\ensuremath{\Gamma}$ such that the average is nonzero only for $\ensuremath{\Gamma}=0$. Composition rules for this quantum number are derived and used to develop an algebra for the multiplication of complex variables at the critical point. Arguments are given to suggest the identifications of elements of the algebra as the spin, the energy density, the Kaufman spinors, and a stress density. The result of this calculation is the operator algebra which formed the starting point of the previous paper.

Physical review. B, Solid stateJun 1, 1971555 citations

Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region

Tai Tsun Wu, Barry M. McCoy, Craig A. Tracy +1 more

We compute exactly the spin-spin correlation functions $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉$ for the two-dimensional Ising model on a square lattice in zero magnetic field for $T>{T}_{c}$ and $T<{T}_{c}$ . We then analyze the correlation functions in the scaling limit $T\ensuremath{\rightarrow}{T}_{c},{M}^{2}+{N}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}$ such that $(T\ensuremath{-}{T}_{c})$ is fixed. In this scaling limit $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉={R}^{\ensuremath{-}\frac{1}{4}}{F}_{\ifmmode\pm\else\textpm\fi{}}(t)+{R}^{\ensuremath{-}\frac{5}{4}}{F}_{1\ifmmode\pm\else\textpm\fi{}}(t)+o({R}^{\ensuremath{-}\frac{5}{4}})$, where $t$ is the scaling variable $\frac{R}{\ensuremath{\xi}}$ and ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ and ${F}_{1\ifmmode\pm\else\textpm\fi{}}(t)$ are the scaling functions ($\ensuremath{\xi}$ is the correlation length). We derive exact expressions for these scaling functions, in terms of a Painlev\'e function of the third kind and analyze both the small- and large- $t$ behavior. A table of values for ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ (good to ten significant digits) is also given. As an application we computer the coefficients ${C}_{0\ifmmode\pm\else\textpm\fi{}}$ and ${C}_{1\ifmmode\pm\else\textpm\fi{}}$ in the expansion ${k}_{B}T\ensuremath{\chi}(T)={C}_{0\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{7}{4}}+{C}_{1\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{3}{4}}+O(1)$ of the zero-field susceptibility $\ensuremath{\chi}(T)$ as $T\ensuremath{\rightarrow}{T}_{c}^{\ifmmode\pm\else\textpm\fi{}}$.We compute exactly the spin-spin correlation functions $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉$ for the two-dimensional Ising model on a square lattice in zero magnetic field for $T>{T}_{c}$ and $T<{T}_{c}$. We then analyze the correlation functions in the scaling limit $T\ensuremath{\rightarrow}{T}_{c},{M}^{2}+{N}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}$ such that $(T\ensuremath{-}{T}_{c})$ is fixed. In this scaling limit $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉={R}^{\ensuremath{-}\frac{1}{4}}{F}_{\ifmmode\pm\else\textpm\fi{}}(t)+{R}^{\ensuremath{-}\frac{5}{4}}{F}_{1\ifmmode\pm\else\textpm\fi{}}(t)+o({R}^{\ensuremath{-}\frac{5}{4}})$, where $t$ is the scaling variable $\frac{R}{\ensuremath{\xi}}$ and ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ and ${F}_{1\ifmmode\pm\else\textpm\fi{}}(t)$ are the scaling functions ($\ensuremath{\xi}$ is the correlation length). We derive exact expressions for these scaling functions, in terms of a Painlev\'e function of the third kind and analyze both the small- and large-$t$ behavior. A table of values for ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ (good to ten significant digits) is also given. As an application we computer the coefficients ${C}_{0\ifmmode\pm\else\textpm\fi{}}$ and ${C}_{1\ifmmode\pm\else\textpm\fi{}}$ in the expansion ${k}_{B}T\ensuremath{\chi}(T)={C}_{0\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{7}{4}}+{C}_{1\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{3}{4}}+O(1)$ of the zero-field susceptibility $\ensuremath{\chi}(T)$ as $T\ensuremath{\rightarrow}{T}_{c}^{\ifmmode\pm\else\textpm\fi{}}$.

Physical review. B, Solid stateJan 1, 1976694 citations

Correlations along a Line in the Two-Dimensional Ising Model

Leo P. Kadanoff

The $2n$ -spin correlation function for the two-dimensional Ising model at $T={T}_{c}$ is evaluated for the special case in which all the spins lie along a straight line, separated by many lattice constants. The resulting $2n$ -spin function is simply a quotient of products of two-spin correlations. A hypothesis of reducibility of fluctuations in the critical state is introduced. This hypothesis asserts that the product of any two local fluctuating quantities in the same neighborhood of space may be effectively replaced by a finite sum of local fluctuating quantities in this neighborhood. As a result, the previously found form for the $2n$ -spin function may be used to evaluate the correlation function of $n$ energy densities when all $n$ points lie on a line. The $n$ -energy correlation function is simply a sum of products of two-energy correlations. The quotient form for the spin correlation plus scaling is shown to immediately imply the logarithmic specific heat.The $2n$-spin correlation function for the two-dimensional Ising model at $T={T}_{c}$ is evaluated for the special case in which all the spins lie along a straight line, separated by many lattice constants. The resulting $2n$-spin function is simply a quotient of products of two-spin correlations. A hypothesis of reducibility of fluctuations in the critical state is introduced. This hypothesis asserts that the product of any two local fluctuating quantities in the same neighborhood of space may be effectively replaced by a finite sum of local fluctuating quantities in this neighborhood. As a result, the previously found form for the $2n$-spin function may be used to evaluate the correlation function of $n$ energy densities when all $n$ points lie on a line. The $n$-energy correlation function is simply a sum of products of two-energy correlations. The quotient form for the spin correlation plus scaling is shown to immediately imply the logarithmic specific heat.

Physical ReviewDec 10, 196951 citations

On the low temperature series expansion for the diagonal correlation functions in the two-dimensional Ising model

Ghosh, Ranjan Kumar

arXiv:cond-mat/0505166May 6, 20054 citations

We study products of spin and disorder variables placed along a diagonal of the Ising model, using the path formulation of correlation functions. A number

\ensuremath{\Gamma}

used in a previous publication, associated with their ordering on the line, characterizes the correlation, in the sense that only those arrangements having

\ensuremath{\Gamma}=0

, \ifmmode\pm\else\textpm\fi{} 1/2 can have nonvanishing average for all temperatures

T\ensuremath{\ne}{T}_{c}

; at

{T}_{c}

, only those with

\ensuremath{\Gamma}=0

are different from zero. A formal, exact expression for the

2n

-spins correlation function is presented.

We present exact explicit expressions for the diagonal spin-spin correlation functions $〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{nn}}〉$ in the

d=2

Ising model, in terms of elliptic integrals, for

n

up to 6. We also give a general structural formula for $〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{\mathrm{nn}}〉$ and discuss its properties.

{\mathit{s}}_{0}

{\mathit{s}}_{\mathit{n}}

〉-〈

{\mathit{s}}_{0}

〉〈

{\mathit{s}}_{\mathit{n}}

〉 in the case that 2J/h is not an integer. The result is a discontinuous function of 2J/h. When p=1/2, we also place a bound on the correlation length of the quenched average of the correlation function 〈

{\mathit{s}}_{0}

{\mathit{s}}_{\mathit{n}}

〉.

T>{T}_{c}

and

T<{T}_{c}

. We then analyze the correlation functions in the scaling limit $T\ensuremath{\rightarrow}{T}_{c},{M}^{2}+{N}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}$ such that $(T\ensuremath{-}{T}_{c})$ is fixed. In this scaling limit $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉={R}^{\ensuremath{-}\frac{1}{4}}{F}_{\ifmmode\pm\else\textpm\fi{}}(t)+{R}^{\ensuremath{-}\frac{5}{4}}{F}_{1\ifmmode\pm\else\textpm\fi{}}(t)+o({R}^{\ensuremath{-}\frac{5}{4}})$, where

t

is the scaling variable $\frac{R}{\ensuremath{\xi}}$ and ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ and ${F}_{1\ifmmode\pm\else\textpm\fi{}}(t)$ are the scaling functions ($\ensuremath{\xi}$ is the correlation length). We derive exact expressions for these scaling functions, in terms of a Painlev\'e function of the third kind and analyze both the small- and large-

t

behavior. A table of values for ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ (good to ten significant digits) is also given. As an application we computer the coefficients ${C}_{0\ifmmode\pm\else\textpm\fi{}}$ and ${C}_{1\ifmmode\pm\else\textpm\fi{}}$ in the expansion ${k}_{B}T\ensuremath{\chi}(T)={C}_{0\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{7}{4}}+{C}_{1\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{3}{4}}+O(1)$ of the zero-field susceptibility $\ensuremath{\chi}(T)$ as $T\ensuremath{\rightarrow}{T}_{c}^{\ifmmode\pm\else\textpm\fi{}}$.

The

2n

-spin correlation function for the two-dimensional Ising model at

T={T}_{c}

is evaluated for the special case in which all the spins lie along a straight line, separated by many lattice constants. The resulting

2n

-spin function is simply a quotient of products of two-spin correlations. A hypothesis of reducibility of fluctuations in the critical state is introduced. This hypothesis asserts that the product of any two local fluctuating quantities in the same neighborhood of space may be effectively replaced by a finite sum of local fluctuating quantities in this neighborhood. As a result, the previously found form for the

2n

-spin function may be used to evaluate the correlation function of

n

energy densities when all

n

points lie on a line. The

n

-energy correlation function is simply a sum of products of two-energy correlations. The quotient form for the spin correlation plus scaling is shown to immediately imply the logarithmic specific heat.