Physical review. B, Condensed matter

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

Ranjan K. Ghosh, Robert E. Shrock

Published Oct 1, 1984• 24 citations

Abstract

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.

DOI: 10.1103/physrevb.30.3790

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Physical review. B, Condensed matter

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

Ranjan K. Ghosh, Robert E. Shrock

Published Oct 1, 1984• 24 citations

Publisher

Abstract

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.

DOI: 10.1103/physrevb.30.3790

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Off-axis correlation functions in the isotropicd=2 Ising model

Robert E. Shrock, Ranjan K. Ghosh

We present exact explicit expressions for the spin-spin correlation functions 〈${\ensuremath{\sigma}}_{0}$,0${\mathrm{\ensuremath{\sigma}}}_{\mathrm{m}}$,n〉 for (m,n)=(2,1), (3,1), (3,2), (4,1), (4,2), and (4,3) in the isotropic d=2 Ising model. We also infer a general structural formula for arbitrary 〈${\ensuremath{\sigma}}_{0}$,0${\mathrm{\ensuremath{\sigma}}}_{\mathrm{m}}$,n〉 in terms of complete elliptic integrals K and E.We present exact explicit expressions for the spin-spin correlation functions 〈${\ensuremath{\sigma}}_{0}$,0${\mathrm{\ensuremath{\sigma}}}_{\mathrm{m}}$,n〉 for (m,n)=(2,1), (3,1), (3,2), (4,1), (4,2), and (4,3) in the isotropic d=2 Ising model. We also infer a general structural formula for arbitrary 〈${\ensuremath{\sigma}}_{0}$,0${\mathrm{\ensuremath{\sigma}}}_{\mathrm{m}}$,n〉 in terms of complete elliptic integrals K and E.

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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 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.

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

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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

Exact Expressions of Spin-Spin Correlation Functions of the Two-Dimensional Rectangular Ising Model on a Finite Lattice

T. Mei

We employ the spinor analysis method to evaluate exact expressions of spin-spin correlation functions of the two-dimensional rectangular Ising model on a finite lattice, special process enables us to actually carry out the calculation process. We first present some exact expressions of correlation functions of the model with periodic-periodic boundary conditions on a finite lattice. The corresponding forms in the thermodynamic limit are presented, which show the short-range order. Then, we present the exact expression of the correlation function of the two farthest pair of spins in a column of the model with periodic-free boundary conditions on a finite lattice. Again, the corresponding form in the thermodynamic limit is discussed, from which the long-range order clearly emerges as the temperature decreases.We employ the spinor analysis method to evaluate exact expressions of spin-spin correlation functions of the two-dimensional rectangular Ising model on a finite lattice, special process enables us to actually carry out the calculation process. We first present some exact expressions of correlation functions of the model with periodic-periodic boundary conditions on a finite lattice. The corresponding forms in the thermodynamic limit are presented, which show the short-range order. Then, we present the exact expression of the correlation function of the two farthest pair of spins in a column of the model with periodic-free boundary conditions on a finite lattice. Again, the corresponding form in the thermodynamic limit is discussed, from which the long-range order clearly emerges as the temperature decreases.

EntropyApr 12, 20181 citation

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.

{T}_{c}

, only those with $\ensuremath{\Gamma}=0$ are different from zero. A formal, exact expression for the

2n

-spins correlation function is presented.

For both

T>{T}_{c}

and

T<{T}_{c}

, we report exact expressions for the

n

-spin correlation function of the two-dimensional Ising model suitable for studying large separations between spins. In particular, the scaling limit of these correlation functions can be shown to exist and yields

n

-point Schwinger functions of some relativistic quantum field theories.

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 $T\ensuremath{\rightarrow}{T}_{c}^{\ifmmode\pm\else\textpm\fi{}}$.