Physical Review Letters

Two-Spin Correlation Functions of an Ising Model with Continuous Exponents

Barry M. McCoy, Jacques H. H. Perk

Published Mar 31, 1980• 120 citations

Abstract

An Ising model on a square lattice is studied, where one row of horizontal bonds has an energy ${{E}_{1}}^{\ensuremath{'}}$ different from all other horizontal bonds. The correlation of two spins is calculated in this row, resulting in exponents $\ensuremath{\beta}$ and $\ensuremath{\eta}$ which depend on ${{E}_{1}}^{\ensuremath{'}}$. The long-distance behavior of the correlation for fixed $T\ensuremath{\ne}{T}_{c}$ is found to have different forms depending upon the value of ${{E}_{1}}^{\ensuremath{'}}$.

DOI: 10.1103/physrevlett.44.840

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Physical Review Letters

Two-Spin Correlation Functions of an Ising Model with Continuous Exponents

Barry M. McCoy, Jacques H. H. Perk

Published Mar 31, 1980• 120 citations

Publisher

Abstract

An Ising model on a square lattice is studied, where one row of horizontal bonds has an energy

{{E}_{1}}^{\ensuremath{'}}

different from all other horizontal bonds. The correlation of two spins is calculated in this row, resulting in exponents

\ensuremath{\beta}

and

\ensuremath{\eta}

which depend on

{{E}_{1}}^{\ensuremath{'}}

. The long-distance behavior of the correlation for fixed

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

is found to have different forms depending upon the value of

{{E}_{1}}^{\ensuremath{'}}

DOI: 10.1103/physrevlett.44.840

Related papers

Theory of Toeplitz Determinants and the Spin Correlations of the Two-Dimensional Ising Model. III

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We study the asymptotic behavior, for large separations, of the spin-spin correlation function $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉$ in the two-dimensional Ising model, where the two spins are not necessarily on the same row. Besides the limiting value for infinite separation, which is the square of the spontaneous magnetization, we evaluate the two leading terms in the asymptotic expression in each of the two cases $T<{T}_{c}$ and $T>{T}_{c}$ . It is found that the nearest singularity of the generating function for the correlation is quite simple in the case $T>{T}_{c}$ , but much more complicated for $T<{T}_{c}$ . In an Appendix, we also give exactly in a very simple form the correlation $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{N,N}〉$ for symmetrical Ising lattice at the critical temperature ${T}_{c}$ .We study the asymptotic behavior, for large separations, of the spin-spin correlation function $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉$ in the two-dimensional Ising model, where the two spins are not necessarily on the same row. Besides the limiting value for infinite separation, which is the square of the spontaneous magnetization, we evaluate the two leading terms in the asymptotic expression in each of the two cases $T<{T}_{c}$ and $T>{T}_{c}$. It is found that the nearest singularity of the generating function for the correlation is quite simple in the case $T>{T}_{c}$, but much more complicated for $T<{T}_{c}$. In an Appendix, we also give exactly in a very simple form the correlation $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{N,N}〉$ for symmetrical Ising lattice at the critical temperature ${T}_{c}$.

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An approximation for calculating the thermodynamic properties of spin-\textonehalf{} Ising models is described. The method is based on the formulation of Kramers and Wannier for the square lattice, and involves expressing the free energy in terms of the limiting maximum eigenvalue of a matrix sequence which represents the contribution from a single spin of the lattice. The application of this formulation to various two- and three-dimensional lattices shows that many of these are characterized by an identically structured matrix sequence with exactly two nonvanishing elements in each row and column. To explore the potential of using this similarity and its simplicity for analyzing Ising models we consider the two-dimensional square lattice in an external field and approximate the doubly infinite lattice by a semi-infinite one consisting of $n (\ensuremath{\le}10)$ infinitely long rows. Calculations are carried out and compared with (1) Onsager's formula for the free energy, (2) the Onsanger-Yang formula for the spontaneous magnetization, and (3) values calculated from series. Except for the region around the critical point where the correlation length exceeds $n$ lattice spacings, substantial agreement---to better than 12 places of decimals in some cases---is obtained. As an additional measure of the potential of this method, a value for the critical exponent $\ensuremath{\delta}$ is computed by the extrapolation of the magnetization along the critical isotherm. The result is found to be in excellent agreement with the known value for $\ensuremath{\delta}$.An approximation for calculating the thermodynamic properties of spin-\textonehalf{} Ising models is described. The method is based on the formulation of Kramers and Wannier for the square lattice, and involves expressing the free energy in terms of the limiting maximum eigenvalue of a matrix sequence which represents the contribution from a single spin of the lattice. The application of this formulation to various two- and three-dimensional lattices shows that many of these are characterized by an identically structured matrix sequence with exactly two nonvanishing elements in each row and column. To explore the potential of using this similarity and its simplicity for analyzing Ising models we consider the two-dimensional square lattice in an external field and approximate the doubly infinite lattice by a semi-infinite one consisting of $n (\ensuremath{\le}10)$ infinitely long rows. Calculations are carried out and compared with (1) Onsager's formula for the free energy, (2) the Onsanger-Yang formula for the spontaneous magnetization, and (3) values calculated from series. Except for the region around the critical point where the correlation length exceeds $n$ lattice spacings, substantial agreement---to better than 12 places of decimals in some cases---is obtained. As an additional measure of the potential of this method, a value for the critical exponent $\ensuremath{\delta}$ is computed by the extrapolation of the magnetization along the critical isotherm. The result is found to be in excellent agreement with the known value for $\ensuremath{\delta}$.

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Spin-spin correlations in central rows of Ising models with holes

Helen Au-Yang, Jacques H. H. Perk

In our previous works on infinite horizontal Ising strips of width $m$ alternating with layers of strings of Ising chains of length $n$ , we found the surprising result that the specific heats are not much different for different values of $N$ , the separation of the strings. For this reason, we study here for $N=1$ the spin-spin correlation in the central row of each strip, and also the central row of a strings layer. We show that these can be written as a Toeplitz determinants. Their generating functions are ratios of two polynomials, which in the limit of infinite vertical size become square roots of polynomials whose degrees are $m+1$ where $m$ is the size of the strips. We find the asymptotic behaviors near the critical temperature to be two-dimensional Ising-like. But in regions not very close to criticality the behavior may be different for different $m$ and $n$ . Finally, in the appendix we shall present results for generating functions in more general models.In our previous works on infinite horizontal Ising strips of width $m$ alternating with layers of strings of Ising chains of length $n$, we found the surprising result that the specific heats are not much different for different values of $N$, the separation of the strings. For this reason, we study here for $N=1$ the spin-spin correlation in the central row of each strip, and also the central row of a strings layer. We show that these can be written as a Toeplitz determinants. Their generating functions are ratios of two polynomials, which in the limit of infinite vertical size become square roots of polynomials whose degrees are $m+1$ where $m$ is the size of the strips. We find the asymptotic behaviors near the critical temperature to be two-dimensional Ising-like. But in regions not very close to criticality the behavior may be different for different $m$ and $n$. Finally, in the appendix we shall present results for generating functions in more general models.

arXiv:1907.04748Jul 10, 20190 citations

An Ising model on a square lattice is studied, where one row of horizontal bonds has an energy

{{E}_{1}}^{\ensuremath{'}}

different from all other horizontal bonds. The correlation of two spins is calculated in this row, resulting in exponents

\ensuremath{\beta}

and

\ensuremath{\eta}

which depend on

{{E}_{1}}^{\ensuremath{'}}

. The long-distance behavior of the correlation for fixed

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

is found to have different forms depending upon the value of

{{E}_{1}}^{\ensuremath{'}}

T<{T}_{c}

and

T>{T}_{c}

. It is found that the nearest singularity of the generating function for the correlation is quite simple in the case

T>{T}_{c}

, but much more complicated for

T<{T}_{c}

. In an Appendix, we also give exactly in a very simple form the correlation $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{N,N}〉$ for symmetrical Ising lattice at the critical temperature

{T}_{c}

R

is the distance between the points in question and

{K}_{0}

and

{K}_{1}

are modified Bessel functions of the second kind. Very similar results hold if the coupling constants are different in the two directions. The above results agree in form with the structure predicted by the "scaling-law" approach.

n

lattice spacings, substantial agreement---to better than 12 places of decimals in some cases---is obtained. As an additional measure of the potential of this method, a value for the critical exponent $\ensuremath{\delta}$ is computed by the extrapolation of the magnetization along the critical isotherm. The result is found to be in excellent agreement with the known value for $\ensuremath{\delta}$.

In our previous works on infinite horizontal Ising strips of width

m

alternating with layers of strings of Ising chains of length

n

, we found the surprising result that the specific heats are not much different for different values of

N

, the separation of the strings. For this reason, we study here for

N=1

the spin-spin correlation in the central row of each strip, and also the central row of a strings layer. We show that these can be written as a Toeplitz determinants. Their generating functions are ratios of two polynomials, which in the limit of infinite vertical size become square roots of polynomials whose degrees are

m+1

where

m

is the size of the strips. We find the asymptotic behaviors near the critical temperature to be two-dimensional Ising-like. But in regions not very close to criticality the behavior may be different for different

m

and

n

. Finally, in the appendix we shall present results for generating functions in more general models.