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Physical Review

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

Barry M. McCoy, Tai Tsun Wu

Published Oct 10, 1968• 30 citations
Publisher

Abstract

In a previous paper of this series, we studied $\mathfrak{P}(\overline{\ensuremath{\sigma}})$, the probability that the average boundary spin for a half-plane of Ising spins is $\overline{\ensuremath{\sigma}}$. We extend our study of such functions by generalizing the previous derivation to any magnetic system and using them to relate the magnetic behavior at the critical isotherm to the spin-spin correlations at the critical temperature. We also clarify the meaning of the secondary maxima previously found in $\mathfrak{P}(\overline{\ensuremath{\sigma}})$ by a more accurate calculation and by examining the magnetization on interior rows of the half-plane. These considerations show that $\mathfrak{P}(\overline{\ensuremath{\sigma}})$ is more properly to be considered as the sum of two separate spin probability functions. The various ways in which the thermodynamic limit may be taken and the effects that these different limiting procedures have on $\mathfrak{P}(\overline{\ensuremath{\sigma}})$ are discussed in detail. The influence of boundary conditions on spin probability functions is studied by reversing the sign of one column of horizontal bonds. We compute the additional free energy resulting from the misfit bonds and use this to show that $\mathfrak{P}(\overline{\ensuremath{\sigma}})$ is no longer bimodal but is rectangular for $|\overline{\ensuremath{\sigma}}|$ less than the boundary spontaneous magnetization. Finally, we present in graphical form numerical integrations of the boundary magnetization and susceptibility.

DOI
10.1103/physrev.174.546

Related papers

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

Barry M. McCoy, Tai Tsun Wu

We consider the rectangular Ising model on a half-plane of infinite extent and study some of the consequences connected with the presence of the boundary. Only the spins on the boundary row are allowed to interact with a magnetic field $\mathfrak{H}$. The method of Pfaffians is employed to obtain exact expressions for the partition function. It is found that the free energy is the sum of two terms, one of which is independent of $\mathfrak{H}$ and proportional to the total number of lattice sites, while the other depends on $\mathfrak{H}$ and is proportional to the number of lattice sites on the boundary. This separation makes it possible to define various thermodynamic quantities associated with the boundary. In particular, the boundary magnetization is shown to be discontinuous, in the ferromagnetic case, at zero magnetic field for temperatures below the bulk critical temperature ${T}_{c}$. This discontinuity, which is the spontaneous boundary magnetization, goes to zero as ${(1\ensuremath{-}\frac{T}{{T}_{c}})}^{\frac{1}{2}}$ as $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$. For $T={T}_{c}$, the discontinuity is of course absent, and the boundary magnetization behaves as $\ensuremath{-}\mathfrak{H}\mathrm{ln}\mathfrak{H}$ for small $\mathfrak{H}$. The boundary susceptibility at zero magnetic field in the ferromagnetic case exhibits a logarithmic singularity at $T={T}_{c}$, both above and below transition. An interesting feature is that the ferromagnetic boundary magnetization, although discontinuous for $T<{T}_{c}$, may be analytically continued beyond the point $\mathfrak{H}=0$. We interpret this as a hystersis phenomenon which we study in detail by computing the probability distribution function for the average boundary spin. The correlation function for two spins, both on the boundary row, is also obtained exactly and its asymptotic behavior is given. Finally, we derive an expression for the magnetization in any row and explicitly evaluate it for the second row, i.e., the row next to the boundary.

Physical Review•Oct 10, 1967•195 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

The Ising model in two dimensions: summary of results

Barry M. McCoy

Abstract This chapter summarizes the many results of the exactly solved Ising model in two dimensions at H = 0 including: the partition function, free energy, spontaneous magnetization, the Toeplitz determinant representations of the correlation functions, the Painlevé VI differential equation for the diagonal two spin correlations, the form factor expansion of the magnetic susceptibility, the scaling limit, the boundary properties of the half plane lattice, and the random layered lattice. For H ≠ 0 the Lee-Yang circle theorem, the free energy and magnetization for H/kBT = iπ/2 expansions for small H, the results of Zamolodchikov at T = Tc and a discussion of extended analyticity are given.

•Dec 1, 2009•0 citations

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

Hung Cheng, Tai Tsun Wu

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 ${T}_{c}$.

Physical Review•Dec 10, 1967•91 citations

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

Tai Tsun Wu

We study in detail the asymptotic behavior, for large separations, of the correlation between two spins in the case of the two-dimensional Ising model without magnetic field. In the limit of infinite separation, this correlation is equal to the square of the spontaneous magnetization per spin. This paper is devoted to answering the question how, for fixed temperature, the correlation approaches this limiting value. The investigation is restricted to the situation where the two spins under consideration lie on the same row of the two-dimensional lattice. There are three distinct cases where the fixed temperature is above, below, or equal to the critical temperature or the Curie temperature. In the first two cases, the limiting value is approached exponentially as a function of the separation, while at the critical temperature the correlation behaves asymptotically as the inverse fourth root of the separation. In this paper, we evaluate exactly the first four terms of the asymptotic expansion in the first case, the first three terms in the second case, and the coefficients of the terms proportional to the -\textonequarter{} and the $\ensuremath{-}\frac{9}{4}$ powers of the separation in the third case. All these results are obtained by first expressing the correlation as a Toeplitz determinant, and the method used is capable of giving in principle the entire asymptotic expansion for large classes of such determinants, when the size of the determinant approaches infinity. The explicit results for the two-dimensional Ising model also serve as an example where the prescription of summing the leading terms, or the most divergent terms, fails.

Physical Review•Sep 9, 1966•317 citations

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
In a previous paper of this series, we studied P(\ensuremathσ‾)\mathfrak{P}(\overline{\ensuremath{\sigma}})P(\ensuremathσ​), the probability that the average boundary spin for a half-plane of Ising spins is \ensuremathσ‾\overline{\ensuremath{\sigma}}\ensuremathσ​. We extend our study of such functions by generalizing the previous derivation to any magnetic system and using them to relate the magnetic behavior at the critical isotherm to the spin-spin correlations at the critical temperature. We also clarify the meaning of the secondary maxima previously found in P(\ensuremathσ‾)\mathfrak{P}(\overline{\ensuremath{\sigma}})P(\ensuremathσ​) by a more accurate calculation and by examining the magnetization on interior rows of the half-plane. These considerations show that P(\ensuremathσ‾)\mathfrak{P}(\overline{\ensuremath{\sigma}})P(\ensuremathσ​) is more properly to be considered as the sum of two separate spin probability functions. The various ways in which the thermodynamic limit may be taken and the effects that these different limiting procedures have on P(\ensuremathσ‾)\mathfrak{P}(\overline{\ensuremath{\sigma}})P(\ensuremathσ​) are discussed in detail. The influence of boundary conditions on spin probability functions is studied by reversing the sign of one column of horizontal bonds. We compute the additional free energy resulting from the misfit bonds and use this to show that P(\ensuremathσ‾)\mathfrak{P}(\overline{\ensuremath{\sigma}})P(\ensuremathσ​) is no longer bimodal but is rectangular for ∣\ensuremathσ‾∣|\overline{\ensuremath{\sigma}}|∣\ensuremathσ​∣ less than the boundary spontaneous magnetization. Finally, we present in graphical form numerical integrations of the boundary magnetization and susceptibility.
We consider the rectangular Ising model on a half-plane of infinite extent and study some of the consequences connected with the presence of the boundary. Only the spins on the boundary row are allowed to interact with a magnetic field H\mathfrak{H}H. The method of Pfaffians is employed to obtain exact expressions for the partition function. It is found that the free energy is the sum of two terms, one of which is independent of H\mathfrak{H}H and proportional to the total number of lattice sites, while the other depends on H\mathfrak{H}H and is proportional to the number of lattice sites on the boundary. This separation makes it possible to define various thermodynamic quantities associated with the boundary. In particular, the boundary magnetization is shown to be discontinuous, in the ferromagnetic case, at zero magnetic field for temperatures below the bulk critical temperature Tc{T}_{c}Tc​. This discontinuity, which is the spontaneous boundary magnetization, goes to zero as ${(1\ensuremath{-}\frac{T}{{T}_{c}})}^{\frac{1}{2}}$ as $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$. For T=TcT={T}_{c}T=Tc​, the discontinuity is of course absent, and the boundary magnetization behaves as $\ensuremath{-}\mathfrak{H}\mathrm{ln}\mathfrak{H}$ for small H\mathfrak{H}H. The boundary susceptibility at zero magnetic field in the ferromagnetic case exhibits a logarithmic singularity at T=TcT={T}_{c}T=Tc​, both above and below transition. An interesting feature is that the ferromagnetic boundary magnetization, although discontinuous for T<TcT<{T}_{c}T<Tc​, may be analytically continued beyond the point H=0\mathfrak{H}=0H=0. We interpret this as a hystersis phenomenon which we study in detail by computing the probability distribution function for the average boundary spin. The correlation function for two spins, both on the boundary row, is also obtained exactly and its asymptotic behavior is given. Finally, we derive an expression for the magnetization in any row and explicitly evaluate it for the second row, i.e., the row next to the boundary.
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.
Abstract This chapter summarizes the many results of the exactly solved Ising model in two dimensions at H = 0 including: the partition function, free energy, spontaneous magnetization, the Toeplitz determinant representations of the correlation functions, the Painlevé VI differential equation for the diagonal two spin correlations, the form factor expansion of the magnetic susceptibility, the scaling limit, the boundary properties of the half plane lattice, and the random layered lattice. For H ≠ 0 the Lee-Yang circle theorem, the free energy and magnetization for H/kBT = iπ/2 expansions for small H, the results of Zamolodchikov at T = Tc and a discussion of extended analyticity are given.
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<TcT<{T}_{c}T<Tc​ and T>TcT>{T}_{c}T>Tc​. It is found that the nearest singularity of the generating function for the correlation is quite simple in the case T>TcT>{T}_{c}T>Tc​, but much more complicated for T<TcT<{T}_{c}T<Tc​. 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 Tc{T}_{c}Tc​.
We study in detail the asymptotic behavior, for large separations, of the correlation between two spins in the case of the two-dimensional Ising model without magnetic field. In the limit of infinite separation, this correlation is equal to the square of the spontaneous magnetization per spin. This paper is devoted to answering the question how, for fixed temperature, the correlation approaches this limiting value. The investigation is restricted to the situation where the two spins under consideration lie on the same row of the two-dimensional lattice. There are three distinct cases where the fixed temperature is above, below, or equal to the critical temperature or the Curie temperature. In the first two cases, the limiting value is approached exponentially as a function of the separation, while at the critical temperature the correlation behaves asymptotically as the inverse fourth root of the separation. In this paper, we evaluate exactly the first four terms of the asymptotic expansion in the first case, the first three terms in the second case, and the coefficients of the terms proportional to the -\textonequarter{} and the $\ensuremath{-}\frac{9}{4}$ powers of the separation in the third case. All these results are obtained by first expressing the correlation as a Toeplitz determinant, and the method used is capable of giving in principle the entire asymptotic expansion for large classes of such determinants, when the size of the determinant approaches infinity. The explicit results for the two-dimensional Ising model also serve as an example where the prescription of summing the leading terms, or the most divergent terms, fails.
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 $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$.