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Physical Review

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

Barry M. McCoy

Published Dec 10, 1969• 118 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.188.1014

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arXiv:cond-mat/9310045•Journal of Physics A Mathematical and General•Dec 21, 1993•0 citations
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→Tc\ensuremath−T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}T\ensuremath→Tc​\ensuremath− the average boundary spontaneous magnetization vanishes as Tc\ensuremath−T{T}_{c}\ensuremath{-}TTc​\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\ensuremath{-}{N}^{\ensuremath{-}1}\ensuremath−N\ensuremath−1 sign (H)[lnN\ensuremathβc∣H∣]\ensuremath−1(\mathfrak{H}){[\mathrm{ln} N{\ensuremath{\beta}}_{c}|\mathfrak{H}|]}^{\ensuremath{-}1}(H)[lnN\ensuremathβc​∣H∣]\ensuremath−1, where N\ensuremath−1{N}^{\ensuremath{-}1}N\ensuremath−1 is a measure of the width of the distribution of random bonds. Whenever T\ensuremath−Tc=O(N\ensuremath−2)T\ensuremath{-}{T}_{c}=O({N}^{\ensuremath{-}2})T\ensuremath−Tc​=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(lnmN\ensuremath−2)\ensuremath−1{N}^{\ensuremath{-}2} {(\mathrm{ln}m{N}^{\ensuremath{-}2})}^{\ensuremath{-}1}N\ensuremath−2(lnmN\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.
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 $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$.
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.
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.
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 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​.
The self-consistent approach to the two-dimensional Ising model with quenched random bonds is extended to the full lattice theory described by four real fermions. A calculation of the averaged spin-spin correlation function for large separation of the spins in the disorder-dominated phase indicates an exponential decay of this quantity and therefore a vanishing spontaneous magnetization. The corresponding correlation length is proportional to 1/ eta 2 , where eta denotes the order parameter of the new phase introduced by Ziegler (1990).