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<Tc and
T>Tc. It is found that the nearest singularity of the generating function for the correlation is quite simple in the case
T>Tc, but much more complicated for
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.
We study in detail the asymptotic behavior, for large separations, of the correlation between two spins in the same row of the two-dimensional Ising model with the particular value of a purely imaginary magnetic field $H=\frac{1}{2}i\ensuremath{\pi}\mathrm{kT}$. We derive the partition function and the magnetization per spin first proposed by Lee and Yang and find that the correlation function approaches its limiting value exponentially. In contrast to the case of no magnetic field, the approach is shown to be oscillatory rather than monotonic. The first three terms of the asymptotic behavior are explicitly given. The correlation function is obtained by expressing it as a block Toeplitz determinant whose behavior for large separation is studied by solving a pair of coupled Wiener-Hopf sum equations.
The asymptotic behavior of the pair correlation ω2(r) = 〈σ0σr〉 between two spins at sites 0 and r on an axis of an isotropic antiferromagnetic triangular lattice is investigated with the aid of the theory of Toeplitz determinants as developed by Wu. The leading terms in the asymptotic expansion are obtained for large spin separation at fixed nonzero temperature. Evidence is presented that the zero-point behavior of the correlation is of the form ω2(r) ∼ ε0r−½ cos ⅔πr, where r = |r| is the spin separation and ε0=212(E0T)2=0.632226080…,E0T being the decay amplitude of the pair correlation at the Curie point (critical point) of an isotropic ferromagnetic triangular lattice. A special class of fourth-order correlations ω4(r) = 〈σ0σδσr σr+δ〉 − 〈σ0σδ〉 〈σrσr+δ〉 between the four spins at sites 0, δ, r, and r + δ on the same lattice axis, where δ is a lattice vector, is reconsidered. The asymptotic form of the correlation for large separation of pairs of spins r = |r| is obtained for all fixed temperatures.
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.
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. 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 and proportional to the total number of lattice sites, while the other depends on
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. 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=Tc, the discontinuity is of course absent, and the boundary magnetization behaves as $\ensuremath{-}\mathfrak{H}\mathrm{ln}\mathfrak{H}$ for small
H. The boundary susceptibility at zero magnetic field in the ferromagnetic case exhibits a logarithmic singularity at
T=Tc, both above and below transition. An interesting feature is that the ferromagnetic boundary magnetization, although discontinuous for
T<Tc, may be analytically continued beyond the point
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.
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.