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Physical Review Letters

Behavior of Two-Point Correlation Functions at High Temperatures

William J. Camp, Michael Ellis Fisher

Published Jan 11, 1971• 53 citations
Publisher

Abstract

The asymptotic decay of the general two-point correlation function ${G}_{\mathrm{AB}}(\stackrel{\ensuremath{\rightarrow}}{R})$ at high temperatures is analyzed on the basis of the $d$-dimensional, spin-\textonehalf{} Ising model in general field $H$. For $H\ensuremath{\ne}0$ the Ornstein-Zernike form $\ensuremath{\approx}\frac{{D}_{A}{D}_{B}{e}^{\ensuremath{-}\ensuremath{\kappa}R}}{{R}^{\frac{(d\ensuremath{-}1)}{2}}}$ is found for general $\stackrel{^}{A}$ and $\stackrel{^}{B}$. However for certain operators, including the energy, the amplitude of the Ornstein-Zernike term vanishes as ${H}^{2}$ and in zero field only the higher order decay $\ensuremath{\sim}\frac{{e}^{\ensuremath{-}2\ensuremath{\kappa}R}}{{R}^{d}}$ remains. The relation to approximate treatments and to critical-point phenomena is discussed briefly.

DOI
10.1103/physrevlett.26.73

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The asymptotic decay of the general two-point correlation function GAB(R\ensuremath→){G}_{\mathrm{AB}}(\stackrel{\ensuremath{\rightarrow}}{R})GAB​(R\ensuremath→​) at high temperatures is analyzed on the basis of the ddd-dimensional, spin-\textonehalf{} Ising model in general field HHH. For H\ensuremath≠0H\ensuremath{\ne}0H\ensuremath=0 the Ornstein-Zernike form \ensuremath≈DADBe\ensuremath−\ensuremathκRR(d\ensuremath−1)2\ensuremath{\approx}\frac{{D}_{A}{D}_{B}{e}^{\ensuremath{-}\ensuremath{\kappa}R}}{{R}^{\frac{(d\ensuremath{-}1)}{2}}}\ensuremath≈R2(d\ensuremath−1)​DA​DB​e\ensuremath−\ensuremathκR​ is found for general \stackrel{^}{A} and \stackrel{^}{B}. However for certain operators, including the energy, the amplitude of the Ornstein-Zernike term vanishes as H2{H}^{2}H2 and in zero field only the higher order decay \ensuremath∼e\ensuremath−2\ensuremathκRRd\ensuremath{\sim}\frac{{e}^{\ensuremath{-}2\ensuremath{\kappa}R}}{{R}^{d}}\ensuremath∼Rde\ensuremath−2\ensuremathκR​ remains. The relation to approximate treatments and to critical-point phenomena is discussed briefly.
We employ the transfer-matrix formalism expounded in Paper I to study the decay of pair correlation functions at high temperatures in the ddd-dimensional Ising model in an arbitrary magnetic field HHH. A general correlation function decays according to $〈\ensuremath{\delta}L(\stackrel{\ensuremath{\rightarrow}}{0})\ensuremath{\delta}Q(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})〉\ensuremath{\approx}{R}^{\frac{(d\ensuremath{-}1)}{2}} {e}^{\ensuremath{-}\ensuremath{\kappa}R}({A}_{0}+{A}_{1}{R}^{\ensuremath{-}1}+\ensuremath{\cdots})+{R}^{\ensuremath{-}x}{e}^{\ensuremath{-}{\ensuremath{\kappa}}^{\ensuremath{'}}R}({B}_{0}+{B}_{1}{R}^{\ensuremath{-}1}+\ensuremath{\cdots})+\ensuremath{\cdots}$ as $R\ensuremath{\rightarrow}\ensuremath{\infty}$. For sufficiently small HHH and high temperature TTT, the exponent xxx is equal to the dimension of the lattice and ${\ensuremath{\kappa}}^{\ensuremath{'}}=2\ensuremath{\kappa}$. The coefficients An{A}_{n}An​ and Bn{B}_{n}Bn​ factor as An(Q)An(L){A}_{n}(Q){A}_{n}(L)An​(Q)An​(L) and Bn(Q)Bn(L){B}_{n}(Q){B}_{n}(L)Bn​(Q)Bn​(L), respectively. If $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ is an operator involving an odd number of closely spaced spins, An(Q){A}_{n}(Q)An​(Q) tends to a finite limit and Bn(Q){B}_{n}(Q)Bn​(Q) tends to zero as HHH tends to zero. In contrast, if $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ involves an even number of closely spaced spins, An(Q){A}_{n}(Q)An​(Q) tends to zero and Bn(Q){B}_{n}(Q)Bn​(Q) to a finite limit as HHH tends to zero. Thus, for finite HHH an arbitrary pair correlation function verifies the Ornstein-Zernike (OZ) prediction; whereas in the zero field, (i) if both $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{L}}$ are products of odd numbers of spin operators, ${G}_{\mathrm{LQ}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})$ will verify the OZ prediction in zero field; (ii) if both $\stackrel{\ensuremath{\rightarrow}}{\mathrm{L}}$ and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ involve even numbers of spins, ${G}_{\mathrm{LQ}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})$ fails to verify the OZ prediction in zero field; and (iii) if $\stackrel{\ensuremath{\rightarrow}}{\mathrm{L}}$ involves an odd number of spins while $\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}$ involves an even number of spins, ${G}_{\mathrm{LQ}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})$ is identically zero in zero field for all $\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}$. The failure of the zero-field Ising model to completely verify the OZ prediction at high temperatures is due to the symmetry of this model about the H=0H=0H=0 line.
In this work the decay of correlation in the ddd-dimensional Ising model is studied at low temperatures as a function of dimensionality of the lattice and magnetic field hhh. Except for the special case of the two-dimensional zero-field nearest-neighbor lattices, the decay of correlation verifies the Ornstein-Zernike prediction ${G}_{\mathrm{AB}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})\ensuremath{\approx}{D}_{\mathrm{AB}}(d,h){R}^{\ensuremath{-}\frac{(d\ensuremath{-}1)}{2}}{e}^{\ensuremath{-}\ensuremath{\kappa}R}$. For the two-dimensional zero-field case, the Ornstein-Zernike form is replaced by the "anomalous" form ${G}_{\mathrm{AB}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})\ensuremath{\approx}{D}_{\mathrm{AB}}{R}^{\ensuremath{-}2}{e}^{\ensuremath{-}\ensuremath{\kappa}R}$. This "anomalous" result is shown to arise from the peculiarities of the spectrum of the transfer matrix in this case and is replaced by the Ornstein-Zernike result when further-neighbor forces are present. The results presented herein agree with the previously obtained exact results for the zero-field two-dimensional Ising model.
The zero-field two-point correlation function of an nnn-vector system in ddd dimensions is calculated to order 1n\frac{1}{n}n1​ for $T\ensuremath{\ge}{T}_{c}$ and 2<d<42<d<42<d<4. The critical-point scaling function is obtained as a closed cutoff-independent integral. As $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}\ensuremath{\rightarrow}0$ at fixed wave vector qqq, the variation of the Fourier transform of the correlation function is ${\stackrel{^}{E}}_{1}(q){t}^{1\ensuremath{-}\ensuremath{\alpha}}+{\stackrel{^}{E}}_{2}(q)t+{\stackrel{^}{E}}_{3}(q){t}^{2}+\ensuremath{\cdots}$, where $\ensuremath{\alpha}$ is the specific-heat exponent and ${\stackrel{^}{E}}_{2}$, ${\stackrel{^}{E}}_{3}$ are of order 1n\frac{1}{n}n1​. At d=3d=3d=3 the term t2{t}^{2}t2 is modified by logarithmic corrections. As $q\ensuremath{\rightarrow}0$ at fixed nonzero ttt, deviations of the scaling function from the Ornstein-Zernike form ${(1+{\ensuremath{\xi}}^{2}{q}^{2})}^{\ensuremath{-}1}$ are of order $\frac{{10}^{\ensuremath{-}3}{\ensuremath{\xi}}^{4}{q}^{4}}{n}$. Results are compared with high-temperature series-expansion estimates and with the $\ensuremath{\epsilon}$-expansion results of previous work.
A detailed calculation, to order ${\ensuremath{\epsilon}}^{2} (\ensuremath{\epsilon}=4\ensuremath{-}d)$, of the two-point-correlation function of an Ising-like system in the whole critical region is presented. The scaling function is shown to be a cut-off-independent function of two variables which is universal in the context of a sharp cut off. Explicit asymptotic expansions in the large and small momentum (relative to the inverse correlation length) are given. Particular attention is paid to the corrections from the Ornstein-Zernike theory. These corrections are two orders of magnitude larger below Tc{T}_{c}Tc​ than above. Numerical comparison with series-expansion results agree surprisingly well. A powerful technique of evaluating diagrams using the Fourier transform of the propagator is also presented.
Power-series expansions of the spin-pair correlation functions of the square, sc, and bcc Ising lattices have been obtained using the semi-invariant approach for general field HHH and temperature TTT. In three dimensions P\'ade-approximant analyses indicate $2{v}^{\ensuremath{'}}=1.285\ifmmode\pm\else\textpm\fi{}0.020$, in agreement with scaling, but $2{v}^{c}\ensuremath{\simeq}0.835\ifmmode\pm\else\textpm\fi{}0.02$ which is some (1-3)% above the scaling prediction ${v}^{c}=\frac{v}{\ensuremath{\beta}\ensuremath{\delta}}\ensuremath{\simeq}0.823$. However, ratio techniques reveal that this discrepancy can be attributed to significant confluent critical singularities. Cubic and quintic parametric representations of the critical equation of state, and corresponding expressions for the correlation length, ${\ensuremath{\xi}}_{1}(H, T)$, are developed, which are considerable improvements over the linear model. The universality and spherical symmetry of the critical scattering intensity $\stackrel{^}{\ensuremath{\chi}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}, H, T)$ is confirmed to within (1-2)% by estimating suitable invariant combinations of amplitudes. The deviations from Ornstein-Zernike theory for general HHH and TTT are found to be considerably greater than for H=0H=0H=0, T>TcT>{T}_{c}T>Tc​. Complete parametric scaling representations of $\stackrel{^}{\ensuremath{\chi}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}, H, T)$ are developed for three dimensions; corresponding scaling approximants are constructed in two dimensions but only for T=TcT={T}_{c}T=Tc​ and for $T\ensuremath{\gtrless}{T}_{c}$ with H=0H=0H=0.
High-temperature series for the inverse spin-spin correlation function of the s=12s=\frac{1}{2}s=21​ Ising ferromagnet are presented for the three cubic lattices. Series are tabulated to twelfth order in $K=\ensuremath{\beta}J$ in zero magnetic field and to eighth order in finite field. Direct series expansions for the true correlation length are derived. The critical correlations are analyzed for scaling behavior in both position and momentum space. The critical amplitudes which characterize the correlations both at Tc{T}_{c}Tc​ and in the Ornstein---Zernike region are determined. The Fisher---Burford scattering approximant is found to be consistent with the data. Finally, the consequences of the universality hypothesis are formulated for the correlations. Correlation-amplitude data for the three cubic lattices are shown to be in remarkable agreement with a formulation of universality which requires that the free energy in a region of diameter given by the coherence length be scale independent.