Decay of Order in Classical Many-Body Systems. II. Ising Model at High Temperatures

We employ the transfer-matrix formalism expounded in Paper I to study the decay of pair correlation functions at high temperatures in the

d

-dimensional Ising model in an arbitrary magnetic field

H

. A general correlation function decays according to

〈 δ L (\vec{0}) δ Q (\vec{R}) 〉 \approx R^{\frac{(d - 1)}{2}} e^{- κ R} (A_{0} + A_{1} R^{- 1} + \dots) + R^{- x} e^{- κ^{'} R} (B_{0} + B_{1} R^{- 1} + \dots) + \dots

R \to \infty

. For sufficiently small

H

and high temperature

T

, the exponent

x

is equal to the dimension of the lattice and

κ^{'} = 2 κ

. The coefficients

A_{n}

and

B_{n}

factor as

A_{n} (Q) A_{n} (L)

and

B_{n} (Q) B_{n} (L)

, respectively. If

\vec{Q}

is an operator involving an odd number of closely spaced spins,

A_{n} (Q)

tends to a finite limit and

B_{n} (Q)

tends to zero as

H

tends to zero. In contrast, if

\vec{Q}

involves an even number of closely spaced spins,

A_{n} (Q)

tends to zero and

B_{n} (Q)

to a finite limit as

H

tends to zero. Thus, for finite

H

an arbitrary pair correlation function verifies the Ornstein-Zernike (OZ) prediction; whereas in the zero field, (i) if both

\vec{Q}

and

\vec{L}

are products of odd numbers of spin operators,

G_{LQ} (

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