Journal of the Physical Society of Japan

Asymptotic Behavior of the Two-Position Correlation Functions of Classical Systems

Tohru Morita

Published Jul 15, 1969• 3 citations

Abstract

An argument based on diagram analysis is given to show that various two-position correlation functions of a classical system have the same asymptotic behavior. It is illustrated for the examples of the density-density and energy-energy correlation functions of classical fluids and of the spin-spin and energy-energy correlation functions of the Ising model.

DOI: 10.1143/jpsj.27.19

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Journal of the Physical Society of Japan

Asymptotic Behavior of the Two-Position Correlation Functions of Classical Systems

Tohru Morita

Published Jul 15, 1969• 3 citations

Publisher

Abstract

DOI: 10.1143/jpsj.27.19

Related papers

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

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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}

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.

R

is the distance between the points in question and

{K}_{0}

and

{K}_{1}

are modified Bessel functions of the second kind. Very similar results hold if the coupling constants are different in the two directions. The above results agree in form with the structure predicted by the "scaling-law" approach.

{Z}_{2}

symmetry. As an illustration, we analyze both the

XXZ

and the transverse field Ising models. Finite size as well as infinite chains are investigated and the quantum criticality is discussed. Moreover, we identify the spin functions that govern the correlations. As a further example, we also consider correlations in the Hartree-Fock ground state of the Lipkin-Meshkov-Glick model. It is then shown that both classical correlation and quantum discord exhibit signatures of the quantum phase transitions.