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Journal of Physics C Solid State Physics

Selfconsistent singularities of the static correlation function at the critical point

M R Bhagavan, Clas Blomberg

Published Jun 26, 1972• 3 citations
Publisher

Abstract

The singularity of the static correlation function at the critical temperature in a ferromagnetic system is analysed by means of a set of coupled nonlinear integral equations which are obtained from the density field formalism of Sherrington (1966) and its extension to general order by Bhagavan and Lambert (1969). With the assumption of the onset of a phase transition, these equations yield a singular correlation function. Its singularity, which is determined in a selfconsistent way, shows a logarithmic deviation from the Ornstein-Zernike results. The singularity obtained is therefore fundamentally different from that given by the Orstein-Zernike theory.

DOI
10.1088/0022-3719/5/12/017

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The singularity of the static correlation function at the critical temperature in a ferromagnetic system is analysed by means of a set of coupled nonlinear integral equations which are obtained from the density field formalism of Sherrington (1966) and its extension to general order by Bhagavan and Lambert (1969). With the assumption of the onset of a phase transition, these equations yield a singular correlation function. Its singularity, which is determined in a selfconsistent way, shows a logarithmic deviation from the Ornstein-Zernike results. The singularity obtained is therefore fundamentally different from that given by the Orstein-Zernike theory.
The thermodynamic properties of the noninteracting Bose gas in the neighborhood of its transition are examined in detail. The order parameter is a complex extensive variable, but the thermodynamic properties depend only on its amplitude under simple boundary conditions. As the dimensionality or the single-particle energy spectrum is varied, the critical singularity displays a variety of forms. The equation of state has a simple structure, different from the homogeneous form often discussed for critical systems but asymptotically reducing to the latter except when logarithmic singularities are involved. The correlation function in the critical region is a homogeneous function of the distance and a correlation length. Only for a quadratic energy spectrum is the Ornstein-Zernike theory result valid at the critical temperature. A precise correspondence is noted between the asymptotic properties of the ideal Bose gas transition and those of the spherical model of ferromagnetism.
The energy-density-energy-density correlation function is studied in the three-dimensional Ising model in zero external field, for various spins on the fcc lattice. An extension of the linked-cluster technique is used to obtain high-temperature series expansions for the correlation function and its associated functions, to at least eighth order in the high-temperature interaction variable. Analysis of the series provides values for their critical indices and amplitudes. These values are used to verify the predictions of several theories of critical behavior. The critical index of the coherence length of the energy density correlation function is found to be equal to that of the spin-spin correlation function for all spins within the accuracy obtained here. For spins higher than ~ this critical index appears to obey strong scaling. An analysis of universality indicates that only one scale factor is needed to transform the energy density correlation function from one spin to another. This prediction is checked here and found to be good within our present accuracy. The current-algebra hypothesis predicts definite relations between the leading singularities of various correlation functions at the critical point. This prediction is checked directly and consistent values of the leading coefficients are obtained. The shape of the energy density correlation function at large distances from the origin, both at and above the critical point, is determined. The results are sufficiently accurate to rule out the Ornstein-Zernike form in zero field for T > T , but not sufficient to eliminate the possibility c of anisotropy at T = T .
Introduction We noted in the previous chapter that the singular behavior of thermodynamic functions at a critical point (the termination of a coexistence line) can be characterized by a set of critical exponents {α, β, γ, …}. Experimental observations indicate that these exponents are quite universal , i.e. independent of the material under investigation, and to some extent, of the nature of the phase transition. For example, the vanishing of the coexistence boundary in the condensation of CO 2 has the same singular behavior as that of the phase separation of protein solutions into dilute and dense components. This universality of behavior needs to be explained. We also noted that the divergence of the response functions, as well as direct observations of fluctuations via scattering studies, indicate that fluctuations have long wavelengths in the vicinity of the critical point, and are correlated over distances ξ ≫ a , where a is a typical interparticle spacing. Such correlated fluctuations involve many particles and a coarse-graining approach, in the spirit of the theory of elasticity, may be appropriate to their description. Here we shall construct such a statistical field theory . We shall frame the discussion in the language of a magnetic system whose symmetries are more transparent, although the results are of more general applicability. Consider a material such as iron, which is experimentally observed to be ferromagnetic below a Curie temperature T c , as in Fig. 1.4.
This chapter presents a simple approximation scheme — the mean field approximation — which allows a study of the neighbourhood of the transition temperature. It distinguishes between first and second order phase transitions. In the latter case, the correlation length diverges at the critical (transition) temperature. This has important physical consequences: in the mean field approximation, the singular behaviour of thermodynamic quantities at the critical temperature, and more generally in the neighbourhood of the critical temperature and in small magnetic field, is universal, that is, does not depend on the details of the interactions, on the dimension of space and, in the high temperature phase at least, on the symmetry of the models.
A previously proposed generalization of the usual Ornstein-Zernike equation for near-critical correlations is examined. It is shown that near the critical point, to leading order, this equation is compatible with realistic critical behavior in both spatial and thermal variables. The scaling predictions of this equation are compared to known results for d=2d=2d=2 and d=3d=3d=3 Ising models. The generalized equation changes the usual Ornstein-Zernike values of various quantities in qualitatively the correct way; however, quantitative discrepancies remain.