Corrections to scaling in real and model magnetic systems

Stuart S. C Burnett, Purdue University

Abstract

In this thesis, the corrections to scaling predicted by the Renormalization Group (RG) are studied for both a real and a model system. These corrections are generally expected to be of two types: (1) analytic (and. sometimes nonanalytic) corrections due to the nonlinearity of the scaling fields and (2) confluent singularities due to the presence of irrelevant variables. The real system that we consider is the (S = 1/2) insulating Heisenberg ferromagnet Copper Ammonium Bromide for which we have compared experimental data for the spontaneous magnetization, the zero-field susceptibility and the zero-field specific heat with two sets of theoretical formulas. The first of these makes use of a certain phenomenology, proposed by Gartenhaus, which is equivalent to the RG in the absence of irrelevant variables. This phenomenology, which accounts for the leading analytic corrections due to the nonlinearity of the scaling fields and which includes explicit formulas for the spontaneous magnetization and the zero-field susceptibility, is extended here to include the zero-field specific heat. The second set of formulas is obtained by appending to the asymptotic scaling formulas the lowest order confluent singularities due to the leading irrelevant variable. It is found that both sets of formulas can be made to agree with the data but that the phenomenology provides a better fit for the spontaneous magnetization and overall requires fewer parameters. Some differences between the theoretical and experimental values of the susceptibility exponent, for certain ferromagnetic materials, are also discussed. The model system that we consider is the antiferromagnetic square Ising lattice. The leading two terms in the expansion about the critical point for the zero-field susceptibility are known and we have carried out a differential approximant analysis of Nickel's 55 term high temperature series to determine the next four terms. These four terms are found to vary as $t, t\sp2{\rm ln} \vert t\vert, t\sp2$ and $t\sp3{\rm ln} \vert t\vert,$ respectively, and values for their coefficients are obtained. Comparison of the resulting extended formula for the susceptibility with the analytic structure predicted by the RG leads us to conclude that, to the order considered, there are no confluent singularities due to irrelevant variables, a result which parallels earlier work by Gartenhaus and McCullough on the corresponding ferromagnetic lattice. The results are also consistent with RG arguments, put forward by Aharony and Fisher, that the lowest order corrections to scaling for the planar Ising systems are due to the nonlinearity of the scaling fields rather than the leading irrelevant variable.

Degree

Ph.D.

Advisors

Gartenhaus, Purdue University.

Subject Area

Condensation

COinS