Statistics of the Two-Dimensional Ferromagnet. Part I

Abstract

In an effort to make statistical methods available for the treatment of cooperational phenomena, the Ising model of ferromagnetism is treated by rigorous Boltzmann statistics. A method is developed which yields the partition function as the largest eigenvalue of some finite matrix, as long as the manifold is only one dirnensionally infinite. The method is carried out fully for the linear chain of spins which has no ferromagnetic properties. Then a sequence of finite matrices is found whose largest eigenvalue approaches the partition function of the twodimensional square net as the matrix order gets large.It is shown that these matrices possess a symmetry property which permits location of the Curie temperature if it exists and is unique. It lies at J/kT. = 0.8814 if we denote by Jthe coupling energy between neighboring spins. The symmetry relation also excludes certain forms of singularities at T"as, e.g. , a jump in the specific heat.However, the information thus gathered by rigorous analytic methods remains incomplete. RANSITION temperatures of various typesare a well-known phenomenon in the study of matter, and the statistical distribution laws form a generally accepted piece of theory. It is also generally believed that the former are a consequence of the latter. This is, however, by no means immediately obvious and an examination of the literature shows that there is not more than one case in which a proof of this fact has been attempted. The case which has received successful treatment is the condensation of vapors. ' This paper wishes to carry out a similar treatment for the Curie transition of ferromagnets.The problem has a mechanical and a statistical aspect. On the mechanical side we wish to improve our understanding of the responsible coupling forces. On the statistical side we wish to derive with certainty the thermal properties from a reasonably accurate mechanical model. Both aspects have received extensive attention. Quantumtheory has explained satisfactorily the origin and nature of the coupling forces. There are also several theories available which explain in terms of them the thermal behavior ' Owing to communication difhculties, one of the authors (G. H. W. ) is entirely responsible for the printed text.

Twinning of Polymer Crystals Suppressed by Entropy

“…The rotational isomeric state (RIS) theory is based on the generator matrix method of Kramers and Wannier [38,39] and was first applied to polymers by Volkenstein (see [40] and refs. therein).…”

Geometrical vs. Fortuin–Kasteleyn clusters in the two-dimensional q-state Potts model

“…By the well-known Kramers-Wannier duality [37], the HT graphs form Peierls domain walls [38] separating spin clusters of opposite orientation on the dual lattice. Each bond in a HT graph intersects a nearest neighbor pair on the dual lattice of unlike spins perpendicular to it.…”

“…On an infinite lattice, the Kramers-Wannier duality implies that observables calculated at an inverse temperature β in the original Ising model can be transcribed to those of the dual model at an inverse temperatureβ. The relation between the two temperatures follows from noting that, an occupied HT bond represents a factor v, while a nearest neighbor pair on the dual lattice of unlike spin on each side of the HT bond carries a Boltzmann weight exp(−2β), so that [37] tanh β = e −2β ,…”

Geometric approach to a massivepform duality

“…In reference [3] it was shown that the "electric-magnetic" duality of abelian gauge theories allows to describe their physical phase space in terms of a pair of non-local observables that are dual in the Kramers-Wannier sense [4]. The algebra that they obey results to be invariant under spatial diffeomorphisms.…”