Renormalization Techniques in the Study of Critical Phenomena. I Lattices of Effectively Nonintegral Dimensionality. II: a Model of the Melting Transition.

Abstract

This thesis is divided into two parts. In Part I, we give an explicit construction for a class of lattices with effectively non-integral dimensionality. A reasonable definition of dimensionality applicable to lattice systems is proposed. The construction is illustrated by several examples. We calculate the effective dimensionality of some of these lattices. The attainable values of the dimensionality d, using our construction, are densely distributed in the interval 1 The variation of critical exponents with dimensionality is studied for a variety of Hamiltonians. It is shown that the critical exponents for the spherical model, for all d, agree with the values derived in literature using formal arguments only. We also study the critical behavior of the classical p-vector Heisenberg model and the Fortuin-Kasteleyn cluster model for lattices with d<2. It is shown that no phase transition occurs at nonzero temperatures. The renormalization procedure is used to determine the exact values of the connectivity constants and the critical exponents α, γ, v for the self-avoiding walk problem on some multiply connected lattices with d<2. It is shown by explicit construction that the critical exponents are not functions of dimensionality alone, but depend on detailed connectivity properties of the lattice. In Part II, we investigate a model of the melting transition in solids. Melting is treated as a layer phenomenon, the onset of melting being characterized by the ability of layers to slip past each other. We study the variation of the root-mean-square deviation of atoms in one layer as the temperature is increased. The adjacent layers are assumed held fixed and provide an external periodic potential. The coupling between atoms within the layer is assumed to be simple harmonic. The model is thus equivalent to a lattice version of the Sine-Gordon field theory in two dimensions. Using an exact equivalence, the partition function for this problem is shown to be related to the grand partition function of a two-species classical lattice Coulomb gas. We use the renormalization procedure to determine the critical behavior of the lattice Coulomb gas problem. Translating the results back to the original problem, it is shown that there exists a phase transition in the model at a finite temperature Tc. Below Tc, the root mean square deviation of atoms in the layer is finite, and varies as (Tc-T)-1/4 near the phase transition. Above Tc, the root mean square deviation is infinite. The specific heat shows an essential singularity at the phase transition, varying as exp(-|Tc -T|-1/2) near Tc.