Phd records
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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenWed, 31 Jan 2024 19:02:20 +0000Renormalization Techniques in the Study of Critical Phenomena. I. Lattices of Effectively Nonintegral Dimensionality. II. A Model of the Melting Transition
https://resolver.caltech.edu/CaltechTHESIS:06192018-091151486
Authors: {'items': [{'email': 'deepakdhar1951@gmail.com', 'id': 'Dhar-Deepak', 'name': {'family': 'Dhar', 'given': 'Deepak'}, 'show_email': 'YES'}]}
Year: 1978
DOI: 10.7907/4CS0-RZ23
<p>This thesis is divided into two parts.</p>
<p>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<d<∞.</p>
<p>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.</p>
<p>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 T<sub>c</sub>. Below T<sub>c</sub>, the root
mean square deviation of atoms in the layer is finite, and
varies as (T<sub>c</sub>-T)<sup>-1/4</sup> near the phase transition. Above T<sub>c</sub>, the
root mean square deviation is infinite. The specific heat
shows an essential singularity at the phase transition,
varying as exp(-|Tc -T|<sup>-1/2</sup>) near T<sub>c</sub>.</p>
https://thesis.library.caltech.edu/id/eprint/11090