Citation
Dhar, Deepak (1978) Renormalization Techniques in the Study of Critical Phenomena. I. Lattices of Effectively Nonintegral Dimensionality. II. A Model of the Melting Transition. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/4CS0RZ23. https://resolver.caltech.edu/CaltechTHESIS:06192018091151486
Abstract
This thesis is divided into two parts.
In Part I, we give an explicit construction for a class of lattices with effectively nonintegral 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<∞.
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 pvector Heisenberg model and the FortuinKasteleyn 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 selfavoiding 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 rootmeansquare 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 SineGordon 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 twospecies 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_{c}. Below T_{c}, the root mean square deviation of atoms in the layer is finite, and varies as (T_{c}T)^{1/4} near the phase transition. Above T_{c}, 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 T_{c}.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  Physics 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Physics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  25 May 1978 
NonCaltech Author Email:  deepakdhar1951 (AT) gmail.com 
Record Number:  CaltechTHESIS:06192018091151486 
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:06192018091151486 
DOI:  10.7907/4CS0RZ23 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  11090 
Collection:  CaltechTHESIS 
Deposited By:  Benjamin Perez 
Deposited On:  25 Jun 2018 20:19 
Last Modified:  21 Dec 2019 03:00 
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