Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

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The renormalization group (RG) approach is largely responsible for the considerable success which has been achieved in developing a quantitative theory of phase transitions. This work investigates various spectral properties of the RG map for Ising-type classical lattice systems. It consists of four parts. The first part carries out some explicit calculations of the spectrum of the linearization of the RG at infinite temperature, and discovers that it is of an unusual kind: dense point spectrum for which the adjoint operators have no point spectrum at all, but only residual spectrum. The second part presents a rigorous justification of the existence and differentiability of the RG map in the infinite volume limit at high temperature by a cluster expansion approach. The third part continues the theme of the third part, and shows that the matrix of partial derivatives of the RG map displays an approximate band property for finite-range and translation-invariant Hamiltonians at high temperature. The last part justifies the differentiability of the RG map in the infinite volume limit at the critical temperature under a certain condition. In summary, the first part deals with special cases where exact computations can be done, whereas the remaining parts are concerned with a general theory and provide a mathematically sound base.

Thesis (Ph. D.)--University of Adelaide, Dept. of Physics and Mathematical Physics, 1992. / Includes bibliographical references (p. 105-111).

Thesis (Ph. D.)--Florida State University, 2005. / Advisor: Dr. Kun Yang, Florida State University, College of Arts and Sciences, Dept. of Physics. Title and description from dissertation home page (viewed Sept. 19, 2005). Document formatted into pages; contains viii, 69 pages. Includes bibliographical references.

Renormalisation is an important aspect of Quantum Field Theory. It is used to create physically meaningful theories and some major developments took place in the 1970's and onwards. We consider Renormalisation in its application to the theories of ψ⁴ , Quantum Electrodynamics, Quantum Chromodynamics and the Background Field Method. Feynman diagrams are used to illustrate many of the concepts.

During my thesis work I have investigated the problem of polyelectrolyte characterization and in particular how to interpret the experimental data to obtain the mass and gyration radius of short polyelectrolytes. This is usually a challenging problem for experimentalists to deal with. For example, the interpretation of the static light scattering data to obtain the gyration radius becomes increasingly inaccurate as the size of the chain becomes very much smaller then the light wavelength. Also, the interpretation of membrane osmomometry data is complicated by the leakage of the solute of low molecular weight polymers and so forth. There are, however, a number of approaches to deal with these problems. In the first chapter of my thesis I obtained a crossover formula of the second virial coefficient of polyelectrolytes that correctly reproduces the perturbative and asymptotic polymer regimes in addition to the salt concentration dependence at high-to-medium salt concentrations. This formula will then be combined with similar crossover formula for the radius of gyration to interpret the later from the second virial coefficient measurement. On the technical side of the story, the crossover formula was obtained by combining the renormalization group equation(to the first nontrivial order in epsilon) with the direct d=3 computation of the perturbative expansion( to the second order in the two coupling constants) obtained from double inverse Laplace transform. The second chapter of my dissertation is about the translocation phenomena. Translocation is a phenomena of threading a polymer through narrow pores and/or channels. This is very promising technique to measure the molecular weight of every individual polymer in the solution. Indeed, the polymer chain threading through the pore blocks the flow of electric current that also flows through the pore. By the duration of the current blockade the length of polymer chain can be obtained. Unfortunately, there are a number of problems this approach encounters. One of them is that the only so far practically obtainable nanometer-size pore is the alpha hemolysine one which has a complicated internal layout- a spherical(more or less) vestibule. This nasty feature makes current blockade vs time data harder to interpret. There is a way to bypass this problem. Recently a number of research groups began to modify the pore via the directed mutagenesis to reduce the time the chain spends in the vestibule. In my work I theoretically investigated translocation of the polyelectrolyte chain through a spherical cavity with tunable charge. The results provide some guidelines on how to reduce the influence of the vestibule on the translocation time if we are to medify the chain in addition to the decorating the pore with charges. This work includes a number of interesting techniques. It is based on the self-consistent field theory which gives us nonlinear Schroedinger and Poisson-Boltzmann equations. These equations are then solved numerically via a finite difference schemes. Lets point out possible extensions of this work. The SCFT technique is a primary computational tool for polyelectrolyte brushes and melts. Those things can be useful for the rapidly developing technology of pore gating, brush filtration and brush lubrication, just to name a few.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. Johannesburg, 2015. / Recently there has been signi cant interest in new types of metals called non-Fermi liquids, which cannot be described by Landau Fermi liquid theory. Landau Fermi liquid theory is a theoretical model used to describe low energy interacting fermions or quasiparticles. There is a growing interest in constructing an e ective eld theory for these types of metals. One of the paradigms to understand these metals is by the use of Wilsonian renormalization group (RG) to study a theoretical toy model consisting of fermions coupled to a gapless order parameter eld. Here we will study fermions coupled to gapless bosons (order parameter) below the upper critical dimension (d = 3). We will treat both fermions and bosons on equal footing and construct an e ective eld theory which only integrates out high momentum modes. Then we compute the one-loop RG ows for the Yukawa coupling and four-Fermi interaction. We will discuss log2 and log3 subleties associated with the one loop RG ows for the four-Fermi interaction and how they can be circumvented.

Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

We use the density matrix renormalization group to study the quantum transitions that occur in the half-filled one-dimensional fermionic Hubbard model with onsite potential disorder. We find a transition from the gapped Mott phase with algebraic spin correlations to a gapless spin-disordered phase beyond a critical strength of the disorder 1 c ss U= 2. Both the transitions in the charge and spin sectors are shown to be coincident. We also establish the finite-size corrections to the charge gap and the spin-spin correlation length in the presence of disorder and using a finite-size-scaling analysis we obtain the zero temperature phase diagram of the various quantum phase transitions that occur in the disorder-interaction plane.