In this thesis, we study several different renormalization group (RG) methods, including the conventional Wilson renormalization group, Monte Carlo renormalization group (MCRG), exact renormalization group (ERG, or sometimes called functional RG), and tensor renormalization group (TRG). We use the two dimensional nearest neighbor Ising model to introduce many conventional yet important concepts. We then generalize the model to Dyson's hierarchical model (HM), which has rich phase properties depending on the strength of the interaction. The partition function zeros (Fisher zeros) of the HM model in the complex temperature plane is calculated and their connection with the complex RG flows is discussed. The two lattice matching method is used to construct both the complex RG flows and calculate the discrete β functions. The motivation of calculating the discrete β functions for various HM models is to test the matching method and to show how physically relevant fixed points emerge from the complex domain. We notice that the critical exponents calculated from the HM depend on the blocking parameter b. This motivated us to analyze the connection between the discrete and continuous RG transformation. We demonstrate numerical calculations of the ERG equations. We discuss the relation between Litim and Wilson-Polchinski equation and the effect of the cut-off functions in the ERG calculation. We then apply methods developed in the spin models to more complicated and more physically relevant lattice gauge theories and lattice quantum chromodynamics (QCD) like theories. Finite size scaling (FSS) technique is used to analyze the Binder cumulant of the SU(2) lattice gauge model. We calculate the critical exponent nu and omega of the model and show that it is in the same universality class as the three dimensional Ising model. Motivated by the walking technicolor theory, we study the strongly coupled gauge theories with conformal or near conformal properties. We compare the distribution of Fisher zeros for lattice gauge models with four and twelve light fermion flavors. We also briefly discuss the scaling of the zeros and its connection with the infrared fixed point (IRFP) and the mass anomalous dimension. Conventional numerical simulations suffer from the critical slowing down at the critical region, which prevents one from simulating large system. In order to reach the continuum limit in the lattice gauge theories, one needs either large volume or clever extrapolations. TRG is a new computational method that may calculate exponentially large system and works well even at the critical region. We formulate the TRG blocking procedure for the two dimensional O(2) (or XY ) and O(3) spin models and discuss possible applications and generalizations of the method to other spin and lattice gauge models. We start the thesis with the introduction and historical background of the RG in general.

The physics that result in the decoupling of a molecule from a bosonic solvent at 0 K are studied. Fixed-node diffusion Monte Carlo (FNDMC) coupled with a Genetic Algorithm is used to perform simulations of the bosonic droplets doped with various molecules. The efficacy and accuracy of this approach is tested on a strongly coupled 2-dimensional quartic oscillator with excellent results. This algorithm is then applied to 4He-CO and 4He-HCN clusters respectively in an effort to determine the factors that result in the onset of microscopic superfluidity. The decoupling of the doped molecule from the bosonic solvent is found to be, primarily, a result of the combined effect of the repulsive interaction between the helium atoms and bose symmetry. The effects of rotor size versus molecular anisotropy in a NH3 molecule seeded into a 4He droplet is studied as well. Simulations are done using the accurate rotational constants (B0=9.945 cm-1, C0=6.229 cm-1) and using "fudged" versions of the rotational constants (Bfudged=0.9945 cm-1, Cfudged=0.6229 cm-1) for the |0011〉state. The simulations done with the fudged rotational constants experience a slightly smaller reduction than those done using the accurate rotational constants. This is attributed to the importance of molecular anisotropy versus the size of larger rotational constants in molecules whose rotational constants fall in an intermediate regime.

In this thesis we develop and use a continuum random walk framework to solve problems that are usually studied using a discrete random walk on a discrete lattice. Problems studied include; the time it takes for a random walker to be absorbed at a trap on a fractal lattice, the calculation of the spectral dimension for several different classes of networks, the calculation of the density of states for a multi-layered Bethe lattice and the relationship between diffusion exponents and a resistivity exponent that occur in relevant power laws. The majority of the results are obtained by deriving an expression for a Laplace transformed Green’s function or first passage time, and then using Tauberian theorems to find the relevant asymptotic behaviour. The continuum framework is established by studying the diffusion equation on a 1-d bar with non-homogeneous boundary conditions. The result is extended to model diffusion on networks through linear algebra. We derive the transformation linking the Green’s functions and first passage time results in the continuum and discrete settings. The continuum method is used in conjunction with renormalization techniques to calculate the time taken for a random walker to be absorbed at a trap on a fractal lattice and also to find the spectral dimension of new classes of networks. Although these networks can be embedded in the d- dimensional Euclidean plane, they do not have a spectral dimension equal to twice the ratio of the fractal dimension and the random walk dimension when the random walk on the network is transient. The networks therefore violate the Alexander-Orbach law. The fractal Einstein relationship (a relationship relating a diffusion exponent to a resistivity exponent) also does not hold on these networks. Through a suitable scaling argument, we derive a generalised fractal Einstein relationship which holds for our lattices and explains anomalous results concerning transport on diffusion limited aggregates and Eden trees.

In this thesis we investigate the employ of the renormalization group for glassy systems. More precisely, we focus on models of spin glasses and structural glasses. Spin-glass models represent disordered uniaxial magnetic materials, such as a dilute solution of Mn in Cu, modeled by an array of spins on the Mn arranged at random in the matrix of Cu, and interacting with a potential which oscillates as a function of the separation of the spins. Structural glasses are liquids that have been cooled fast enough to avoid crystallization, like o-Terphenyl or Glycerol. Spin and structural glasses are physically interesting because their critical properties are known only in the limit where the space dimensionality tends to infinity, i. e. in the mean-field approximation. A fundamental question is whether the physical properties characterizing these systems in the mean-field case still hold for real spin or structural glasses, which live in a space with a finite number of dimensions. The spin and structural glasses that we study in this thesis are models built up on hierarchical lattices, which are the simplest non-mean field systems where the renormalisation group approach can be implemented in a natural way. The features emerging from this implementation clarify the critical behavior of these systems. As far as the finite-dimensional spin glass studied in this thesis is concerned, we developed a new technique to implement the renormalization group transformation for finite-dimensional spin glasses. This technique shows that the system has a finite-temperature phase transition characterized by a critical point where the system's correlation length is infinite. As far as the structural glass studied in this thesis is concerned, this is the first structural glass model where we showed the existence of a phase transition beyond mean field. The ideas introduced in this work can be further developed in order to understand the structure of the low-temperature phase of these systems, and in order to establish whether the properties of the low-temperature phase holding in the mean-field case still hold for finite-dimensional glassy systems.

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

The low-energy properties of quantum dot systems are dominated by the Kondo effect. We study the dependence of the characteristic energy scale of the effect, the Kondo temperature, on the gate voltage, which controls the number of electrons in the strongly blockaded dot. We show that in order to obtain the correct Kondo temperature as a function of the gate voltage, it is crucial to take into account the presence of many energy levels in the dot. The dependence turns out to be very different from that in the conventional single-level Anderson impurity model. Unlike in the latter, the Kondo temperature cannot be characterized by a single parameter, such as the ratio of the tunneling-induced width of the energy levels in the dot and the charging energy.

Thesis (Ph. D.)--University of Washington, 2003. / Vita. Includes bibliographical references (p. 86-89).

Thesis (Ph. D.)--University of Washington, 2003. / Vita. Includes bibliographical references (leaves 86-90).

Not available / text

We present the techniques for the calculation of one- and two-loop integrals contributing to the virtual corrections to 2→2 scattering of massless particles. First, tensor integrals are related to scalar integrals with extra powers of propagators and higher dimension using the Schwinger representation. Integration By Parts and Lorentz Invariance recurrence relations reduce the number of independent scalar integrals to a set of master integrals for which their expansion in є = 2 — D/2 is calculated using a combination of Feynman parameters, the Negative Dimension Integration Method, the Differential Equations Method, and Mellin-Barnes integral representations. The two-loop matrix-elements for light-quark scattering are calculated in Conventional Dimensional Regularisation by direct evaluation of the Feynman diagrams. The ultraviolet divergences are removed by renormalising with the MS scheme. Finally, the infrared singular behavior is shown to be in agreement with the one anticipated by the application of Catani's formalism for the infrared divergences of generic QCD two-loop amplitudes.