Renormalization procedures for families of dynamical systems have been used to prove many interesting results. Examples of results include that the bifurcation rate for the attractors of an analytic one-parameter family of quadratic-like maps is universal for all such families, unique ergodicity for almost every interval exchange mapping, a unique ergodicity criterion for the vertical translation flow of a flat surface in terms of its ``renormalization dynamics", known as Masur's criterion, and the classification of circle diffeomorphisms up to $C^{\infty}$ conjugation. We introduce renormalization procedures for $C^{*}$-algebras and étale groupoids using the concepts of $C_{0}(X)$-algebras and Morita equivalence for the former, and groupoid bundles and groupoid equivalence, in the sense of Muhly, Renault and Williams, for the latter. We focus on proving analogs to Masur's criterion in both cases using $C^{*}$-algebraic methods. Applying our criterion to our examples of renormalization procedures provides a unique trace criterion for unital AF algebras extending the one provided by Treviño in the setting of flat surfaces and the one provided by Veech in the setting of interval exchange mappings. Also, we recover the old fact that rotation of the circle by an irrational angle is uniquely ergodic, and the new fact that interesting groupoids associated to certain iterated function systems, recently introduced by Korfanty, have unique invariant probability measures whenever they are minimal. Lastly, we show how an étale groupoid renormalization procedure arises from an étale groupoid which factors down onto a groupoid associated to its renormalization dynamics, whenever it is a local homeomorphism. / Graduate

Ph.D., Faculty of Science, University of Witwatersrand, 2011 / Operators in N = 4 super Yang-Mills theory with an R-charge of O(N2) are dual to backgrounds which are asymtotically AdS5 S5. In this thesis we develop e cient techniques that allow the computation of correlation functions in these backgrounds. We nd that (i) contractions between elds in the string words and elds in the operator creating the background are the eld theory accounting of the new geometry, (ii) correlation functions of probes in these backgrounds are given by the free eld theory contractions but with rescaled propagators and (iii) in these backgrounds there are no open string excitations with their special end point interactions; we have only closed string excitations. Furthermore, these correlation functions are not well approximated by the planar limit. The non-planar diagrams, which in the bulk spacetime correspond to string loop corrections, are enhanced by huge combinatorial factors. We show how these loop corrections can be resummed. As a typical example of our results, in the half-BPS background of M maximal giant gravitons we nd the usual 1=N expansion is replaced by a 1=(M +N) expansion. Further, we nd that there is a simple exact relationship between amplitudes computed in the trivial background and amplitudes computed in the background of M maximal giant gravitons. We also nd strong evidence for the BMN-type sectors suggested in arXiv:0801.4457. The problem of computing the anomalous dimensions of (nearly) half-BPS operators with a large R-charge is reduced to the problem of diagonalizing a Cuntz oscillator chain. Due to the large dimension of the operators we consider, non-planar corrections must be summed to correctly construct the Cuntz oscillator dynamics. These non-planar corrections do not represent quantum corrections in the dual gravitational theory, but rather, they account for the backreaction from the heavy operator whose dimension we study. Non-planar corrections accounting for quantum corrections seem to spoil integrability, in general. It is interesting to ask if non-planar corrections that account for the backreaction also spoil integrability. We nd a limit in which our Cuntz chain continues to admit extra conserved charges suggesting that integrability might survive.

A general exact matrix renormalization group method is developed for solving quantum transmission through networks. Using this method transmission of spinless electrons is calculated for a Hanoi network and a (newly introduced) fully connected Bethe lattice. Plots of the transmission and wavefunctions are obtained through application of the derived Renormalization Group recursion relations. The plots reveal band gaps (which has possible application in nano devices) in HN3 networks while no band gaps are observed in HN5 networks. With the fully connected Bethe lattice a drastic reduction in the transmission (in comparison to the normal Bethe lattice) is observed. This reduction can be found to be a purely quantum mechanical effect.

Fluctuation driven phenomena refer to a broad class of physical systems that are shaped and influenced by randomness. These fluctuations can manifest in various forms such as thermal noise, stochasticity, or even quantum fluctuations. The importance of understanding these phenomena lies in their ubiquity in natural systems, from the formation of patterns in biological systems, to the behavior of phase transitions and universality classes, to quantum computers. In this dissertation, we delve into the peculiar phenomena driven by fluctuations in the following scenarios: We study the near-equilibrium critical dynamics of the O(3) nonlinear sigma model describing isotropic antiferromagnets with a non-conserved order parameter reversibly coupled to the conserved total magnetization. We find that in equilibrium, the dynamics is well-separated from the statics and the static response functions are recovered in the limit ω → 0, at least to one-loop order in a perturbative treatment with respect to the static and dynamical nonlinearities. Since the static nonlinear sigma model must be analyzed in a dimensional d = 2 + ε expansion about its lower critical dimension d_lc = 2, whereas the dynamical mode-coupling terms are governed by the upper critical dimension d_c = 4, a simultaneous perturbative dimensional expansion is not feasible, and the reversible critical dynamics for this model cannot be accessed at the static critical renormalization group fixed point. However, in the coexistence limit addressing the long-wavelength properties of the low-temperature ordered phase, we can perform an ε = 4 − d expansion near dc. This yields anomalous scaling features induced by the massless Goldstone modes, namely sub-diffusive relaxation for the conserved magnetization density with asymptotic scaling exponent z_Γ= d − 2 which may be observable in neutron scattering experiments. We investigate the influence of spatial disorder on coined quantum walks. Coined quantum walks describe the time evolution of a quantum particle that is controlled by a quantum coin degree of freedom. We consider one-dimensional walks and use a two- level system as quantum coin. Each time step thus consists of the iterative application of a quantum coin toss and a conditional shift operator. Qualitative differences with classical random walks arise due to superpositioned states and entanglement between walker and coin. We consider spatially inhomogeneous coin tosses with every lattice site having a tossing amplitude. These amplitudes are noisy such that the walk is spatially disordered. We find that disorder deteriorates the ballistic transport properties of non-noisy quantum walks. This leads to an extremely slow spreading of the quantum walker and potentially induces localization behavior. We investigate this slow dynamics and compare the disordered quantum walk with the standard coined Hadamard walk. Special focus is given to the influence of disorder on entanglement-related properties. We apply a perturbative field-theoretical analysis to the symmetric Rock-Paper-Scissors (RPS) model and the symmetric May-Leonard (ML) model, in which three species compete cyclically. We demonstrate that the qualitative features of the ML model are insensitive to intrinsic reaction noise. In contrast, and although not yet observed in numerical simulations, we find that the RPS model acquires significant fluctuation- induced renormalizations in the perturbative regime. We also study the formation of spatio-temporal structures in the framework of stability analysis and provide a clearcut explanation for the absence of spatial patterns in the RPS model, whereas the spontaneous emergence of spatio-temporal structures features prominently in the ML model. We delve into the action-to-absorbing phase transition in the Pair Contact Process with Diffusion (PCPD), which naturally generalizes the Directed Percolation (DP) reactions. We revisit the single-species PCPD model in the Doi-Peliti formalism and propose a possible perturbative solution for the model. In addition, we investigate the two-species effective model of PCPD and demonstrate its equivalence to the single- species PCPD at tree-level effective field theory. We also examine the fixed point of the model where all relevant parameters are set to zero. Our analysis reveals that the fixed-point theory is inconsistent with the PCPD critical condition. Thus, combining the effective field theory argument, this inconsistency suggests that the critical theory should already be completely encoded in the single-species model. / Doctor of Philosophy / Fluctuations are a ubiquitous aspect of the real world. For instance, even though a train schedule may be set, the train may arrive two minutes ahead of schedule or two hours late. Similarly, if you were to flip a coin ten times, you would expect to get five heads and five tails based on simple probability, but in reality, you may not even come close to this result. In classical situations, these fluctuations are a result of our lack of knowledge about the details of the system. However, in quantum mechanics, scientists have demonstrated that fluctuations are inherent to the system, even when every single detail of the system is known. Therefore, understanding fluctuations is crucial to gaining insight into the fundamental laws of the universe. In most cases, fluctuations are insignificant and the world can be accurately described by a set of deterministic equations. However, there are situations in which fluctuations play a significant role and can greatly deviate the system from the predictions of deterministic equations. In this dissertation, we study the following scenarios where fluctuations dominate and lead to peculiar phenomena: Near continuous phase transitions, due to the divergence of the characteristic length, most systems become long-range correlated. This means that the changes at one point can affect another point very far away. We study the critical dynamics of two systems near their phase transitions: antiferromagnetic system in Chapter 2 and a simplified population dynamics model in Chapter 5. Through our analysis, we demonstrate how fluctuations significantly alter the behavior of these systems near their critical points. In chapter 3, we examine the impact of spatial disorder on the quantum random walk, a quantum counterpart of the classical random walk or "drunkard's walk". Given that the quantum random walk has been shown to have universal quantum computing capabilities, this disorder can be considered as errors in the control of the system. We reveal how disorder effects drastically change the dynamics of the system. The formation of patterns is typically studied in deterministic nonlinear systems. In Chapter 4, we analyze pattern formation in stochastic population dynamics models, and demonstrate emergent behavior that goes beyond what is seen in their deterministic counterparts.

In this thesis we study the scaling properties of unconventional metals that arise at quantum critical points using low-energy effective field theories. Due to high rate of scatterings between electrons and critical fluctuations of the order parameter associated with spontaneous symmetry breaking, Landau’s Fermi liquid theory breaks down at the critical points. The theories that describe these critical points generally flow into strong coupling regimes at low energy in two space dimensions. Here we develop and utilize renormalization group methods that are suitable for the interacting non-Fermi liquids. We focus on the critical points arising at excitonic, and commensurate spin and charge density wave transitions. By controlled analyses we find stable non-Fermi liquid and marginal Fermi liquid states, and extract the scaling behaviour. The field theories for the non-Fermi liquids are characterized by symmetry groups, local curvature of the Fermi surface, the dispersion of the order parameter fluctuations, and dimensions of space and Fermi surface. / Thesis / Doctor of Philosophy (PhD)

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PhD Thesis / The superconductivity in heavy-fermion compounds, iron pnictides and cuprates has been intensively studied for over thirty years. Amongst some of these materials, the common denominator is the presence of strong antiferromagnetic fluctuations in their normal state, signaling an underlying quantum phase transition between a paramagnetic metal and a metal with antiferromagnetic long-range order. Although the quantum critical point is experimentally inaccessible due to the presence of superconducting order, it determines the physical properties of the normal state of the metal in a wide range of temperatures. In this thesis we study the low-energy theory for the critical metallic state that arises at the aforementioned quantum critical point. We present a nonperturbative study of the theory in spatial dimensions between two and three. We pay special attention to two dimensions where we show that our physical predictions are in qualitative agreement with experiments in electron-doped cuprates. We further develop a field theoretic functional renormalization group scheme that is analytically tractable. It provides a general framework to study the low-energy theory of metallic states with or without a quasiparticle description. Within this formalism we characterize the single-particle properties of the antiferromagnetic quantum critical metal. This allows one to study the superconducting instability triggered by critical antiferromagnetic quantum fluctuations quantitatively. / Thesis / Doctor of Science (PhD)

This work details how to use the Point-Particle Effective Field Theory (PPEFT) framework to make predictions for the nuclear-size contributions to spectroscopic transitions of atoms without the overbearing large uncertainties generally associated with such effects. After a lightning review of Quantum Field Theories, Effective Field Theories and their model-building algorithms, the backbones of the PPEFT formalism are laid down by considering the low-energy effective theories of lumps. Then, by drawing an analogy between a certain type of lumps and a freely propagating point-particle we build a PPEFT for nuclei, which we gradually couple to gauge and fermionic fields. We find that the consequences of having a nucleus in our theory are captured by a set of new near-nucleus boundary conditions its action implies for the surrounding fields, set up on a Gaussian spherical boundary with arbitrary radius, $\epsilon$. Afterwards, we use this formalism to derive the effects of the finite size of the nucleus on bound-state energies in terms of Renormalization Group (RG)-invariant parameters that characterize the running of the PPEFT couplings in $\epsilon$, implied by these new boundary conditions in order to keep physical quantities independent of this fictitious scale. Surprisingly, when comparing to formulae from the literature that express these same energy shifts in terms of nuclear moments there always appear to be fewer RG-invariants than moments. By fitting these handful of parameters using experimental data we then reduce the errors in nuclear-size effect predictions for other transitions by writing them in terms of differences between spectroscopic measurements and their corresponding energy differences predicted by those bound-state Quantum Electrodynamics calculations that assume nuclei to be point-like. Finally, we apply this algorithm to the systems: ${}^4_2 {\rm He}^+$, $\mu \, {}^4_2 {\rm He}^+$, H, and $\mu$H, where we make such predictions. / Thesis / Doctor of Philosophy (PhD) / The finite size of the nucleus shifts the bound-state energy of electrons (or muons) in atoms. Although these effects had been captured through a large number of nuclear-model independent ``nuclear moments'' closely related to the extent of the nucleus in the past, they introduce large uncertainties into theoretical predictions, which hinders testing fundamental subatomic processes in spectroscopic measurements. In this work it is shown that there is a more manageable number of parameters that control these effects because the above moments always appear in specific combinations. This allows for trading these combinations for differences between experimental values and their theoretically expected ones that assume the nucleus to have no size, which is the key in making predictions for atomic transitions that do not suffer from the large nuclear errors. A large set of such predictions are made for Hydrogen and the principles are applied to its muonic cousin as well.

Yes / A total set of states for which we have no resolution of the identity (a `pre-basis'), is considered in a finite dimensional Hilbert space. A dressing formalism renormalizes them into density matrices which resolve the identity, and makes them a `generalized basis', which is practically useful. The dresssing mechanism is inspired by Shapley's methodology in cooperative game theory, and it uses Mobius transforms. There is non-independence and redundancy in these generalized bases, which is quantifi ed with a Shannon type of entropy. Due to this redundancy, calculations based on generalized bases, are sensitive to physical changes and robust in the presence of noise. For example, the representation of an arbitrary vector in such generalized bases, is robust when noise is inserted in the coeffcients. Also in a physical system with ground state which changes abruptly at some value of the coupling constant, the proposed methodology detects such changes, even when noise is added to the parameters in the Hamiltonian of the system.