In this work we explore how lattice gauge theory stands to benefit from new developments in machine learning, and look at two specific examples that illustrate this point. We begin with a brief overview of selected topics in machine learning for those who may be unfamiliar, and provide a simple example that helps to show how these ideas are carried out in practice. After providing the relevant background information, we then introduce an example of renormalization group (RG) transformations, inspired by the tensor RG, that can be used for arbitrary image sets, and look at applying this idea to equilibrium configurations of the two-dimensional Ising model. The second main idea presented in this thesis involves using machine learning to improve the efficiency of Markov Chain Monte Carlo (MCMC) methods. Explicitly, we describe a new technique for performing Hamiltonian Monte Carlo (HMC) simulations using an alternative leapfrog integrator that is parameterized by weights in a neural network. This work is based on the L2HMC ('Learning to Hamiltonian Monte Carlo') algorithm introduced in [1].

The analysis of the error of perturbative series by comparing it to the exact solution is an important tool to understand the non-perturbative physics of statistical models. For some toy models, a new method can be used to calculate higher order weak coupling expansion and modified perturbation theory can be constructed. However, it is nontrivial to generalize the new method to understand the critical behavior of high dimensional spin and gauge models. Actually, it is a big challenge in both high energy physics and condensed matter physics to develop accurate and efficient numerical algorithms to solve these problems. In this thesis, one systematic way named tensor renormalization group method is discussed. The applications of the method to several spin and gauge models on a lattice are investigated. theoretically, the new method allows one to write an exact representation of the partition function of models with local interactions. E.g. O(N) models, Z2 gauge models and U(1) gauge models. Practically, by using controllable approximations, results in both finite volume and the thermodynamic limit can be obtained. Another advantage of the new method is that it is insensitive to sign problems for models with complex coupling and chemical potential. Through the new approach, the Fisher's zeros of the 2D O(2) model in the complex coupling plane can be calculated and the finite size scaling of the results agrees well with the Kosterlitz-Thouless assumption. Applying the method to the O(2) model with a chemical potential, new phase diagram of the models can be obtained. The structure of the tensor language may provide a new tool to understand phase transition properties in general.

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

Thesis (Ph. D.)--University of Texas at Austin, 2004. / Supervisor: Philip J. Morrison. Vita. Includes bibliographical references.

O comportamento critico de sistemas unidimensionais de spin do tipo Z(N) na ausência de campos magnéticos, é estudado sob a luz da teoria do grupo de renormalização. Os modelos são resolvidos exatamente pelo método da matriz de transferência e expressões para as funções de correlação e susceptibilidade (a campo zero) por si tio são também calculadas. As transformações do grupo de renormalização são efetuadas através de um traço parcial na função de partição, obtendo- se um conjunto de relações de recorrência que podem ser escritas de maneira simples para qualquer valor inteiro do fator de reescala espacial, mediante o uso de campos de escala convenientes. Tirando vantagem de um ponto fixo inteiramente atrativo, calculamos uma expressão para a energia livre por sitio, exata para T ¢ O. Analisamos o comportamento de nossos modelos no espaço de parâmetros, onde identificamos em particular as ~s ferro e antiferromagnéticas. O problema de correções às previsões de escala em termos de campos de escala não lineares é discutido. Aventamos também a possibilidade de calcular os auto valores da matriz de transferência através dos campos não lineares / In this work we study the criticai behaviour of one dimensional Z(N) spin systems in zero magnetic fields, using the approach of the renormalization group (RG) theory. The models are solved by the transfer matrix method and expressions for the correlation functions and zero field susceptibility per site are found. The RG transformations are carried out via a partial trace over the partition function and one obtains a set of recursion relations which, with the use of a convenient set of scaling fields, are written out in a simple manner for any integer value of the spatial rescaling factor. Using a totaly attractive fixed point we calculate an expression for the free energy per site, valid exactly for non zero values of the temperature. We analyse the behaviour of our models in the space of parameters, identifying in particular ferro and antiferromagnetic regions. The problem of corrections to scaling in terms of nonlinear scaling fields is discussed and a possibility of finding the eigen values of the transfer matrix from such fields is contemplated

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)