Yao, Hong2023-05-162023-05-162023-05-15vt_gsexam:37297http://hdl.handle.net/10919/115048Fluctuation 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.ETDenIn CopyrightField TheoryRenormalization GroupDisorderCritical DynamicsUniversal BehaviorDissertation