We propose a general quantum Hamiltonian formalism of a renormalization group (RG) flow with an emphasis on generalized symmetry by interpreting the elementary relationship between homomorphism, quotient ring, and projection. In our formalism, the noninvertible nature of the ideal of a fusion ring realizing the generalized symmetry of an ultraviolet (UV) theory plays a fundamental role in determining condensation rules between anyons, resulting in the infrared (IR) theories. Our algebraic method applies to the domain wall problem in
2+1 dimensional topologically ordered systems and the corresponding classification of
1+1 dimensional gapped phase, for example. An ideal decomposition of a fusion ring provides a straightforward but strong constraint on the gapped phase with noninvertible symmetry and its symmetry-breaking (or emergent symmetry) patterns. Moreover, even in several specific homomorphisms connected under massless RG flows, less familiar homomorphisms appear, and we conjecture that they correspond to partially solvable models in recent literature. Our work demonstrates the fundamental significance of the abstract algebraic structure, ideal, for the RG in physics.
This dissertation focuses on theoretical studies of relativistic Bose-Einstein condensation (BEC).Based on the complex scalar field theory, which serves as a fundamental model for relativistic BEC, we provide a detailed explanation of the analyses using the functional renormalization group (FRG), including both its formalism and the underlying physics.We also present the numerical implementation of the FRG flow equation and the results obtained from it, as well as the derivations of various related equations.Although previous studies rely on approaches that are not well justified, we develop a novel and physically consistent FRG framework.In this framework, we impose a condition on the FRG flow equation, analogous to that in the non-interacting case.By solving the equation under this condition, we find that quantum fluctuations significantly affect the phase transition of relativistic BEC in the strong coupling regime.We further investigate the dependence of the condensate on spatial dimensions, and in particular, confirm that the FRG analysis is consistent with the Mermin-Wagner theorem from both numerical and analytical perspectives.These results contribute to the development of the theoretical foundation of the FRG, and uncover an aspect of relativistic BEC under non-perturbative interactions and for various spatial dimensions.