Theta Function Framework for Quantum Systems with Magnetic Flux

We develop an analytic framework for quantum systems defined on a circle threaded by magnetic flux, generalizing the Bargmann representation via Theta functions.By introducing flux-dependent coherent states and displacement operators, we construct entire analytic wavefunctions and derive a reproducing kernel with quasi-periodic boundary conditions.This approach captures topological effects such as the Aharonov-Bohm phase and provides tools relevant for mesoscopic rings and topological quantum computation.Compared to flux-free analytic approaches, the present framework directly encodes quasi-periodic boundary conditions, making it suitable for simulating observables such as persistent currents and phase shifts.Future work includes numerical implementations of the Theta function framework for flux qubits, extensions to multi-particle systems, and exploration of connections to non-Abelian topological phases.