Fixed Points, Floquet Entanglement Asymmetry, and Quantum Mpemba Effects

We investigate the dynamics of entanglement asymmetry in periodically driven two-dimensional conformal field theories with a global U (1) symmetry, using a dynamical-system description based on iterated conformal maps, in particular Möbius maps. Starting from a symmetry-breaking excited state prepared by a local operator insertion, we show that a broad range of nonequilibrium phenomena, including local symmetry restoration, the quantum Mpemba effect, and its inverse, admit a simple geometric description in terms of the invariant data of the conformal map. We explicitly demonstrate this using Möbius maps, for which the invariant structure is described by the conjugacy classes. In particular, the dynamics of entanglement asymmetry is determined by the relative positions of the state-preparing operator insertion, the subsystem, and the fixed points, yielding distinct patterns of growth, decay, oscillation, and saturation. We further argue that this geometric picture extends to general holomorphic maps, establishing the conformal-map dynamics as a natural framework for organizing the phenomenology of entanglement asymmetry in periodically driven two-dimensional conformal field theories.