Non-equilibrium Phase Transition in Aperiodically and Randomly Driven Conformal Theories

Aperiodically driven quantum systems can exhibit novel phenomena absent in stationary systems, but drive-induced heating limits the timescales over which they persist. In this work, we investigate heating dynamics in SL(2, R)-spatially deformed conformal field theories driven by Thue-Morse (TM) sequences, which can be analytically reduced to a series of Möbius transformations. Our results show an exponentially long prethermal lifetime and a prethermal-to-heating phase transition, characterized by a stability analysis of the TM trace map. To realize a robust non-heating phase, we generalize the SL(2, R)-deformed Hamiltonian to SU(2) deformation, which remains resilient to heating under arbitrary random sequences. By designing the combination of SL(2, R) and SU(2)-deformed Hamiltonians, we demonstrate the non-heating to heating phase transition under TM driving and random driving. Additionally, we analyze the stability of the phase diagram through perturbative methods and randomized truncation.