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We study the energy and entanglement dynamics of $(1+1)$D conformal fieldtheories (CFTs) under a Floquet drive with the sine-square deformed (SSD)Hamiltonian. Previous work has shown this model supports both a non-heating anda heating phase. Here we analytically establish several robust and`super-universal' features of the heating phase which rely on conformalinvariance but not on the details of the CFT involved. First, we show theenergy density is concentrated in two peaks in real space, a chiral andanti-chiral peak, which leads to an exponential growth in the total energy. Thepeak locations are set by fixed points of the M\"obius transformation. Second,all of the quantum entanglement is shared between these two peaks. In eachdriving period, a number of Bell pairs are generated, with one member pumped tothe chiral peak, and the other member pumped to the anti-chiral peak. TheseBell pairs are localized and accumulate at these two peaks, and can serve as asource of quantum entanglement. Third, in both the heating and non-heatingphases we find that the total energy is related to the half system entanglemententropy by a simple relation $E(t)\propto c \exp \left( \frac{6}{c}S(t)\right)$ with $c$ being the central charge. In addition, we show that thenon-heating phase, in which the energy and entanglement oscillate in time, isunstable to small fluctuations of the driving frequency in contrast to theheating phase. Finally, we point out an analogy to the periodically drivenharmonic oscillator which allows us to understand global features of thephases, and introduce a quasiparticle picture to explain the spatial structure,which can be generalized to setups beyond the SSD construction.

Emergent Spatial Structure and Entanglement Localization in Floquet Conformal Field Theory