Classification of $SL_2$ deformed Floquet Conformal Field Theories

Classification of the non-equilibrium quantum many-body dynamics is a challenging problem in condensed matter physics and statistical mechanics. In this work, we study the basic question that whether a (1+1) dimensional conformal field theory (CFT) is stable or not under a periodic driving with $N$ non-commuting Hamiltonians. Previous works showed that a Floquet (or periodically driven) CFT driven by certain $SL_2$ deformed Hamiltonians exhibit both non-heating (stable) and heating (unstable) phases. In this work, we show that the phase diagram depends on the types of driving Hamiltonians. In general, the heating phase is generic, but the non-heating phase may be absent in the phase diagram. For the existence of the non-heating phases, we give sufficient and necessary conditions for $N=2$, and sufficient conditions for $N>2$. These conditions are composed of $N$ layers of data, with each layer determined by the types of driving Hamiltonians. Our results also apply to the single quantum quench problem with $N=1$.