Krylov Complexity in $2d$ CFTs with SL$(2,\mathbb{R})$ deformed Hamiltonians

In this study, we analyze Krylov Complexity in two-dimensional conformal field theories subjected to deformed SL$(2,\mathbb{R})$ Hamiltonians. In the vacuum state, we find that the K-complexity exhibits a universal phase structure. The phase structure involves the K-complexity exhibiting an oscillatory behaviour in the non-heating phase, which contrasts with the exponential growth observed in the heating phase, while it displays polynomial growth at the phase boundary. Furthermore, we extend our analysis to compute the K-complexity of a light operator in excited states, considering both large-c CFT and free field theory. In the free field theory, we find a state-independent phase structure of K-complexity. However, in the large-c CFT, the behavior varies, with the K-Complexity once again displaying exponential growth in the heating phase and polynomial growth at the phase boundary. Notably, the precise exponent governing this growth depends on the heaviness of the state under examination. In the non-heating phase, we observe a transition in K-complexity behavior from oscillatory to exponential growth, akin to findings in [1], as it represents a special case within the non-heating phase.