Operator growth in 2d CFT

We investigate and characterize the dynamics of operator growth in irrational two-dimensional conformal field theories. By employing the oscillator realization of the Virasoro algebra and CFT states, we systematically implement the Lanczos algorithm and evaluate the Krylov complexity of simple operators (primaries and the stress tensor) under a unitary evolution protocol. Evolution of primary operators proceeds as a flow into the 'bath of descendants' of the Verma module. These descendants are labeled by integer partitions and have a one-to-one map to Young diagrams. This relationship allows us to rigorously formulate operator growth as paths spreading along the Young's lattice. We extract quantitative features of these paths and also identify the one that saturates the conjectured upper bound on operator growth.

Automated summary

In this study, the dynamics of operator growth in irrational two-dimensional conformal field theories is systematically characterized using the oscillator realization of the Virasoro algebra and CFT states. The evolution of primary operators is found to flow into the 'bath of descendants' of the Verma module, which are labeled by integer partitions and have a one-to-one map to Young diagrams. The relationship between these descendants and Young diagrams rigorously formulates operator growth as paths spreading along the Young's lattice, with quantitative features extracted and a specific path identified that saturates the conjectured upper bound on operator growth.