Operator growth in 2d CFT

TL;DR

This paper studies how operators evolve and grow in complexity within 2D irrational conformal field theories, revealing a structured growth pattern linked to Young diagrams and saturating theoretical bounds.

Contribution

It introduces a systematic method to analyze operator growth in 2D CFTs using the Virasoro algebra and Krylov complexity, connecting growth paths to Young diagrams.

Findings

Operator growth follows paths in Young's lattice.

Identified the growth path saturating the upper bound.

Quantified the dynamics of primary and descendant operators.

Abstract

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.