Operator growth in 2d CFT

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.

Quantum chaos and the complexity of spread of states

“…Our approach to state complexity is related to the notion of Krylov operator complexity, which has been put forward in [46], and developed in [47][48][49][50][51][52][53][54][55][56][57][58][59][60][61], based on the Lanczos approach [13] to operator dynamics in manybody systems. This approach starts with a Hamiltonian H and a time-dependent operator O(t), determined by…”

“…In fact, however, Krylov operator complexity has a further ambiguity that is not present in our approach, arising from the choice of inner product. Starting with [46], recent literature [47][48][49][50][51][52][53][54][55][56][57][58][59][60][61] has mainly focused on the Wightman inner product…”

“…The amplitudes ψ n (t) of the time-evolved state in the Krylov basis |K n are obtained by expanding these states in an orthonormal basis. The link with coherent states allows us to geometrize the notion of complexity following [55,57].…”

“…II B and the variance of the distribution. In the context of operator growth, these were studied in [47,57] respectively, where they were dubbed K-entropy and K-variance.…”

Operator Growth and Symmetry-Resolved Coefficient Entropy in Quantum Maps

“…Among the many operational diagnostics of quantum chaos, operator growth, or operator spreading, has played a vital role in characterizing dynamical chaos in the operator Hilbert space. It describes the process where a "simple" operator becomes increasingly complicated under chaotic Heisenberg time evolution, and can be intuitively quantified by out-of-time-ordered correlator (OTOC) [8; 10], operator entanglement [11][12][13], Krylov complexity [14][15][16], and coefficient entropy [1]. The common belief, which has been tested in various discrete and continuous models, is that for a quantum dynamics to be chaotic it must satisfy certain form of maximal operator growth.…”