QSLs on Operator Flows

Conventional QSLs typically address quantum dynamics in the Schrödinger picture, where the state of the system undergoes time evolution while quantum operators are kept fixed. However, this framework proves inadequate to deal with many applications, both theoretical and experimental, in which physical phenomena are naturally described in terms of operator flows. Notably, the buildup of complexity in quantum systems is well-captured by the operator growth of its typical observables in the Heisenberg picture, as extensively discussed in the first part of this thesis. Moreover, two-point correlation functions of operators characterize the nonequilibrium physics of many-body systems. In these contexts, the dynamics is formulated as a unitary flow of the relevant observable and cannot be constrained in terms of the standard QSLs. In this chapter, we generalize the notion of QSL to unitary operator flows, which are ubiquitous in physics, ranging from the quantum evolution of an observable in the Heisenberg picture to renormalization group flows and Lax pair equations in classical integrable systems [1, 2].