Information Scrambling with Conservation Laws

The delocalization or scrambling of quantum information has emerged as a central ingredient in the understanding thermalization isolated many-body systems. Recently, significant progress been made analytically by modeling non-integrable systems stochastic systems, lacking Hamiltonian picture, while honest dynamics are frequently limited to small system sizes due computational constraints. In this paper, we address investigating role conservation laws (including energy conservation) process from an information-theoretic perspective. For general models, use equilibrium approximation show that maximal amount is scrambled (as measured tripartite mutual time-evolution operator) at late times even when conserves energy. contrast, explicate how additional symmetries lead degeneracies spectrum, must decrease. This theory exemplified case studies holographic conformal field theories (CFTs) and Sachdev-Ye-Kitaev (SYK) model. Due large Virasoro symmetry 1+1D CFTs, argue that, sense, these not maximally chaotic, which explicitly seen non-saturation second R\'enyi information. roles particle-hole U(1) SYK model milder being only two-fold, confirm both large- small-$N$. We reinterpret operator entanglement terms growth local operators, connecting our results with described out-of-time-ordered correlators, identifying mechanism for suppressed Heisenberg