Emergent Lie Algebra and Quantum information dynamics in Brownian SYK models

The out-of-time ordered correlator plays an important role in understanding how the information of the initial state scrambles over the entire system under unitary time evolution. Incorporating additional symmetries, such as charge conservation, can have profound effects on information scrambling. We study the out-of-time ordered correlator of generic chaotic systems with charge conservation, using the Brownian SYK models as examples. We show that the averaging of the uncorrelated random couplings at different times gives rise to an emergent SU(2) x SU(2) algebra structure in the Super-Hamiltonian, which is enlarged to SU(4) when charge conservation is included. The algebra structure drastically reduces the size of the Hilbert space from exponential to linear in N, providing us full access to the quantum dynamics away from the large N limit, where the quantum fluctuations become important. The generalization of using the formalism to examine other observables and higher dimensional models will also be discussed.