  The evolution of complex correlated quantum systems such as random circuit
networks is governed by the dynamical buildup of both entanglement and entropy.
We here introduce a real-time field theory approach -- essentially a fusion of
the $G \Sigma$-functional of the SYK-model and the field theory of disordered
systems -- enigneered to microscopically describe the full range of such
crossover dynamics: from initial product states to a maximum entropy ergodic
state. To showcase this approach in the simplest nontrivial setting, we
consider a tensor product of coupled random matrices, and compare to exact
diagonalization.
