  Our current understanding of quantum chaos hinges on the random matrix
behavior (RMT) of typical states in quantum many-body systems, particularly
eigenstates and their energy level statistics. Although RMT has been remarkably
successful in describing `coarse' features of quantum states in chaotic
regimes, it fails to capture their `finer' features, particularly those arising
from spatial locality and symmetries. Here, we show that we can accurately
describe the behavior of eigenstate ensembles in physical systems by using RMT
ensembles with constraints that capture the key features of the physical
system. We demonstrate our approach on local spin Hamiltonians with a scalar
U(1) charge. By constructing constrained RMT ensembles that account for two
local scalar charges playing the role of energy and magnetization, we describe
the patterns of entanglement of mid-spectrum eigenstates at all lengthscales
and beyond their average behavior, analytically and numerically. When defining
the correspondence between quantum chaos and RMT, our work clarifies that RMT
ensembles must be constrained to account for all the features of the underlying
Hamiltonian, particularly spatial locality and symmetries.
