  Variational wave function ansatze are an invaluable tool to study the
properties of strongly correlated systems. We propose such a wave function,
based on the theory of auxiliary fields and combining aspects of
auxiliary-field quantum Monte Carlo and modern variational optimization
techniques including automatic differentiation. The resulting ansatz,
consisting of several slices of optimized projectors, is highly expressive and
systematically improvable. We benchmark this form on the two-dimensional
Hubbard model, using both cylindrical and large, fully periodic supercells. The
computed ground-state energies are competitive with the best variational
results. Moreover, the optimized wave functions predict the correct
ground-state order with near full symmetry restoration (i.e. translation
invariance) despite initial states with incorrect orders. The ansatz can become
a tool for local order prediction, leading to a new paradigm for variational
studies of bulk systems. It can also be viewed as an approach to produce
accurate and systematically improvable wave functions in a convenient form of
non-orthogonal Slater determinants (e.g., for quantum chemistry) at polynomial
computational cost.
