I posted a paper on arXiv! Titled Automatic Order Detection and Restoration Through Systematically Improvable Variational Wave Functions, I alongside Miguel and Shiwei at Flatiron CCQ developed a new form of VAFQMC (originally introduced by Sorella in 2021 ). What is special about this, besides its great energetic performance and symmetry restoration through optimization, is the notable ability of the ansatz to predict the local order of the ground state without presupposing an order. Take a look!
Ryan Levy, Miguel Morales, and Shiwei Zhang
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.
 See Phys. Rev. B 107, 115133