  The Mott-insulating phase of the two-dimensional (2d) Bose-Hubbard model is
expected to be characterized by a non-local brane parity order. Parity order
captures the presence of microscopic particle-hole fluctuations and
entanglement, whose properties depend on the underlying lattice geometry. We
realize 2d Bose-Hubbard models in dynamically tunable lattice geometries, using
neutral atoms in a novel passively phase-stable tunable optical lattice in
combination with programmable site-blocking potentials. We benchmark the
performance of our system by single-particle quantum walks in the square,
triangular, kagome and Lieb lattice. In the strongly correlated regime, we
microscopically characterize the geometry dependence of the quantum
fluctuations and experimentally validate the brane parity as a proxy for the
non-local order parameter signaling the superfluid-to-Mott insulating phase
transition.
