Fractional corner charges in a 2D super-lattice Bose-Hubbard model

We study a two dimensional super-lattice Bose-Hubbard model with alternating hoppings in the limit of strong on-site interactions. We evaluate the phase diagram of the model around half-filling using the density matrix renormalization group method and find two gapped phases separated by a gapless superfluid region. We demonstrate that the gapped states realize two distinct higher order symmetry protected topological phases, which are protected by a combination of charge conservation and $C_4$ lattice symmetry. The phases are distinguished in terms of a quantized fractional corner charge and a many-body topological invariant that is robust against arbitrary, symmetry preserving edge manipulations. We support our claims by numerically studying the full counting statistics of the corner charge, finding a sharp distribution peaked around the quantized values. These results are experimentally observable in ultracold atomic settings using state of the art quantum gas microscopy.