Bose-Einstein condensation in exotic lattice geometries

Modern quantum engineering techniques allow for synthesizing quantum systems in exotic lattice geometries, from self-similar fractal networks to negatively curved hyperbolic graphs. We demonstrate that these structures profoundly reshape Bose-Einstein condensation. Fractal lattices dramatically lower the condensation temperature, while hyperbolic lattices cause it to increase as the system grows - a behavior not seen in ordinary two-dimensional arrays, where the condensation temperature vanishes in the large-size limit. The underlying geometry also controls condensate fluctuations, enhancing them on fractal networks but suppressing them on hyperbolic graphs compared with regular one-dimensional or two-dimensional lattices. When strong repulsive interactions are included, the gas enters a Mott insulating state. A multi-site Gutzwiller approach finds a smooth interpolation between the characteristic insulating lobes of one-dimensional and two-dimensional systems. Re-entrant Mott transitions are seen within a first-order resummed hopping expansion. Our findings establish lattice geometry as a powerful tuning knob for quantum phase phenomena and pave the way for experimental exploration in photonic waveguide arrays and Rydberg-atom tweezer arrays.