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A bosonic model on a 3 dimensional fcc lattice with emergent low energy excitations, with the same polarization and gauge symmetries as gravitons is constructed. The novel phase obtained is a stable gapless boson liquid phase, with algebraic boson density correlations. The stability of this phase is protected against the instanton effect and superfluidity by self-duality and large gauge symmetries. The gapless collective excitation of this phase closely resembles gravitons, although they have a soft ω k² dispersion relation. The dynamics of this novel phase is described by new set of Maxwell equations. This phase also possesses an intricate topological order, requiring 18 winding numbers to specify each topological sector.
Cenke Xu (Sun,) studied this question.