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Bosonic Analogue of Dirac Composite Fermi Liquid

We introduce a particle-hole-symmetric metallic state of bosons in a magnetic field at odd-integer filling. This state hosts composite fermions whose energy dispersion features a quadratic band touching and corresponding $2π$ Berry flux protected by particle-hole and discrete rotation symmetries. We also construct an alternative particle-hole symmetric state---distinct in the presence of inversion symmetry---without Berry flux. As in the Dirac composite Fermi liquid introduced by Son, breaking particle-hole symmetry recovers the familiar Chern-Simons theory. We discuss realizations of this phase both in 2D and on bosonic topological insulator surfaces, as well as signatures in experiments and simulations.