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Thermodynamics of a two-dimensional dipolar Bose gas with correlated disorder in the roton regime

We sudy the impact of a weak random potential with a Gaussian correlation function on the thermodynamics of a two-dimensional (2D) dipolar bosonic gas. Analytical expressions for the quantum depletion, anomalous density, the ground state energy, the equation of state and the sound velocity are derived in the roton regime within the framework of the Bogoliubov theory. Surprisingly, we find that the condensate depletion and the anomalous density are comparable. The structure factor and the superfluid fraction are also obtained analytically and numerically. We show that these quantities acquire dramatically modified profiles when the roton is close to zero yielding the transition to an unusual quantum state.