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Delocalization effects, entanglement entropy and spectral collapse of boson mixtures in a double well

We investigate the ground-state properties of a two-species condensate of interacting bosons in a double-well potential. Each atomic species is described by a two-space-mode Bose-Hubbard model. The coupling of the two species is controlled by the interspecies interaction $W$. To analyze the ground state when $W$ is varied in both the repulsive ($W>0$) and the attractive ($W<0$) regime, we apply two different approaches. First we solve the problem numerically i) to obtain an exact description of the ground-state structure and ii ) to characterize its correlation properties by studying (the appropriate extensions to the present case of) the quantum Fisher information, the coherence visibility and the entanglement entropy as functions of $W$. Then we approach analytically the description of the low-energy scenario by means of the Bogoliubov scheme. In this framework the ground-state transition from delocalized to localized species (with space separation for $W>0$, and mixing for $W<0$) is well reproduced. These predictions are qualitatively corroborated by our numerical results. We show that such a transition features a spectral collapse reflecting the dramatic change of the dynamical algebra of the four-mode model Hamiltonian.