Charge singlets and orbital-selective charge density wave transitions

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The possibility of ``orbitally selective Mott transitions'' within a multiband Hubbard model, in which one orbital with large on-site electron-electron repulsion ${U}_{1}$ is insulating and another orbital, to which it is hybridized, with small ${U}_{\ensuremath{-}1}$, is metallic, is a problem of long-standing debate and investigation. In this paper we study an analogous phenomenon, the coexistence of metallic and insulating bands in a system of orbitals with different electron-phonon coupling. To this end, we examine two variants of the bilayer Holstein model: a uniform bilayer and a ``Holstein-metal interface'' where the electron-phonon coupling, $\ensuremath{\lambda}$, is zero in the ``metallic'' layer. In the uniform bilayer Holstein model, charge density wave (CDW) order dominates at small interlayer hybridization ${t}_{3}$, but decreases and eventually vanishes as ${t}_{3}$ grows, providing a charge analog of singlet (spin liquid) physics. In the interface case, we show that CDW order penetrates into the metal layer and forms long-range CDW order at an intermediate ratio of inter- to intralayer hopping strengths, $1.4\ensuremath{\lesssim}{t}_{3}/t\ensuremath{\lesssim}3.4$. This is consistent with the occurrence of an ``orbitally selective CDW'' regime at weak ${t}_{3}$ in which the layer with ${\ensuremath{\lambda}}_{1}\ensuremath{\ne}0$ exhibits long-range charge order, but the ``metallic layer'' with ${\ensuremath{\lambda}}_{\ensuremath{-}1}=0$, to which it is hybridized, does not.