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The possibility of "orbitally selective Mott transitions" within a multibandHubbard model, in which one orbital with large on-site electron-electronrepulsion $U_1$ is insulating and another orbital, to which it is hybridized,with small $U_{-1}$, is metallic, is a problem of long-standing debate andinvestigation. In this paper we study an analogous phenomenon, the co-existenceof metallic and insulating bands in a system of orbitals with differentelectron-phonon coupling (EPC). To this end, we examine two variants of thebilayer Holstein model: a uniform bilayer and a "Holstein-Metal interface"where the electron-phonon coupling, $\lambda$, is zero in the "metallic" layer.In the uniform bilayer Holstein model, charge density wave (CDW) orderdominates at small interlayer hybridization $t_3$, but decreases and eventuallyvanishes as $t_3$ grows, providing a charge analog of singlet (spin liquid)physics. In the interface case, we show that CDW order penetrates into themetal layer and forms long-range CDW order at intermediate ratio of inter- tointra-layer hopping strengths, $1.4 \lesssim t_3/t \lesssim 3.4$. This isconsistent with the occurrence of an "orbitally selective CDW" regime at weak$t_3$ in which the layer with $\lambda_{1} \neq 0$ exhibits long-range chargeorder, but the "metallic layer" with $\lambda_{-1}=0$, to which it ishybridized, does not.

Charge Singlets and Orbital-Selective Charge Density Wave Transitions