Layer-Resolved Topological Metals in the Bilayer Lieb Lattice

We identify a two-dimensional time-reversal-invariant topological metallic phase on a bilayer Lieb lattice, characterized by a quantized layer--resolved pseudo-spin Chern number. Without the orbital-angular-momentum-dependent (OAM-dependent) coupling, the system gives rise to a time-reversal-invariant topological semimetal with a zero indirect gap and quantized pseudo-spin Chern number. Opposite-sign intralayer OAM-dependent coupling immediately converts the zero-indirect-gap semimetal into a metal, in which the global spectrum is metallic while the layer--resolved pseudo-spin Chern number remains well defined as long as the direct gap at each crystal momentum and the pseudo-spin gap remain open. The model also exhibits asymmetric boundary states: in the semimetallic regime, one edge hosts perfectly flat bands, whereas the opposite edge supports gapless counter-propagating modes forming a one-dimensional Dirac cone. An edge-localized interlayer coupling gaps only the counter-propagating edge states, leaving the flat-band edge essentially intact, while intralayer OAM-dependent coupling bends the exact flat band into a dispersive boundary mode without affecting the gapped Dirac edge. These results open a route toward the controlled engineering of layer--resolved topological gapless phases in synthetic and quantum materials.