Layer-Resolved Topological Metals in the Bilayer Lieb Lattice

This paper investigates the existence and properties of topological metallic phases in a bilayer Lieb lattice. Specifically, the authors explore whether a system that is globally metallic can still host robust topological invariants and asymmetric boundary states, and how these features evolve under the influence of orbital-angular-momentum-dependent (OAM-dependent) coupling.

The researchers construct a two-layer parent Hamiltonian based on the Lieb lattice, where each layer is a Chern semimetal with opposite Chern numbers. They introduce an OAM-dependent coupling term that preserves time-reversal symmetry. By analyzing the bulk spectrum, they define a layer-resolved pseudo-spin Chern number. They further investigate the boundary physics by applying open boundary conditions and introducing an edge-localized interlayer coupling to manipulate the boundary states.

The study demonstrates that the system transitions from a topological semimetal to a topological metal as the OAM-dependent coupling is increased. Despite the global metallic nature, the occupied subspace remains locally separated in momentum space, allowing the pseudo-spin Chern number to remain quantized. The boundary states are notably asymmetric: one edge hosts perfectly flat bands, while the opposite edge supports gapless counter-propagating modes forming a Dirac cone. The authors show that these boundary modes can be selectively gapped or deformed using edge-specific perturbations without destroying the bulk topological invariant.

This work provides a framework for engineering topological gapless phases in synthetic and quantum materials. By demonstrating that topological protection can persist in metallic systems through direct-gap-protected markers, the findings offer new strategies for controlling electronic transport and boundary-localized phenomena, such as flat-band physics and Dirac-like edge states, which are of significant interest for future quantum devices.