Experimental Realization of Two-Dimensional Buckled Lieb Lattice.

Two-dimensional (2D) materials with a Lieb lattice host exotic electronic band structures. Such a system does not exist in nature, and it is also difficult to obtain in the laboratory due to its structural instability. Here, we experimentally realized a 2D system composed of a tin overlayer on an aluminum substrate by molecular beam epitaxy. The specific arrangement of Sn atoms on the Al(100) surface, which benefits from favorable interface interactions, forms a stabilized buckled Lieb lattice. Theoretical calculations indicate a partially broken nodal line loop and a topologically nontrivial insulating state with a spin-orbital coupling effect in the band structure of this Lieb lattice. The electronic structure of this system is experimentally characterized by angle-resolved photoemission spectroscopy, in which the hybridized states between topmost Al atoms and Sn atoms are revealed. Our work provides an appealing method for constructing 2D quantum materials based on the Lieb lattice.

Topological Band Engineering of Lieb Lattice in Phthalocyanine-Based Metal-Organic Frameworks.

Topological properties of the Lieb lattice, i.e., the edge-centered square lattice, have been extensively studied and are, however, mostly based on theoretical models without identifying real material systems. Here, based on tight-binding and first-principles calculations, we demonstrate the Lieb-lattice features of the experimentally synthesized phthalocyanine-based metal-organic framework (MPc-MOF), which holds various intriguing topological phase transitions through band engineering. First, we show that the MPc-MOFs indeed have a peculiar Lieb band structure with 1/3 filling, which has been overlooked because of its unconventional band structure deviating from the ideal Lieb band. The intrinsic MPc-MOF presents a trivial insulating state, with its gap size determined by the on-site energy difference (ΔE) between the corner and edge-center sites. Through either chemical substitution or physical strain engineering, one can tune ΔE to close the gap and achieve a topological phase transition. Specifically, upon closing the gap, topological semimetallic/insulating states emerge from nonmagnetic MPc-MOFs, while magnetic semimetal/Chern insulator states arise from magnetic MPc-MOFs, respectively. Our discovery greatly enriches our understanding of the Lieb lattice and provides a guideline for experimental observation of the Lieb-lattice-based topological states.

Effect of a flat band on a multiband two-dimensional Lieb lattice with intra- and interband interactions.

In this study, we explore the effect of a single flat band in the electronic properties of a ferromagnetic two-dimensional Lieb lattice using the multiband Hubbard model with polarized carriers, spin-up and spin-down. We employ the self-consistent dynamical mean field theory and a Green functions cumulant expansion around the atomic limit to obtain the correlated densities of states while varying the intra- and interband interactions. Our findings demonstrate a renormalization of the correlated density of states in both the spin-up and spin-down carriers as we varied the intra- and interband interactions. We conclude that the presence of a flat band enables the system to maintain a metal state with itinerant ferromagnetism in the spin-up carrier.

Realization of a 2D Lieb Lattice in a Metal-Inorganic Framework with Partial Flat Bands and Topological Edge States.

Flat bands and Dirac cones in materials are the source of the exotic electronic and topological properties. The Lieb lattice is expected to host these electronic structures, arising from quantum destructive interference. Nevertheless, the experimental realization of a 2D Lieb lattice remained challenging to date due to its intrinsic structural instability. After computationally designing a Platinum-Phosphorus (Pt-P) Lieb lattice, it has successfully overcome its structural instability and synthesized on a gold substrate via molecular beam epitaxy. Low-temperature scanning tunneling microscopy and spectroscopy verify the Lieb lattice's morphology and electronic flat bands. Furthermore, topological Dirac edge states stemming from pronounced spin-orbit coupling induced by heavy Pt atoms are predicted. These findings convincingly open perspectives for creating metal-inorganic framework-based atomic lattices, offering prospects for strongly correlated phases interplayed with topology.

Topological phases of monolayer and bilayer depleted Lieb lattices.

Existence of nontrivial topological phases in a tight binding Haldane-like model on the depleted Lieb lattice is reported. This two-band model is formulated by considering the nearest-neighbor (NN), next-NN (NNN) and next-NNN hopping terms along with complex phase which breaks the time reversal symmetry of this semi-metallic system. Topological feature of this model is studied along with the presence of sublattice symmetry breaking staggered onsite energy. Combined effect of these two broken symmetries is found crucial for an additional transition between nontrivial and trivial phases. System exhibits two types of phase transitions, say, between two nontrivial phases and nontrivial to trivial phases. Nonzero Chern numbers, existence of Hall plateau and symmetry protected edge states confirm the presence of the nontrivial phases. This two-band system hosts four different types of phases where two are topological. Additionally topological properties of stacked bilayer of the depleted Lieb lattices is also studied with similar Haldane-like Hamiltonian. This four-band system is found to host Chern insulating phases, with higher values of Chern numbers supported by in-gap edge states.