Graphene nanoribbons (GNRs)—narrow stripes of graphene—have emerged as promising building blocks for nanoelectronic devices. Recent advances in bottom-up synthesis have allowed production of atomically well-defined armchair GNRs with different widths and doping. While all experimentally studied GNRs have exhibited wide bandgaps, theory predicts that every third armchair GNR (widths of N=3m+2, where m is an integer) should be nearly metallic with a very small bandgap. Here, we synthesize the narrowest possible GNR belonging to this family (five carbon atoms wide, N=5). We study the evolution of the electronic bandgap and orbital structure of GNR segments as a function of their length using low-temperature scanning tunnelling microscopy and density-functional theory calculations. Already GNRs with lengths of 5 nm reach almost metallic behaviour with ∼100 meV bandgap. Finally, we show that defects (kinks) in the GNRs do not strongly modify their electronic structure.

Topological materials exhibit protected edge modes that have been proposed for applications in, for example, spintronics and quantum computation. Although a number of such systems exist [1][2][3][4][5][6] , it would be desirable to be able to test theoretical proposals in an artificial system that allows precise control over the key parameters of the model 7 . The essential physics of several topological systems can be captured by tight-binding models, which can also be implemented in artificial lattices 6,7 . Here, we show that this method can be realized in a vacancy lattice in a chlorine monolayer on a Cu(100) surface. We use low-temperature scanning tunnelling microscopy (STM) to fabricate such lattices with atomic precision and probe the resulting local density of states (LDOS) with scanning tunnelling spectroscopy (STS). We create analogues of two tight-binding models of fundamental importance: the polyacetylene (dimer) chain with topological domain-wall states, and the Lieb lattice with a flat electron band. These results provide an important step forward in the ongoing e ort to realize designer quantum materials with tailored properties.The topological theory of matter has its roots in the 70s and 80s studies of polyacetylene 8 , the quantum Hall effect 9 and superfluid 3 He (ref. 10). The interest in the field exploded after the 2006 discovery of topological insulators 2,3 . Crystalline solids exhibit a spectrum of energy bands constrained by the material's symmetries. Topological properties of the band structure give rise to protected boundary states that are the hallmark of topological materials. Well-known examples include chiral and helical states in quantum (spin) Hall systems and Majorana modes in topological superconductors 1,9,[11][12][13][14] . A central goal of research has become to identify systems with a topological spectrum.Theories of topological materials 15 are often formulated using tight-binding models describing hopping between localized electronic orbitals. This means that, given sufficient control, one can implement these models by assembling the corresponding structure from individual constituents that are suitably coupled. It is hence possible to build 'designer quantum materials' based on specific Hamiltonians through atomic assemblies. Although there has been considerable success in this direction using ultracold atomic gases, STM can be used to fabricate and characterize structures at the atomic scale in the solid state 7,[16][17][18][19] . Recently, it was shown that vacancy defects in the c(2 × 2) chlorine superstructure on Cu (100) make an excellent system for large-scale atomic assembly 20 . We show that individual Cl vacancies host a well-defined vacancy state below the band edge of the chlorine layer which interact when sufficiently close to each other. We demonstrate that it is possible to construct coupled lattices and implement two Hamiltonians of general interest, the Su-Schrieffer-Heeger (SSH) dimer chain and the two-dimensional Lieb lattice.An overview image of the...

We present a theory of the superfluid weight in multiband attractive Hubbard models within the BardeenCooper-Schrieffer (BCS) mean-field framework. We show how to separate the geometric contribution to the superfluid weight from the conventional one, and that the geometric contribution is associated with the interband matrix elements of the current operator. Our theory can be applied to systems with or without time-reversal symmetry. In both cases the geometric superfluid weight can be related to the quantum metric of the corresponding noninteracting systems. This leads to a lower bound on the superfluid weight given by the absolute value of the Berry curvature. We apply our theory to the attractive Kane-Mele-Hubbard and Haldane-Hubbard models, which can be realized in ultracold atom gases. Quantitative comparisons are made to state of the art dynamical mean-field theory and exact diagonalization results.

Graphene nanostructures, where quantum confinement opens an energy gap in the band structure, hold promise for future electronic devices. To realize the full potential of these materials, atomic-scale control over the contacts to graphene and the graphene nanostructure forming the active part of the device is required. The contacts should have a high transmission and yet not modify the electronic properties of the active region significantly to maintain the potentially exciting physics offered by the nanoscale honeycomb lattice. Here we show how contacting an atomically well-defined graphene nanoribbon to a metallic lead by a chemical bond via only one atom significantly influences the charge transport through the graphene nanoribbon but does not affect its electronic structure. Specifically, we find that creating well-defined contacts can suppress inelastic transport channels.