Mass generation in graphene and kagome metals
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Kagome materials received profound attention in recent years, becoming some of the most prominent structures in condensed matter theory. They have been proposed as hosts for a large variety of quantum phases, such as spin liquids, unconventional superconductivity and topological phases of matter. At specific lattice filling the electronic bands have a semimetallic structure with a vanishing density of states, hosting Dirac, massless quasiparticles. This paradigm is well known in other materials, among which graphene stands first. When specific effects are taken into account, however, a gap is opened at the Dirac points and the quasiparticles become massive: a semimetal- insulator phase transition occurs. Such perturbations in graphene have been widely studied from different perspectives. Instead, the plethora of gap-opening terms in kagome lacks an established classification, and, in most cases, a clear physical interpretation. Recent work however indicates candidate kagome materials as strongly interacting compared to graphene. This suggests that mass terms are more likely to spontaneously originate, motivating interest in this research. In this work we map the sixteen gap-opening terms of kagome materials into physical effects, leveraging information on the relative broken and preserved symmetries. The general features of kagome instabilities are then discussed with reference to graphene ones, picturing a complex and multifaced relation. The anomalous Hall effect and the spin-Hall effect, originally proposed for the honeycomb lattice, find a prominent host in kagome structures: contrarily to graphene, they are already relevant for nearest-neighbour hoppings, which generally represent the leading contribution. Remarkably, lattice deformation instabilities reflect one property of the model, namely that kagome lattice at 1/3 filling (Dirac points) can be mapped into a dimer model on the hexagonal lattice. Consistently, gap-opening distortions of lattice sites in one structure find a related bond-distortion in the other structure, and viceversa. The staggered, on-site potential which makes graphene an insulator has a dimerization pattern of alternating bonds as the corresponding mass in kagome, and similar mappings are found in more sophisticated distortions as Kekul ́e patterns. Lastly, honeycomb materials are known to have antiferromagnetic instability which leads to a trivial, insulating phase; instead, kagome materials suffer from magnetic frustration. At 1/3 filling however, orderings are possible thanks to the empty sites. Antiferromagnetic phases at the Dirac points are therefore gapped in both materials, with one difference: graphene AFM has a simple magnetic cell, while kagome AFM can be realised with enlarged magnetic cell. In conclusion, the established ground of graphene works as one additional benchmark, giving indications about the relevant instabilities of the two structures.