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A kagome lattice naturally features Dirac fermions, flat bands and van Hove singularities in its electronic structure. The Dirac fermions encode topology, flat bands favour correlated phenomena such as magnetism, and van Hove singularities can lead to instabilities towards long-range many-body orders, altogether allowing for the realization and discovery of a series of topological kagome magnets and superconductors with exotic properties. Recent progress in exploring kagome materials has revealed rich emergent phenomena resulting from the quantum interactions between geometry, topology, spin and correlation. Here we review these key developments in this field, starting from the fundamental concepts of a kagome lattice, to the realizations of Chern and Weyl topological magnetism, to various flat-band many-body correlations, and then to the puzzles of unconventional charge-density waves and superconductivity. We highlight the connection between theoretical ideas and experimental observations, and the bond between quantum interactions within kagome magnets and kagome superconductors, as well as their relation to the concepts in topological insulators, topological superconductors, Weyl semimetals and high-temperature superconductors. These developments broadly bridge topological quantum physics and correlated many-body physics in a wide range of bulk materials and substantially advance the frontier oftopological quantum matter.
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A kagome lattice (Fig. la), made of corner-sharing triangles, is a geometrically frustrated two-dimensional (2D) lattice introduced to quantum physics1 in 1951, as an analogy to a type of bamboo basket used in eastern countries. Interestingly, the kagome pattern has no direct analogy to wild nature, but a similar pattern has long been used as the star of David in religious ceremonies and a hexagram in alchemy symbols. In quantum physics research, inspired by Onsager's exact solution of the Ising model for the square lattice2 in the 1940s, researchers extended the study of magnetic phase transitions to triangular, honeycomb and, eventually, kagome lattices. In the 1951 kagome work1, Syôzi demonstrated that, in contrast to the ferromagnetic case, a phase transition does not occur in a kagome lattice with a nearest-neighbour antiferromagnetic Ising interaction. Nowadays, geometrical spin frustration in a kagome lattice under an antiferromagnetic exchange interaction is widely appreciated, and shows a great potential to realize quantum-spin-liquid states3 that feature long-range quantum entanglement, fractionalized excitations and the absence of ordinary magnetic order4. Although these studies are about...