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| kagome lattice | |
|---|---|
| Name | Kagome lattice |
| Type | Two-dimensional crystalline lattice |
| Symmetry | p6m |
| Prototype | None |
kagome lattice The kagome lattice is a two-dimensional periodic network formed by corner-sharing triangles and hexagons, notable for geometric frustration and flat electronic bands. It appears in theoretical models and real materials studied across condensed matter physics, materials science, and mathematics. Prominent research connects the lattice to phenomena investigated by groups at universities and laboratories, and to historic studies by individual researchers and institutions.
The kagome lattice is defined as a planar arrangement of vertices forming interconnected equilateral triangles and regular hexagons with a three-site basis per unit cell in a hexagonal Bravais lattice; typical treatments reference methods from Euclidean geometry, Crystallography and lattice constructions used by Augustin-Jean Fresnel in wave analyses. Its symmetry is described by the wallpaper group p6m, related to symmetry analyses by the International Union of Crystallography and classifications used by Felix Klein and the École Polytechnique. Geometric descriptions employ coordinates and basis vectors familiar from texts by William Rowan Hamilton, Sophus Lie, and in group-theoretic form by Évariste Galois.
The name derives from a Japanese woven bamboo pattern associated with craft traditions in regions such as Kyoto, and the etymology connects to historical artisans and guilds like those documented in records of the Tokugawa shogunate and cultural archives at institutions including the Tokyo National Museum. Scientific adoption of the term occurred in 20th-century literature influenced by researchers affiliated with universities such as University of Tokyo and by collaborations spanning Harvard University and Princeton University. Early mathematical and physical analyses drew on methods developed at the Royal Society and in papers appearing in journals associated with the American Physical Society and the Royal Institution.
Mathematically, the kagome lattice is studied via graph theory, spectral theory, and algebraic topology by researchers working in departments at Massachusetts Institute of Technology, University of Cambridge, and ETH Zurich. Its graph representation is a planar, non-bipartite lattice with coordination number four; studies reference theorems from Paul Erdős, William Tutte, and spectral results linked to the work of Israel Gohberg. Band-limited operators on the lattice connect to analyses by John von Neumann and matrix methods used in linear algebra courses at Columbia University. Eigenvalue degeneracies and flat bands are treated using techniques from the Courant Institute and operator theory influenced by Niels Henrik Abel.
Electronic structure calculations for kagome lattices employ tight-binding models and density functional methods used by groups at Los Alamos National Laboratory, Bell Labs, and Argonne National Laboratory. Tight-binding Hamiltonians produce dispersive Dirac cones and dispersionless flat bands analogous to features studied in graphene research at University of Manchester, while topological classifications reference frameworks developed by researchers at Princeton University and the Institute for Advanced Study. The interplay of spin–orbit coupling, described in contexts by Yoshio Nambu-inspired field theories, leads to Chern insulating phases related to work by Thouless, Kosterlitz, and Haldane.
Magnetic phenomena on the kagome lattice involve strong geometric frustration, studied intensively at experimental centers like Los Alamos National Laboratory and theoretical groups at California Institute of Technology. Models include Heisenberg, Ising, and Kitaev-type Hamiltonians; the latter build on concepts from Alexei Kitaev and spin-liquid paradigms discussed by Philip W. Anderson and P. W. Anderson's contemporaries. Quantum spin liquid candidates, valence-bond crystals, and magnetization plateaus on kagome networks have been reported in collaborations involving researchers from University of California, Berkeley and Stanford University, involving neutron-scattering analyses connected to methods developed at the Institut Laue–Langevin.
Realizations of kagome geometry occur in magnetic and metallic compounds such as minerals and oxides synthesized at facilities like Brookhaven National Laboratory and characterized in studies from Max Planck Society research groups. Notable material families include intermetallics and transition-metal compounds investigated at Argonne National Laboratory and by teams at Rice University and University of Tokyo. Experimental reports link kagome motifs to structures observed in compounds studied by researchers associated with the National Institute for Materials Science and characterizations published by the American Chemical Society.
Applications and techniques tied to kagome-lattice research span spintronics, photonics, and catalysis with contributions from industry labs like IBM Research and national facilities such as Lawrence Berkeley National Laboratory. Experimental probes include angle-resolved photoemission spectroscopy (ARPES) used by groups at Stanford Synchrotron Radiation Lightsource, inelastic neutron scattering at Oak Ridge National Laboratory, and scanning tunneling microscopy pioneered at IBM Research and universities like ETH Zurich. Theoretical and computational tools derive from software and frameworks developed at National Center for Supercomputing Applications and collaborative initiatives funded by agencies such as the National Science Foundation.
Category:Crystallography Category:Frustrated magnetism Category:Condensed matter physics