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Kagome: The Story of the Basketweave Lattice

Bamboo basket woven in kagome pattern

The Japanese word kagome has become popular with the magnetism community in discussing the lattice structure of geometrical spin frustration. One might guess that this word is the name of some Japanese scientist. But kagome means a bamboo-basket (kago) woven pattern (me) that is composed of interlaced triangles whose lattice points each have four neighboring points. Now that bamboo baskets have been largely replaced by plastic containers, the word kagome may seem a bit old-fashioned. Who introduced it to denote a special lattice structure? Those who know the answer to that question may belong to the first generation of researchers in frustration study.

The 1944 paper by Lars Onsager,¹ who solved the square-lattice Ising model exactly, had a great impact on the study of phase transitions; Onsager's work motivated researchers to extend the study to other lattices--triangular and honeycomb, for example--and to antiferromagnetism. G. H. Wannier's famous paper on the antiferromagnetic triangular lattice is one of the results.² Kodi Husimi of Osaka University, and young staff member Itiro Syôzi, also started to explore phase transitions in various lattices. Using a dual transformation, they simplified Onsager's abstract algebraic method and obtained exact solutions for the honeycomb and triangular lattices.³ As soon as Husimi and Syôzi published their results in 1950, there was a rush of papers about phase transitions on those lattices.

Itiro Syôzi

Syôzi studied a decorated honeycomb lattice with an extra spin at the middle point of each bond to obtain the exact solution for an antiferromagnet. He found that the decorated honeycomb lattice turns into a new lattice by star-to-triangle transformation. Husimi's appreciation of art--he painted pictures as a hobby--led him to name this new lattice kagome. The first paper on the subject, with Syôzi as sole author,⁴ appeared in Progress of Theoretical Physics in 1951. In that paper, Syôzi gave a logarithmic temperature dependence for the specific heat and showed the transition temperature for the ferromagnetic kagome lattice. He also demonstrated that a magnetic transition does not occur in the antiferromagnetic kagome lattice. Because the journal in which he published that paper was a fledgling one with low circulation, his work has only gradually been discovered.

Two years after the first kagome paper was published, Kenji Kano and Shigeo Naya of the Husimi group calculated the residual entropy of the Ising spin kagome lattice by using a method different from Syôzi's to solve the eigenvalue problem. In 1972, Syôzi reviewed Ising models on various lattices.⁵ Subsequent theoretical studies of the kagome lattice in the 1980s covered effects of magnetic field, randomness, second neighbor interaction, spin freedom, and combination of interactions.

Experimentally, the mineral jarosite, with Heisenberg spins on stacked kagome lattices, was first discussed as a model compound 17 years after the first theoretical work.⁶ An Ising model run on the kagome lattice has been applied to two-dimensional hydrogen bonding in CsOH�H₂O and the second layer of adsorbed helium-3 on graphite.⁷ A group at Bell Laboratories revealed that SrCr_9-xGa_3+xO₁₉ has unusual magnetic properties that have been associated with those characteristic of the kagome lattice.⁸ Those properties have attracted much interest among both theorists and experimentalists.

On 18 October 2001, Syôzi passed away at the age of 81. The author of the first kagome paper is gone but the word survives among us.