*Citation:* Siddharthan, Rahul, and Antoine Georges. "Square kagome quantum antiferromagnet and the eight-vertex model." Physical Review B 65.1 (2001): 014417.

*Web:* https://journals.aps.org/prb/abstract/10.1103/PhysRevB.65.014417

*Tags:* Physical, Hardware

This paper introduces a new lattice called the *square kagome lattice*. This lattice
has many nice properties, which makes it interesting to study. It is good to be aware of.

In particular, the authors study a quantum antiferromagnet on this lattice. Mathematically, this amounts to choosing a dimer covering. The square kagome lattice can be broken up into chunks consisting of a square surrounded by four equilateral triangles. Dimer coverings of these chunks which cover the center square amounts to choosing an even number of the outer vertices to cover, modulo a Z2 degeneracy. The chunks themselves are fit together in a square lattice.

In the end, modulo the Z2 degeneracy, this model ends up being exactly the same as choosing an orientation on the edges of a square lattice such that each vertex has an even number of edges going into it. This is a FANTASTIC model for how parity, Z2 symmetry, and orientation can naturally appear from dimer models. This model of a square lattice with orientations on edges such that each vertex has an even number of edges going into it is known as the eight-vertex model, and has been studied in detail by Baxter:

> Baxter, Rodney J. Exactly solved models in statistical mechanics. Elsevier, 2016.

Incidentally, this book by Baxter seems like a fantastic place to learn about statistical mechanics for a mathematician interested in topological quantum.

This original study of the square kagome lattice as an interesting model with many nice properties. However, like many objects in this area, it was found to describe a real crystal which really does manifest quantum spin liquid properties, such as the experimental compounds (K Cu_6 Al Bi O4 (SO_4)_5 Cl) and (Na_6 Cu_7 Bi O_4 (PO_4)_4 [Cl,(OH)]3):

> Fujihala, Masayoshi, et al. "Gapless spin liquid in a square-kagome lattice antiferromagnet." Nature communications 11.1 (2020): 3429.

> Yakubovich, Olga V., et al. "Hydrothermal synthesis and a composite crystal structure of Na6Cu7BiO4 (PO4) 4 [Cl,(OH)] 3 as a candidate for quantum spin liquid." Inorganic Chemistry 60.15 (2021): 11450-11457.

The square kagome lattice is now a hot topic amount experimentalists:

> Richter, Johannes, and Jürgen Schnack. "Magnetism of the $ s= 1/2$$ J_1 $-$ J_2 $ square-kagome lattice antiferromagnet." arXiv preprint arXiv:2212.10838 (2022).

> Richter, Johannes, Oleg Derzhko, and Jürgen Schnack. "Thermodynamics of the spin-half square kagome lattice antiferromagnet." Physical Review B 105.14 (2022): 144427.

> Gembé, Martin, et al. "Non-Coplanar Magnetic Orders in Classical Square-Kagome Antiferromagnets." arXiv preprint arXiv:2302.04171 (2023).