What are examples of materials that closely correspond to the Heisenberg model?

Question

I use the antiferromagnetic Heisenberg model all the time:

$ H = J \sum \limits_{\langle i,j \rangle} \vec S_i \cdot \vec S_j$

What are some examples of materials that are well-described by this model in 3D? What about in 1D and 2D?

Answer 1

1D

A famous example of a nearly ideal spin-$1/2$ isotropic Heisenberg antiferromagnetic chain (1D) system is copper pyrazine dinitrate [Cu(C$_4$H$_4$N$_2$)(NO$_3$)$_2$], which was discussed in Hammar et al. Phys. Rev. B 59, 1008 (1999) [arXiv link]. Another excellent realizations include KCuF$_3$, which has stronger (but still low) interchain coupling, and orders at low temperatures. However, the spectrum of magnetic excitations above $\sim J/10$ matches DMRG and Bethe Ansatz calculations very closely. See e.g. Lake et al. Phys. Rev. Lett. 111, 137205 (2013) [arXiv link]. A third example is CuSO$_4\cdot 5$D$_2$O, see Mourigal et al. Nature Physics 9, 435 (2013) [arXiv link].

For $S=1$ the materials I'm aware of seem to have some degree of single-ion anisotropy. The most well-known one is probably NENP [Ni(C$2$H$_8$N$_2$)$_2$NO$_2$(ClO$_4)], as studied in e.g. Avenel et al. Phys. Rev. B 46, 8655 (1992). Earlier this year a molecular coordination complex was introduced and claimed to be one of the most ideal realizations yet, see Williams et al. Phys. Rev. Research 2, 013082 (2020).

There are some higher-spin realizations too, but I'm not sure which are good examples, and which aren't.

2D

For higher dimensions the lattice geometry really needs to be specified. I will here assume you are interested in simple lattices, and not some geometrically frustrated one (though there is a fascinating literature on e.g. triangular, kagome lattices in the pursuit of quantum spin liquids). Spin-$1/2$ examples on the square lattice include

• La$_2$CuO$_4$, see e.g. the review Manousakis Rev. Mod. Phys. 63, 1 (1991)
• Sr$_2$CuO$_2$Cl$_2$, see Greven et al. Phys. Rev. Lett. 72, 1096 (1994)
• certain organic compounds, see Woodward et al. Phys. Rev. B 65, 144412 (2002) [arXiv link].

An $S=5/2$ example is found in Rb$_2$MnF$_4$, see Huberman et al. Phys. Rev. B 72, 014413 (2005) [arXiv link].

3D

I don't know too much about the 3D systems, but the two best realizations of nearest-neighbor only Heisenberg models I'm aware of are

• The $S=5/2$ simple cubic lattice compound RbMnF$_3$, see e.g. Coldea et al. Phys. Rev. B 57, 5281 (1998)
• KMnF$_3$ see eg. Salazar et al. Phys. Rev. B 75, 224428 (2007) [arXiv link]. There are also the related KCoF$_3$ and KNiF$_3$ discussed in e.g. Oleaga et al. J. Alloys and Compounds 629, 178 (2015) [non-paywall link].

Answer 2

Actual examples of 2D magnetic systems are MXenes and metal-organic adsorption monolayers.

Answer 3

The Heisenberg formalism is often used to describe the interaction between molecules adsorbed on a surface (2D) using a cluster expansion. This has nothing to do with magnetism, but the mathematical framework is suitable for this kind of problem. Please take a look at Nielsen et al. J. Chem. Phys. 139 (2013) 224706. The application of the Heisenberg formalism is shown in detail in the supplementary material.