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?

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    $\begingroup$ I love this question! $\endgroup$ – Nike Dattani May 12 '20 at 15:44
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    $\begingroup$ It's unfortunately not something that I have (yet) the knowledge to answer. I do want the most to be achieved from this bounty though, and with 23 hours left I have started to look up authors who might have written papers on this Hamiltonian. I found this book chapter: link.springer.com/chapter/10.1007/978-3-540-85416-6_7 and will try to contact the authors tomorrow (it's almost 1am here and I'm heading to bed soon!). If others are able to help do the same it would be appreciated, as I truly think this question needs more attention from experts in the field. $\endgroup$ – Nike Dattani May 20 '20 at 4:55


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.


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

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


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

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    $\begingroup$ +100 Nicely done, with 3 hours left on the bounty! Excellent first post on this site, and welcome! We hope to see more of you !!! $\endgroup$ – Nike Dattani May 21 '20 at 0:42
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    $\begingroup$ @NikeDattani Thank you! I stumbled on the site yesterday, and found this question. Lucky timing I guess :) $\endgroup$ – Anyon May 21 '20 at 1:03
  • $\begingroup$ Very lucky timing!!! $\endgroup$ – Nike Dattani May 21 '20 at 2:56
  • $\begingroup$ I wonder if you have any insight you could provide for this (related) unanswered question: mattermodeling.stackexchange.com/q/659/5 It has been bothering us for a long time now! $\endgroup$ – Nike Dattani Jul 6 '20 at 20:24
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    $\begingroup$ @NikeDattani Hmm. The fact that the question focuses on ferromagnetism makes it a tricky question to answer properly (both generally and for me personally), but maybe I could say something. I will have to think about how to phrase things though. $\endgroup$ – Anyon Jul 6 '20 at 22:41

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

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    $\begingroup$ This could be the start of a good answer, but could you give a brief summary of the linked papers and how the heisenberg model performs in them? $\endgroup$ – Tyberius May 18 '20 at 20:05
  • $\begingroup$ Thank you for adding these details. I wonder if you have any insight you could provide for this unanswered question: mattermodeling.stackexchange.com/q/659/5 It has been bothering us for a long time now! $\endgroup$ – Nike Dattani Jul 6 '20 at 20:23

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.

  • $\begingroup$ I'm especially interested in 2D realizations. I look at the that reference and I wasn't able to see how what they're doing maps onto a Heisenberg model in the form I described in my question. Are they really arriving at a Heisenberg-like $\vec S_i \cdot \vec S_j$ interaction? $\endgroup$ – taciteloquence May 18 '20 at 4:02
  • $\begingroup$ They don't use spins, but instead an occupation number (denoted σ) in the manuscript, which is an integer indicating the species of molecule. For pair isterations there would be a $\sigma_i \cdot \sigma_j$ term. For three-body interactions there is a $\sigma_i \cdot \sigma_j \cdot \sigma_k$ term.I think that's the main difference to what you described, but the shape of the expression remains similar. $\endgroup$ – Franziska Heß May 18 '20 at 7:19

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