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Does anyone know if magnetism and topological insulating behavior coexist in a material? If yes, can someone refer to a recent work?

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Not a very thorough answer, but it should get the ball rolling.

Spontaneous magnetization or extrinsically magnetic TIs have been achieved through defect engineering in non-magnetic Topological Insulators. This is typically done via doping of 3d magentic atoms (e.g. $\ce{Fe}$, $\ce{Mn}$). A recent review (2019) published in Nature Reviews Physics on the concepts and experimental progress on Magnetic Topological Insulators can be found here.

Also last year, $\ce{MnBi_2Te_4}$ was predicted and discovered. It is a van der Waals layered material, with intralayer ferromagnetic coupling. However, the interlayer coupling is anti-ferromagntic. The nature article can be found here. This is big, so here is a physics news article about the discovery.

Question: Given that these materials are layered, has anyone studied if the monolayer transitions to Ferromagnetic? That would be an intersesting study!

Also, a theoretical study focused on extending the (then predicted) properties of $\ce{MnBi_2Te_4}$ to a general class of compounds $\ce{MB_2X_4}$. Their methodology was based on primarily on DFT, so it will definitely be an interesting read (in reference my question you answered!).

I am sure that there are more studies out by now. I would check recent publications in Physical Review journals to start!

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Magnetic order and topological order can exist simultaneously.

In fact, what one may call the very first proposal of an instrinsic topological material was Haldane's model from 1988, which is an example of this. In this tight binding model, based on a hexagonal 2-dimensional lattice (think graphene), we have a next-nearest-neighbour complex hopping term which represents a staggered magnetic flux. As such, this model breaks time reversal symmetry which leads to an opening of the band gap of graphene (see my answer here) and to a non-zero Chern number, which results in the quantum anomalous Hall effect. The resulting topologically ordered material is called a Chern insulator, and the quantum anomalous Hall effect was realized experimentally in 2013 by doping magnetic chromium into the topological insulator (Bi,Sb)$_2$Te$_3$ as reported in this paper. As Etienne Palos nicely explains in their answer, there has been a lot of work since in trying to get materials that exhibit simultaneous magnetic and topological orders intrinsically.

A final remark about terminology. When people refer to "topological insulators", it is typically understood that one is talking about $\mathbb{Z}_2$ time-reversal invariant topological insulators. As such, breaking time reversal symmetry with magnetic order cannot be included in this classification, but instead leads to other types of topological order.

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  • $\begingroup$ More on terminology: topological insulators are usually said to be symmetry-protected topological phases, not topologically ordered ones. (See e.g. Wen's review.) Not saying you claim they are, but the closing "other types of topological order" is maybe a bit ambiguous. $\endgroup$ – Anyon Jul 26 at 14:42
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    $\begingroup$ @Anyon, thanks for your comment. This is indeed what I mean with "time-reversal invariant topological insulators", which are "protected" by time reversal symmetry, and it is broken when you have magnetic order. Feel free to edit my answer if you think it can be clarified. And +1 for your answer, very nice perspective. $\endgroup$ – ProfM Jul 26 at 15:43
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Topological magnon band structures

Another way of combining topology and magnetism is to consider a magnetic insulator with non-trivial magnon band structure. This setting is somewhat different from the usual picture of topological electron band structure in that i) the band structure represents only quasiparticle excitations, ii) the quasiparticles are bosonic, iii) the quasiparticles are electrically neutral. Nevertheless, we can define and calculate topological invariants for such systems - especially on the level of linear spin wave theory. In this setting, nontrivial topology is typically due to spin-orbit coupling related interactions, e.g. of the Dzyaloshinskii-Moriya type, or to noncollinear magnetic orders. The most straight-forward systems are analogs of Chern insulators where the bands are gapped and separated from each other, with a non-trivial Chern number. This can be linked to an non-quantized magnon thermal Hall effect.

However, it's also possible to have magnon band structures that are analogs of other electronic topological phases. For example, a ferromagnetic model on the honeycomb lattice was proposed in this paper, and shown to be analogous to Haldane's model mentioned in ProfM's answer in one limit. Over the last several years it has also become popular to consider other interesting band structures, including exotic band touching points, and analogs of topological semimetals.

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