# Noncollinear phases in moiré magnets

See allHide authors and affiliations

Contributed by Leon Balents, March 16, 2020 (sent for review January 9, 2020; reviewed by Liang Fu and Oskar Vafek)

## Significance

Moiré patterns, which result when two or more two-dimensional materials with incommensurate or rotated lattices are layered together, create controllable electronic bands that have recently been shown to induce a tremendous wealth of physical phenomena. Here we initiate the theoretical study of moiré patterns’ influence on magnetic states of localized spins. We construct a general formalism using continuum field theory and present thorough analyses in twisted bilayers of antiferromagnets and also ferromagnets, which are within the experimental reach of Van der Waals materials. Two-dimensional magnets, well known for their strong intrinsic spin fluctuations, may serve as a new platform for moiré effects and open the door to a large class of novel phenomena that were once unimaginable.

## Abstract

We introduce a general framework to study moiré structures of two-dimensional Van der Waals magnets using continuum field theory. The formalism eliminates quasiperiodicity and allows a full understanding of magnetic structures and their excitations. In particular, we analyze in detail twisted bilayers of Néel antiferromagnets on the honeycomb lattice. A rich phase diagram with noncollinear twisted phases is obtained, and spin waves are further calculated. Direct extensions to zigzag antiferromagnets and ferromagnets are also presented. We anticipate the results and formalism demonstrated to lead to a broad range of applications to both fundamental research and experiments.

The wealth of new phenomena revealed in incommensurate layered structures of graphene and other two-dimensional (2D) semiconductors and semimetals have sparked major efforts in the study of electronic physics atop moiré patterns. The materials from which these structures are made, Van der Waals (VdW) solids, come in many varieties, inspiring a nascent field going well beyond graphene (1). In particular, a growing family of VdW magnets are being explored both for their magnetism directly as well as for the interplay of that magnetism with electronics (2). Two-dimensional magnets are of particular interest for the fluctuation effects inherent to them. For example, the Mermin–Wagner theorem (3) proves that a strictly 2D magnet with Heisenberg or XY symmetry cannot show long-range order at any nonzero temperature. Exotic quantum phases of magnets, e.g., quantum spin liquids, are widely expected to be more prevalent in two dimensions (4).

In this paper, we introduce a framework to study moiré structures of 2D magnets, under assumptions which are widely applicable and achievable in VdW systems. We present a general methodology to derive continuum models for incommensurate/twisted/strained multilayers including the effects of interlayer coupling, obviating the need to consider thousands or tens of thousands of lattice sites/spins with complicated local environments. We illustrate the method with detailed calculations for the case of a twisted bilayer of two-sublattice Néel antiferromagnets on the honeycomb lattice, a situation realized in

Now we turn to the exposition of the problem and approach, which we illustrate as we go for the simplest case of a two-sublattice Néel order on the honeycomb lattice. First, we detail the assumptions under which a continuum description is possible. We consider structures built from 2D magnets with long-range magnetic order at zero temperature and assume that the interlayer exchange interactions

In this situation the interlayer couplings and the displacement gradients are small perturbations on the intrinsic magnetism of the layers and therefore have significant effects only at low energies. This allows a continuum representation of the magnetism of each layer in terms of its low-energy modes: space–time fluctuations of the order parameters. The order parameter of the two-sublattice antiferromagnet is a Néel vector

Next we consider the first-order effects of displacement gradients upon the intralayer terms in Eq. **2**. As in ref. 8, such terms arise from pure geometry, i.e., carrying out the coordinate transformation from **1**, and from strain-induced changes in energetics. Taking them together, the leading corrections to Eq. **2** are**3** are SU(2) invariant: anisotropic deformation terms must be small in both spin-orbit coupling and in displacement gradients and hence are neglected.

Next we turn to the interlayer coupling terms. By locality and translational symmetry, it is generally of the form**4**. Generally, **4**, with**6** captures the fact that, e.g., for intrinsically ferromagnetic exchange

The full Lagrange density

Eqs. **7**–**9** form the basis for an analysis of the magnetic structure on the moiré scale, as well as for the magnon excitations above them. The magnetic ground state is obtained as the variational minimum of *SI Appendix* for a complete weak coupling analysis), and without loss of generality we can take the spins to lie in the *x*–*z* plane: **10**:

For *SI Appendix*, Section C. In this twisted solution, *SI Appendix*.

For small twist angles, on the other hand, **10** dominates, and the energy is minimized by choosing

Interestingly, however, one can further check that in the same limit of *SI Appendix* for details). This implies a nonzero value for φ so that the twisted-a phase spontaneously breaks the aforementioned Ising symmetry. The value of

Finally, we study the

To summarize, we find three different phases for **12**, Eq. **13**, which is consistent with and in fact interpolates between the perturbative and strong-coupling analyses above. The dashed and dotted lines in Fig. 1 show examples of paths with a fixed ratio *A* and *B*.

Once the minimum energy saddle point is obtained, the full Lagrangian allows for calculation of the magnon spectrum. We define**15** into Eq. **7**, expanding to quadratic order in the fluctuations, and finding the Euler–Lagrange equations for *SI Appendix* for the general result), in which case the four modes decouple immediately,*C* shows the lowest magnon bands when α is at intermediate value. There are three gapless Goldstone modes in the

Finally, we comment on the case of *SI Appendix*.

## Zigzag Antiferromagnet

Having described the case of the Néel antiferromagnet in detail, we give further results more succinctly for other types of 2D magnets. The materials **5**) therefore contains three order parameter “flavors,”

Eq. **19** gives a continuum model to determine the magnetic ordering texture for arbitrary twist angles. The most important difference from the two-sublattice antiferromagnet is that here each spatial harmonic couples to a single flavor, while in the former case, Eq. **18**, the single flavor of order parameter couples to the sum of harmonics. While we do not present a general solution, we note immediate consequences in the strong coupling limit,

## Twisted Ferromagnet

Naïvely, twisting a homobilayer of ferromagnets is relatively innocuous. However, experiments and theory (10⇓⇓⇓⇓⇓–16) for

To this end, for a general twisted bilayer of a ferromagnetic material with the above property, one can use the energy functional shown in Eq. **8** with minimal modifications: 1) the Néel vectors **8** can be performed, which leads to the same set of Euler–Lagrange equations, i.e., Eqs. **12** and **13**. In order to simplify the analysis, we will only consider an infinitesimal β here; its effect is to fix the value of *SI Appendix*. The effects of nonzero β can also be studied in a way similar to the previous case. The mathematical problem is then to obtain the functional form of *SI Appendix*. A plot of the average magnetization in the system is shown in Fig. 4*A* with a transition from collinear to twisted phase at finite α. Unlike the antiferromagnets discussed above, there is a finite interval of twist angles where the collinear phase exists even with infinitesimal anisotropy parameter β. Also a plot of the spatial configuration of a twisted solution is presented in Fig. 4*B*; it shows that there are large regions in real space with maximal magnetization while at the same time there are also other regions exhibiting close to zero magnetization.

## Conclusion

In this work, we have considered moiré 2D magnets and in particular the twisted bilayers of VdW magnetic materials. We have developed a low-energy formalism in the continuum and studied in detail three different examples of twisted bilayers: antiferromagnetic, zigzag antiferromagnetic, and ferromagnetic. Remarkably, a rich phase diagram is obtained as one varies the twist angle and material parameters; there are interesting twisted ground state solutions comprising long-wavelength noncollinear magnetic textures. Such spatial patterns can potentially be observed in experiments, where the twist angle control adds to the tunability of the system. Furthermore, at small twist angles in the noncollinear phases, certain spin waves also exhibit interesting features such as flattening dispersion curves.

Material-wise, *A*, *Bottom*; it can be seen that at large angles, the system is in the collinear phase, but the twisted-s phase (

The present methodology can be utilized with minimal modifications in further analyses of other moiré systems in the vast collection of possible bilayer magnetic materials. For example, here we have mainly presented the examples of homobilayers, but interesting phenomena can also arise for heterobilayers of VdW magnets, such as the stacking of ferromagnets on antiferromagnets (18). The magnetic properties of general moiré systems as well as their interplay with the electronic/transport properties could be the subject of future studies. Given the extremely fruitful research done in the field of moiré electronic systems, one can anticipate that the magnetic moiré systems could play the role of a new platform where novel exciting physics could be pursued.

## Materials and Methods

In this section, we explain our numerical manipulations.

In order to find the ground states, one needs to solve Eqs. **12** and **13** simultaneously. We have done so by two different methods: The first method is solving the equations in real space by the use of overdamped dynamics, i.e., adding fictitious time derivatives of

In Fig. 1, the phase boundaries can be extracted by observing the changes of behavior of **12** and **13** for various combinations of α and β and plot the corresponding functions *A*.

As for Fig. 2*C*, the spin waves are obtained from the Bloch ansatz **17** thus become

The interlayer exchange for

### Data Availability.

No data, materials, or protocols are needed to reproduce the results presented in this paper. All codes are available upon request.

## Acknowledgments

We thank Andrea Young for useful discussions. Z.-X.L. thanks Mengxing Ye for helpful conversations. This work was supported by the Simons Collaboration on Ultra-Quantum Matter, grant 651440 from the Simons Foundation (L.B. and Z.-X.L.), and by the US Department of Energy, Office of Science, Basic Energy Sciences under Award DE-FG02-08ER46524 (K.H.).

## Footnotes

↵

^{1}K.H., Z.-X.L., and L.B. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: balents{at}kitp.ucsb.edu.

Author contributions: L.B. designed research; K.H., Z.-X.L., and L.B. performed research; and K.H., Z.-X.L., and L.B. wrote the paper.

Reviewers: L.F., Massachusetts Institute of Technology; and O.V., Florida State University.

The authors declare no competing interest.

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.2000347117/-/DCSupplemental.

Published under the PNAS license.

## References

- ↵
- K. S. Novoselov,
- A. Mishchenko,
- A. Carvalho,
- A. H. Castro Neto

- ↵
- K. S. Burch,
- D. Mandrus,
- J. G. Park

- ↵
- ↵
- ↵
- R. Brec

- ↵
- ↵
- R. Bistritzer,
- A. H. MacDonald

- ↵
- L. Balents

- ↵
- A. R. Wildes,
- B. Roessli,
- B. Lebech,
- K. W. Godfrey

- ↵
- ↵
- T. Song et al.

- ↵
- ↵
- ↵
- ↵
- ↵
- P. Jiang et al.

- ↵
- L. Chen et al.

- ↵

## Citation Manager Formats

## Article Classifications

- Physical Sciences
- Physics