Square root topological lattice models such as the non-Hermitian 1D Su-Schrieffer-Heeger (SSH) model [1] and the 2D honeycomb lattice [2, 3] have recently been introduced and realised in several platforms [4, 6]. The recipe for the square root construction has paved the way for 2n root models [7], in which the transition from higher to lower roots can be readily made due to the arborescent structure of the resultant lattice by taking the n-th power of the corresponding Hamiltonian ((Hroot)n) and subtracting a constant energy. The present method is only applicable for the square root operation. Here, we introduce a new procedure which enables us to realise any arbitrary n-th root of the lattice, including prime number roots, by simply adding n-1 new sites between two parental sites such that the new directional couplings create a closed loop in either the clockwise or anti-clockwise directions (Fig. 1a). While the Hermiticity is preserved in square root operations, the non-reciprocal couplings in the n-th root method require breaking the Hermiticity so that the arborescent construction can be followed to form (Hroot)n as the direct sum of Hparental and Hresidual where Hparental is the original lattice Hamiltonian, which is topologically non-trivial [8], and Hresidual represents the direct sum of all the residual parts. This form allows us to transition between roots (Fig. 1b) without the need for a constant energy shift in every operation (Fig. 2). Every root operation results in a multiplication of the energy band structure, which depends on the degree of the root. This multiplication in energy also results in the multiplication of the topological states. In contrast to conventional TIs, the n-th root version of the topological lattices possesses multiple topological mid-gap states with finite energy as well as non-Hermitian phenomenon such as the non-Hermitian skin effect, which may be useful for various potential applications such as magnetic sensing due to the extensive sensitivity and robustness of the topological states.

![](https://assets.underline.io/uploads/markdown_image/1/image/7985dfc21e709bda9aa9472f8edf38c1.jpg)

Illustration of systematic shifting between (SSH)1/n lattices.

![](https://assets.underline.io/uploads/markdown_image/1/image/c228455f335d23670adca71013893745.jpg)

Real energy spectrum as a function of site index of cubic root (3n) SSH model with 1 unit cell, the transition from n=3 to n=0.

N-th Root Topological Lattices

Magnetic multilayer films are known to host a variety of novel magnetic configurations such as chiral domain walls and topological magnetic skyrmions with potential nano-electronic applications. However, characterizing these material systems typically involves a combination of several magnetometry, microscopy, and spectroscopy experiments, and some of these techniques can be time-consuming and expensive. Therefore, to rapidly characterize these material systems, it is crucial to maximize the information extracted from results of experiments that are more accessible, such as domain images obtained from magnetic force microscopy. We show that deep convolutional neural networks can extract magnetic parameters such as the exchange interaction, Dzyaloshinskii-Moriya interaction, and uniaxial anisotropy from images of domain configurations. Experimentally realistic training and validation data were generated through micromagnetic simulations. The trained models were consistently able to reach R2 values greater than 0.7 on validation data. By inspecting the intermediate feature maps of the neural network, we find that the network is able to learn features such as domain boundaries. Testing the models on actual experimental data yield values that were consistent with our knowledge of the material systems. Our work thus demonstrates the utility of developing machine models trained on simulation data as a means to accelerate the characterization of magnetic systems.

Estimation of magnetic parameters from domain images with convolutional neural networks

Magnetic skyrmions are nanoscale topological spin structures stabilized by chiral interactions and they are promising candidates for realizing brain-inspired computing devices [1,2]. One important prerequisite of such applications is the controlled formation of skyrmions. Skyrmions could form via one of two pathways - nucleation from uniform magnetization or evolution from magnetic stripes. The latter can be further divided into two distinct mechanisms – the reversible shrinking of a stripe into a skyrmion or the irreversible fission of a stripe into multiple skyrmions [3]. In this work, Ir/Fe/Co/Pt chiral multilayers were characterized with first-order reversal curve (FORC) magnetometry in conjunction with Lorentz transmission electron microscopy (LTEM) over the temperature range of 100 - 350 K. With both techniques, we observed a marked increase in stripe-to-skyrmion fission tendency with increasing temperature. FORC magnetometry consists of measuring a series of minor hysteresis loops (M(H,Hr)) and computing the quantity ρ (=-1/2 dM(H,Hr)/dHdHr) which represents the degree of irreversibility of a magnetic process [4,5]. The fission of stripes into skyrmions at intermediate fields and the annihilation of skyrmions at high fields are two processes that are revealed by FORC as ρ≠0 features [3]. The annihilation-fission separation (AFS, indicated by red arrows in Fig. 1e-h) could be used as a proxy of the favourability of the fission process. In Fig. 1i, one can observe the rise of AFS with temperature. LTEM microscopy captures the real-space evolution of magnetic textures as the field is swept. By tallying the stripe and skyrmion densities and tracking their field dependence, we estimate the average number of skyrmions generated by a stripe, η, using the ratio of the maximum skyrmion to stripe densities. As shown in Fig. 2, η increases 3-fold: from ~1 at 100K to ~3 at 350K. We interpret our results as the consequence of temperature-dependent magnetic interactions such as direct exchange, Dzyaloshinskii–Moriya interaction (DMI), and anisotropy. The relative strengths of these competing interactions will determine the energy barrier of the fission process. With theoretical modeling from geodesic nudged elastic band (GNEB) calculations [6], we found that the barrier height is greatly reduced with increasing temperature. These findings establish temperature as an important tuning parameter and provide insights into skyrmion formation, which is crucial to the realization of skyrmionic devices.

**References:**

1. Fert, A. et al. Nat. Rev. Mater. 2, natrevmats201731 (2017). 2. Soumyanarayanan, A. et al. Nat. Mater. 16, nmat4934 (2017). 3. Tan, A. K. C. et al. Phys. Rev. Mater. 4, 114419 (2020). 4. Pike, C. R. et al. J. Appl. Phys. 85, 6660 (1999). 5. Davies, J. E. et al. Phys. Rev. B 70, 224434 (2004). 6. Bessarab, P. F. et al. Comput. Phys. Commun. 196, 335–347 (2015).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/75d2d0e93437b24489d3975ae58dedb6.jpg)
Fig. 1: Temperature dependence of annihilation-fission separation (AFS). (a-d) Out-of-plane (OP) hysteresis loops of a multilayer sample from 100K to 350K. (e-h) Irreversibility ρ(H,Hr) derived from FORC magnetometry at the same temperatures. Hr, the reversal field, is the starting point of each minor loop. AFS are indicated by red arrows. (i) AFS plotted against temperature, showing a remarkable temperature dependence.

![](https://assets.underline.io/uploads/markdown_image/1/image/e3a535a7a6251b9e645ce19e5a74fc92.jpg)
Fig. 2: Texture density evolution imaged by LTEM. (a-d) Field evolution of stripe (St) and skyrmion (Sk) densities of a multilayer sample from 100 K to 350 K. While the stripe densities remained relatively constant, the skyrmion densities appreciated significantly with temperature. Accompanying LTEM images show the field evolution of stripes into skyrmions over selected fields. White dot lines enclose a typical magnetic domain for 100K and 300K. (e) η, estimated by ratio of maximum skyrmion to stripe densities, increases with temperature.

 Thermal Evolution of Magnetic Skyrmion Formation Mechanism in Chiral Multilayers

Magnetic multilayer films are known to host a variety of novel magnetic configurations such as topological magnetic skyrmions with potential nano-electronic applications. However, characterizing these material systems at film and device-level typically involves a combination of several magnetometry, microscopy, and spectroscopy experiments. Some of these techniques can be time-consuming and expensive, or even inaccessible to materials scientists. Therefore, to rapidly characterize these material systems, it is crucial to maximize the information extracted from results of experiments that are more accessible, such as domain images obtained from magnetic force microscopy (MFM). We show that deep convolutional neural networks are able to extract magnetic parameters, in particular the exchange interaction (A), Dzyaloshinskii-Moriya interaction (D), effective anisotropy (Keff), and the thermodynamic stability (κ), from images of domain configurations. Experimentally realistic training and validation input images, as shown in Fig. 1, were generated through micromagnetic simulations with MuMax3 [1]. The magnetic parameters and structures used in the simulations are based on Ir/Fe/Co/Pt multilayers with tunable magnetic properties [2, 3]. The neural network architecture, implemented with the Keras package [3], consists of multiple convolutional layers followed by a fully connected network with 4 outputs corresponding to the magnetic parameters of interest. The trained models were consistently able to reach R2 values greater than 0.9 on validation data, as shown in Fig. 2. By inspecting the intermediate feature maps of the neural network, we find that the network is able to learn features such as domain boundaries. Notably, testing the models on experimentally obtained MFM images yield magnetic parameters that were consistent with our knowledge of the material systems. Our work thus demonstrates the utility of developing machine models trained on simulation data as a means to accelerate the characterization of magnetic systems.

**References:**

[1] A. Vansteenkiste, J. Leliaert, M. Dvornik et al, AIP Advances 4, 107133 (2014). [2] A. Soumyanarayanan, M. Raju, A. L. Gonzalez Oyarce et al., Natural Materials 16, 898-904 (2017). [3] A. K. C. Tan, J. Lourembam, X. Chen et al., Phys. Rev. Materials 4, 114419 (2020). [4] F. Chollet et al., https://keras.io (2015)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/fb4ee2b3d9e20e768e23cfca8d572833.jpg)
Representative micromagnetic images used as input for the training and validation of neural networks.

![](https://assets.underline.io/uploads/markdown_image/1/image/f46694e5bfcc6346ff1b6bf794d5f03e.jpg)
Parity plots showing the predicted micromagnetic magnetic parameters (exchange stiffness A, Dzyaloshinskii-Moriya interaction D, effective anisotropy Keff, and thermodynamic stability κ) vs the actual parameters on the validation data.

 Estimation of Magnetic Parameters from Domain Images with Convolutional Neural Networks in Chiral Multilayers

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N-th Root Topological Lattices

Estimation of magnetic parameters from domain images with convolutional neural networks

Thermal Evolution of Magnetic Skyrmion Formation Mechanism in Chiral Multilayers

Estimation of Magnetic Parameters from Domain Images with Convolutional Neural Networks in Chiral Multilayers

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