heterostructures (for example with heavy metals), so that switching may only be accomplished with the help of an external magnetic field. Recent reports, however, have shown that taking advantage of the reduced symmetry of the 2D semimetal WTe₂ or β-MoTe₂, torques with this type of symmetry can be obtained[7], [8]. Alternative strategies including wedged structural engineering, the use of ferroelectric substrates, domain wall pinning by geometry engineering or even taking advantage of exchange coupling, have also been considered[3].

Increasing the operation speed further motivates the study of SOT in antiferromagnetic materials[2], [12]. These are especially suited for ultrafast devices because the magnetization dynamics, governed by exchange fields, is in the THz range (whilst in ferromagnets it is limited to GHz). Antiferromagnets are being studied both as the SOT source and as the magnetic layer to be manipulated (Fig. 2d). Additionally, the lack of a macroscopic magnetic moment could help increase the integration density in magnetic memories, as well as field-free switching of PMA nano-objects by taking advantage of exchange coupling between a ferromagnet and an antiferromagnet.

<figure><img src="image_33.png" /><figcaption>Figure 2. Schematic of new materials with potential for spin orbit torque technologies, and their possible use as large spin orbit coupling (SOC) layer (orange arrows). Additionally, 2D materials and antiferromagnets can also be embedded in the MTJ stacks (blue arrows), both as the insulating barrier or the magnetic element. The authors have used VESTA [13] and Crystallography Open Databases for the sketch of 2D crystal structures.</figcaption></figure>

## Advances in Science and Technology to Meet Challenges

The study of SOTs is a young research field and new types of spin torque materials are still flourishing, which accounts for the relatively few antiferromagnets, topological insulators or 2D materials that have been so far investigated. It is thus expected that many new results with novel materials will arise in the near future. Considerable progress is being made in material science towards developing high-quality heterostructures but further advances are required. They include achieving material growth optimization and large-scale integration of, for example, topological insulators, Weyl and Dirac semimetals and 2DMs with controlled thickness[11], as well as improving interfacial quality within the heterostructures' layers. Impurities (source of extrinsic contributions to the SOC), crystalline symmetries, interface oxidation and intermixing greatly influence the observed spin properties[2], [3]. Therefore, a better control of such parameters will help us advance in the reproducibility and engineering of the torques, while facilitating the understanding of the underlying physical mechanisms. Moreover, in view to consider the use of such devices in industry, further research on the compatibility of the new materials with the current Si-based technology is still pending[11].

Progress in the understanding of microscopic torque mechanisms requires the identification of the origin of large discrepancies between the reported SOT efficiencies with similar systems, which are
characterized at different laboratories or with different methods[2], [5]. The wide variety of effects unrelated to SOTs that contribute to the detected signals, from Oersted fields to thermoelectric effects, illustrates the need for comparing the results obtained with alternative experimental techniques using the same heterostructure. The complex nature of the phenomena must be taken into account in the experimental design, motivating systematic measurements as a function of temperature, thickness or interlayers[2] and calling for complementary experiments with advanced characterization material science methods.

## Concluding Remarks

SOTs represent a unique opportunity to electrically manipulate magnetic layers, enabling increased efficiency and speed towards devices such as the SOT-MRAMs exemplified above. Despite the necessary progress required both for their understanding and optimization, the potential of SOTs for applications goes well beyond simple increased MRAM performance, as it can provide novel functionalities within other technologies, for instance memristors and nanooscillators for neuromorphic computing[14].

There exists a wide variety of materials with potential in spin torque structures (for example within the families of 2D materials or topological insulators), each with distinct and unique properties. These materials are being thoroughly studied in other areas of condensed matter physics, offering a rich playground for interaction between fields, with much scope for yet unimaginable new applications.

## Acknowledgements

We acknowledge support from the H2020 European Research Council PoC under Grant Agreement 899896 SOTMEM and from MINECO with grant FJCI-2016-28645.

## References

[1] N. Locatelli, V. Cros, and J. Grollier, "Spin-torque building blocks," *Nat. Mater.*, vol. 13, no. 1, pp. 11–20, Jan. 2014.

[2] A. Manchon et al., "Current-induced spin-orbit torques in ferromagnetic and antiferromagnetic systems," *Rev. Mod. Phys.*, vol. 91, no. 3, p. 035004, Sep. 2019.

[3] R. Ramaswamy, J. M. Lee, K. Cai, and H. Yang, "Recent advances in spin-orbit torques: Moving towards device applications," *Appl. Phys. Rev.*, vol. 5, no. 3, pp. 1–50, 2018.

[4] Y. Wang et al., "Topological Surface States Originated Spin-Orbit Torques in Bi 2 Se 3," *Phys. Rev. Lett.*, vol. 114, no. 25, p. 257202, Jun. 2015.

[5] H. Wu et al., "Room-Temperature Spin-Orbit Torque from Topological Surface States," *Phys. Rev. Lett.*, vol. 123, no. 20, Nov. 2019.

[6] F. Bonell et al., "Control of spin-orbit torques by interface engineering in topological insulator heterostructures," *Nano Lett.*, Jun. 2020. https://dx.doi.org/10.1021/acs.nanolett.0c01850

[7] D. MacNeill, G. M. Stiehl, M. H. D. Guimaraes, R. A. Buhrman, J. Park, and D. C. Ralph, "Control of spin-orbit torques through crystal symmetry in WTe2/ferromagnet bilayers," *Nat. Phys.*, vol. 13, no. 3, pp. 300–305, 2017.

[8] G. M. Stiehl et al., "Layer-dependent spin-orbit torques generated by the centrosymmetric transition metal dichalcogenide beta-MoTe2," *Phys. Rev. B*, vol. 100, p. 184402, 2019.

[9] P. Huang, P. Zhang, S. Xu, H. Wang, X. Zhang, and H. Zhang, "Recent advances in two-dimensional ferromagnetism: Materials synthesis, physical properties and device applications," *Nanoscale*, vol. 12, no. 4. Royal Society of Chemistry, pp. 2309–2327, 28-Jan-2020.

[10] M. Piquemal-Banci et al., "2D-MTJs: Introducing 2D materials in magnetic tunnel junctions," *J. Phys. D. Appl. Phys.*, vol. 50, no. 20, 2017.
[11] Y. Liu and Q. Shao, "Van der Waals materials for energy-efficient spin-orbit torque devices," *arXiv*2003.11966 [cs], Mar. 2020.

[12] J. Železný, P. Wadley, K. Olejník, A. Hoffmann, and H. Ohno, "Spin transport and spin torque in antiferromagnetic devices," *Nat. Phys.*, vol. 14, no. 3, pp. 220–228, Mar. 2018.

[13] K. Momma and F. Izumi, "VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data," *J. Appl. Crystallogr.*, vol. 44, no. 6, pp. 1272–1276, Dec. 2011.

[14] F. Moradi et al., "Spin-Orbit-Torque-based Devices, Circuits and Architectures," *arXiv*1912.01347 [cs], Dec. 2019.

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<figure><img src="image_34.png" /></figure>
# Magnetic skyrmions

Marius V. Costache¹ and Aurélien Manchon²

¹Catalan Institute of Nanoscience and Nanotechnology (ICN2),
CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain

²Aix-Marseille Université, CNRS, CINaM, Marseille, France

## Status

Alike Majorana fermions, the concept of skyrmions originates from elementary particle physics and has inspired physicists to explain various quantum phenomena such as the quantum Hall effects. Over the past years magnetic skyrmions have become a major area of interest within the field of spintronics due to its novel physics, stimulated by the vibrant interest on topological states of matter, and its great potential for applications.

Magnetic skyrmions¹,², bridging topology and magnetism, are local excitations in a homogeneous magnetic domain state with unique real-space topological characteristics³. They are part of a broader family of chiral magnetic objects that includes antiskyrmions⁴, merons, bimerons etc., all featuring various forms of chiral magnetic textures. Due to the nontrivial topology of their spin texture, these skyrmions are expected to be very stable⁵,⁶, can have a diameter of only few nanometers, and are in principle very easy to manipulate by tiny electrical and magnetic forces⁷. Thus, they hold real potential for future low-power and high-density magnetic memory and logic applications⁸,⁹, but also bio-inspired computing.

Magnetic skyrmions can emerge as a ground state, in which case they form a skyrmion lattice¹,² and collective behaviors become important, or as an isolated metastable state that can be manipulated individually.

Magnetic skyrmions were first experimentally observed in bulk non-centrosymmetric crystals¹,² and later on single ultrathin films¹⁰. Current approaches are largely centered on metallic heterostructures where a thin ferromagnetic layer is coupled to materials with large spin-orbit coupling. There, one can obtain skyrmions that are stable against perturbations of nanometer-scale size at room temperature. In these heterostructures, the key ingredient is the Dzyaloshinskii–Moriya magnetic interaction¹¹ (DMI) that emerges at interfaces upon the combination of broken inversion symmetry and strong spin-orbit interaction. Materials quality and interface control (Moriya-Dzyaloshinskii term) are key issues in skyrmions research that will allow controlling skyrmions properties in terms of size, shape, room temperature stability and mobility at low current densities.

Magnetic skyrmions can interact very efficiently with electrons and spin currents due to their non-collinear spin texture. This results in potentially high skyrmions mobility up to 1 km/s for current density of $10^8$ A/cm².
<figure><img src="image_35.png" /><figcaption>**Figure 1.** Typical skyrmion spin textures where the spin texture is indicated by the arrows. (a) Bloch-type skyrmion; (b) Neel-type skyrmion; (c) Bloch-type antiferromagnetic skyrmion; (d) Neel-type antiferromagnetic skyrmion.</figcaption></figure>

## Current and Future Challenges

Although the magnetic skyrmions possess great potential for applications, there are several challenges related to materials quality and interface control, spin sensitive imaging techniques and theoretical modelling. Any utilization of skyrmions in future technological applications will require room temperature all-electrical generation, manipulation, and detection of individual skyrmions.

Magnetic skyrmions can be generated by magnetic and electric fields, electric current, thermal gradients, or geometrical constrictions from domain walls. Here the challenge is to obtain better deterministic nucleation, both in space and time. The manipulation of skyrmions presents at least two challenges. One is related to the improvement of the spin-torque induced motion of skyrmions to obtain large skyrmion velocities. To do so, one can use materials with larger efficiency than heavy metals to produce spin currents such as Rashba/Dirac/valley-Zeeman interfaces. The second challenge is related to the presence of the skyrmion Hall effect that can create an undesired oblique skyrmions motion³. Using narrow nanowires could be a way to mitigate this problem, although skyrmions tend to vanish when reaching the wire's boundary. An alternative route is to use synthetic or real antiferromagnets, where the oblique motion is naturally suppressed¹². There, the challenge is however to control the lateral expansion occurring at high velocity¹³.

For detection, recent works have focused on the electrical read-out of skyrmions by exploiting the characteristic magneto-transport (the conventional type of read-out scheme based on the TMR effect, the anomalous Hall effect), the topological Hall effect, or magnetoresistance effect measured with an STM non-magnetic tip, although the latter seems unpractical for industry.

Despite its substantial success in terms of skyrmion nucleation and manipulation, a reliable and all-electrical design integrating electric write-in, transmission and read-out of magnetic skyrmions remains a challenge. Two possible concepts exist to solve this hurdle; using the spin-transfer torque (STT) based-switching by vertical spin-polarized current injection and/or using a local electric-field effect. The use of local electric fields to create and delete individual skyrmions offers dissipation-less alternatives, compared to STT, for energy efficient skyrmion devices.
In addition to being very attractive for technological applications, magnetic skyrmions exhibit novel phenomena such as topological and skyrmion Hall effects³. Future research might help to understand the dynamics of skyrmions and to account for quantum and lattice effects. These rich phenomena can include Bose-Einstein condensation, localization, Mott-superfluid transition or Wigner crystallization.

Advances in Science and Technology to Meet Challenges

For progress to continue, substantial improvement in the understanding and control of various aspects of magnetic skyrmions are required.

For high-density and fast magnetic memory applications, very small (~nm) and high mobility (1km/s) skyrmions are necessary. Due to their nontrivial spin texture, electrons flowing through skyrmions experience a fictitious magnetic field (order of 100 Tesla) that increases upon scaling down the skyrmion size and leads to topological and skyrmion Hall effects. Hence, small skyrmions are expected to display deteriorated transport properties. A better understanding of this effect is necessary to understand the edge motion of the skyrmions and identify efficient electrical detection methods.

Small and fast skyrmions require advanced engineering of ferromagnetic multilayers (to better control of the interlayer dipolar fields) but also novel structures based on synthetic ferrimagnetic and synthetic antiferromagnetic multilayers¹⁴. In addition, since skyrmions belong to a large family of chiral magnetic objects, the search for novel paradigms with different properties, e.g., the antiskyrmions, could open new perspectives (absence of Hall effect, better topological protection...).

Another beneficial direction is the interaction between skyrmion and thermally and electrically excited magnons. Together with a detailed study of skyrmion-skyrmion and skyrmion-anti-skyrmion interaction that could lead to the design of devices with high skyrmion mobility and novel ways of deleting/annihilating skyrmions.

To take advantage of their huge potential for technological applications, multilayers engineering, new device design, novel materials, and advanced inspection techniques will be highly desirable. To characterize nm size skyrmions, as well as their dynamics and their spin texture high-resolution microscopy techniques such as but not limited to X-ray methods, advance electron and nitrogen-vacancy center microscopy will be required.

On the materials side, as an alternative, one can envision the use of two-dimensional materials, such as van der Waals (vdW) magnets¹⁵, graphene, transition metal dichalcogenides (TMDC) or their combinations i.e. vdW heterostructures with several advantages. Strong spin-orbit coupling (SOC) plays a central role in these systems. For example, in vdW magnets SOC stabilizes the perpendicular magnetic anisotropy, in graphene/TMDC interfaces SOC can efficiently generate spin currents via the Rashba and Zeeman valley couplings¹⁶. These couplings can favour small size skyrmions with high mobility. Graphene can be used as alternative to detect skyrmions. Graphene is sensitive to the properties of the substrate on which it is placed and can inherit substrate properties due to proximity effect.
<figure><img src="image_36.png" /><figcaption>Figure 2. Schematics of three emblematic skyrmionic devices: (a) The skyrmion racetrack memory, inspired from the domain wall racetrack memory<sup>8</sup>. Bits of information are encoded in the skyrmions and manipulated through spin transfer torque. The working principle is similar to that of a shift register. (b) The skyrmion reshuffler realized in Ref. 9 and based on thermal diffusion of skyrmions can be used to destroy unwanted correlations in skyrmion-based probabilistic computing architecture. (c) The electrically-operated skyrmion-based artificial synaptic device can mimic the behavior of a biological synapse.</figcaption></figure>

**Concluding Remarks**

Magnetic skyrmions bring nanometer-scale information size, stability against external perturbations, high speed and performance advantages beyond the theoretical limits of conventional magnetic technology. Rapid advances have been achieved in materials engineering and experimental design that allowed nucleation, manipulating and deleting individual skyrmions. Novel skyrmion devices concepts have been proposed such as racetrack type memory applications, logic devices, skyrmion oscillator, bio-inspired devices, topological based communication and computing designs.

For the future, it will be necessary to substantially advance the understanding and control of the interaction between skyrmions and the interaction of skyrmions with spin currents and magnons. Advance in materials engineering allowing generating nm-size skyrmions that are stable at room temperature and easily manipulated with tiny currents or local electric fields together with advanced imaging methods for the study of spin texture and dynamics skyrmions would be needed. The discovery of novel chiral magnetic objects beyond skyrmions with improved properties will undoubtedly facilitate the translation towards industrial applications. Their special topological properties would allow us to explore fascinating new physics in ways not possible with other particles. More important, given the technological importance of non-volatile magnetic memories today, the research on magnetic skyrmions will continue at least at a similar rate and level.

**Acknowledgements**

M.V.C. acknowledges discussion with Nicolas Reyren and support from the Spanish Ministry of Economy and Competitiveness, MINECO (under contracts FIS2015-62641-ERC, MAT2016-75952-R and SEV-2017-0706 Severo Ochoa).
References

1. Yu, X. Z. et al. Real-space observation of a two-dimensional skyrmion crystal. *Nature* **465**, 901 (2010).

2. Mühlbauer, S. et al. Skyrmion lattice in a chiral magnet. *Science* **323**, 915 (2009).

3. Nagaosa, N. & Tokura, Y. Topological properties and dynamics of magnetic skyrmions. *Nat. Nanotechnol.* **8**, 899 (2013).

4. Nayak, A. K. et al. Magnetic antiskyrmions above room temperature in tetragonal Heusler materials. *Nature* **548**, 561 (2017).

5. Iwasaki, J., Mochizuki, M. & Nagaosa, N. Current-induced skyrmion dynamics in constricted geometries. *Nat. Nanotechnol.* **8**, 742 (2013).

6. Sampaio, J., Cros, V., Rohart, S., Thiaville, A. & Fert, A. Nucleation, stability and current-induced motion of isolated magnetic skyrmions in nanostructures. *Nat. Nanotechnol.* **8**, 839 (2013).

7. Jonietz, F. et al. Spin transfer torques in MnSi at ultralow current densities. *Science* **330**, 1648 (2010).

8. Fert, A., Cros, V. & Sampaio, J. Skyrmions on the track. *Nat. Nanotechnol.* **8**, 152 (2013).

9. Zázvorka, J. et al. Thermal skyrmion diffusion used in a reshuffler device. *Nat. Nanotechnol.* **14**, 658 (2019).

10. Jiang, W. et al. Blowing magnetic skyrmion bubbles. *Science* (80-). **349**, 283 (2015).

11. Belabbes, A., Bihlmayer, G., Bechstedt, F., Blügel, S. & Manchon, A. Hund's rule-driven Dzyaloshinskii-Moriya interaction at 3d-5d interfaces. *Phys. Rev. Lett.* **117**, 247202 (2016).

12. Barker, J. & Tretiakov, O. A. Static and Dynamical Properties of Antiferromagnetic Skyrmions in the Presence of Applied Current and Temperature. *Phys. Rev. Lett.* **116**, 147203 (2016).

13. Salimath, A., Zhuo, F., Tomasello, R., Finocchio, G. & Manchon, A. Controlling the deformation of antiferromagnetic skyrmions in the high-velocity regime. *Phys. Rev. B* **101**, 024429 (2020).

14. Legrand, W. et al. Room-temperature stabilization of antiferromagnetic skyrmions in synthetic antiferromagnets. *Nat. Mater.* **19**, 34 (2020).

15. Deng, Y. et al. Gate-tunable room-temperature ferromagnetism in two-dimensional Fe₃GeTe₂. *Nature* **563**, 94 (2018).

16. Benítez, L. A. et al. Tunable room-temperature spin galvanic and spin Hall effects in van der Waals heterostructures. *Nat. Mater.* **19**, 170 (2020).
# Machine Learning using Experimental Quantum Materials Data

Eun-Ah Kim¹

¹Department of Physics, Cornell University, Ithaca, NY 14850, United States, USA.

## Status

The central goal of modern quantum materials is to search for new systems and technological paradigms that utilize quantum mechanical aspects of matter rather than being limited by them. In particular, there is an active search for new materials that exhibit surprising physical properties because of strong correlations. The key challenge in discovering these new materials lies in understanding strongly correlated quantum matter that defies both the simple theory and the majority of ab-initio computations. Hence any progress has generally been experimentally driven. Decades of efforts in developing new experimental techniques and improving the resolution of instruments and detectors put us in an abundance of complex and comprehensive data. Two particular frontiers are (1) “image”-like real space data obtained through scanning tools and (2) high-dimensional and voluminous momentum space data obtained through scattering tools. The next breakthrough in the field will come from relating vast quantities of collected experimental data to theoretical models using tools of data science.

<figure><img src="image_37.png" /><figcaption>**Figure 1.** Harnessing data revolution in the quantum materials data. Data science tools can provide the critical link between theoretical understanding and complex or voluminous data now accessible through revolutionary developments in experimental techniques and detector technology.</figcaption></figure>

## Current and Future Challenges

Increasingly, data science tools such as machine learning are showing great promise for accelerating discovery of materials through guided searches¹²³ ⁴ and for improving our understanding of simulated synthetic data associated with key models⁵⁶⁷⁸. The next challenge is to tackle the data-driven challenges forming the critical bottlenecks between comprehensive and complex experimental data on quantum materials and a theoretical understanding. Although these data-driven challenges require a fundamentally new data science approaches for two reasons: first, quantum mechanical imaging is probabilistic; and second, inference from data should be subject to fundamental laws governing microscopic interactions. Hence the readily-available data science tools need to be suitably altered to comply with hard and fast rules of fundamental microscopic laws. Moreover, any insight from learning by machines require “sanity check” or “interpretation” on machine’s learning⁹.
Advances in Science and Technology to Meet Challenges

Recently significant first steps were taken towards matching complex experimental data such as "image"-like data to theoretical hypothesis, discriminating multiple hypothesis against each other and hence offering theoretical insight, using machine learning<sup>1011</sup>. In Ref[10], we have developed and demonstrated a new general protocol for machine learning-based identification of symmetry-breaking ordered states in electronic-structure images. Our artificial neural networks were trained to learn the defining motifs of each category, including its topological defects, and to recognize those motifs in experimental data (Fig. 1). Applied to the hole-doped Mott insulator state harbouring the enigmatic pseudogap states<sup>12</sup> over a wide range of doping, the neural networks repeatedly and reliably discover the predominant features of a specific ordered state of unidirectional period 4a₀ density wave modulations at the pseudogap energy that break the global rotational symmetry to generate a nematic state. Hence the machine learning based hypothesis testing revealed that strong coupling mechanism that drives such commensurate order is universally at play in the entire underdoped regime of the cuprate phase diagram. In Ref[11], we showed that the modality developed in Ref. [10] readily extends to a completely different measurement technique of resonant ultra-sound spectroscopy.

<figure><img src="image_38.png" /><figcaption>Figure 2. The machine learning based hypothesis testing on quantum material image data in practice. Figures reproduced from Ref[1]. (a) For machine learning based hypothesis testing, artificial neural networks (ANNs) are trained to recognize the given hypothesis in the synthetic data such as in electronic-structure images representing different ordered states. A training image set is synthesized by appropriately diversifying pristine images of four distinct electronic ordered states. Each translational-symmetry-breaking ordered state is labelled by a category C = 1, 2, 3, 4 associated with a wavelengths of 4.348a₀, 4.000a₀, 3.704a₀, 3.448a₀, respectively. The training images in each category are diversified by appropriate addition of noise, short-correlation length fluctuations in amplitude and phase, and topological defects. (b) The trained ANN is then presented with experimental data to decide among the four hypothesized states. (c) The scanning tunneling spectroscopy data of underdoped cuprate at doping level p=0.08. (d) The ANN outcome exhibits a clear preference for the commensurate period of 4.000a₀ (category 2), especially along the X-direction. (e) A comprehensive analysis of the data set over the entire range of energies reveal the onset of unidirectional commensurate order at the pseudogap energy scale.</figcaption></figure>

Concluding Remarks

The demonstrations that artificial neural networks can process and identify specific broken symmetries of highly complex experimental image arrays[10] and constrain the nature of order parameter from a sequence of peaks [11] are milestones for general scientific techniques. Moving
forward, more progress in unsupervised discovery from high-dimensional and voluminous momentum
space data are to be much anticipated.

Acknowledgements

Eun-Ah Kim was supported by DOE basic energy sciences through Award DE-SC0018946 and by NSF, Institutes for Data-Intensive Research in Science and Engineering – Frameworks through award OAC-1934714; by the Cornell Center for Materials Research with funding from the NSF MRSEC program (DMR-1719875).

References

<footnote n="1">¹ Ouyang, R. et al. SISSO: a compressed-sensing method for identifying the best low-dimensional descriptor in an immensity of offered candidates. *Phys. Rev. Mat.* **2**, 083802 (2018).</footnote>

<footnote n="2">² Stanev, V. et al. Machine learning modeling of superconducting critical temperature. *Npj Comput. Mater.* **4**, 29 (2018).</footnote>

<footnote n="3">³ Kusne, A. G. et al. On-the-fly machine-learning for high-throughput experiments: search for rare-earth-free permanent magnets. *Sci. Rep.* **4**, 6367 (2014).</footnote>

<footnote n="4">⁴ G.R. Schleder et al. From DFT to machine learning: recent approaches to materials science-a review, J. Phys. Materials 2, 032001 (2019).</footnote>

<footnote n="5">⁵ Carrasquilla, J. & Melko, R. G. Machine learning phases of matter. *Nat. Phys.* **13**, 431–434 (2017).</footnote>

<footnote n="6">⁶ Carleo, G. & Troyer, M. Solving the quantum many-body problem with artificial neural networks. *Science* **355**, 602–606 (2017).</footnote>

<footnote n="7">⁷ Zhang, Y. & Kim, E.-A. Quantum loop topography for machine learning. *Phys. Rev. Lett.* **118**, 216401 (2017).</footnote>

<footnote n="8">⁸ J. Venderley, V. Khemani, E.-A. Kim, “Machine Learning Out-of-Equilibrium Phases of Matter”, Phys. Rev. Lett. **120**, 257204 (2018).</footnote>

<footnote n="9">⁹ Gunning et al. XAI—Explainable artificial intelligence, Sci. Robot. **4**, eaay7120 (2019).</footnote>

<footnote n="10">¹⁰ Y. Zhang, A. Mesaros, K. Fujita, S. D. Edkins, M. H. Hamidian, K. Ch' ng, H. Eisaki, S. Uchida, J.C. Davis, E. and Khatami, E.-A. Kim, “Machine learning in electronic-quantum-matter imaging experiments”, *Nature* **570**, 484 (2019).</footnote>

<footnote n="11">¹¹ S. Ghosh, M. Matty, R. Baumbach, E. Bauer, K. A. Modic, A. Shekhter, J. A. Mydosh, E.-A. Kim, B. J. Ramshaw, Science Advances 6, eaaz4074 (2020).</footnote>

<footnote n="12">¹² Keimer, B., Kivelson, S. A., Norman, M. R., Uchida, S. & Zaanen, J. From quantum matter to high-temperature superconductivity in copper oxides. *Nature* **518**, 179–186 (2015).</footnote>
# Machine Learning and DFT simulations of Quantum Materials

Gabriel R Schleder ¹,², Adalberto Fazzio ¹,²

¹Federal University of ABC, 09210-580, Santo André, São Paulo, Brazil

²Brazilian Nanotechnology National Laboratory (LNNano/CNPEM), 13083-970, Campinas, São Paulo, Brazil

## Status

The microscopic understanding of materials started with the paradigm shift from Classical to Quantum Mechanics in the early 20th century. Based on Schrödinger and Dirac equations, the accurate description of the behavior of materials at the fundamental level was for the first time possible. This enabled the understanding of many phenomena that were not possible until then. Although in principle any problem can be described by quantum mechanics, in practice, even for simple cases this is not possible due to the complexity in solving the many-body problems. One example is superconductivity, which was discovered in 1911 but explained only 46 years later. This led to the famous Dirac remark in 1929 stating that "*the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved*". This remark preludes the imminent central role of Computational Materials Science, presenting the main technical challenges for it to overcome. With the impressive computational advances both in methodologies and technologies, especially since the 1970s, the description and simulation of realistic quantum materials have been possible. Density functional theory (DFT) established itself as the standard tool for simulation of materials systems after success in describing many important physical properties such as ground state structures, relative total energies, and electronic band structures. Subsequently, major milestones were achieved such as the description of structural, electronic, optical, magnetic, catalytic, and quantum properties, both for bulk and nanoscale materials.

Nowadays, with the ever-increasing data generation, including high-throughput techniques [1], and storage from sources as diverse as imaginable, and with the developments in computational and information science, a new data-driven paradigm has emerged (Fig. 1a). Building from experimental, theoretical, and computational science paradigms, the data-driven science paradigm aims to extract knowledge from this huge amount of raw data and information now available [2]. In practice, data-driven science is embodied in novel tools and algorithms; among those, the field of statistical learning developed machine learning (ML) techniques, which are increasingly important [3]. These algorithms generate output results given any kind of numerical input data, whether theoretical or experimental. In materials research, many kinds of problems have been tackled, and different strategies have been thought. These can be broadly classified into learning properties of three types (Fig. 1b): i) pre-Schrödinger equation, such as learning the electronic density [4], to be then used as input for DFT or ML; ii) replacing- or accelerating- Schrödinger equation, thus creating machine-learned approximations to solving the quantum problem; and iii) post-Schrödinger equation, such as directly learning the outputs of the problem, as structures, energies, and materials properties. Examples include the discovery of novel 2D materials and properties [5], learning and predicting novel topological materials [6,7], and creating a twistorics database for identifying strongly correlated behavior in two-dimensional systems [8].
<figure><img src="image_39.png" /><figcaption>Figure 1. (a) Approaches in materials discovery and design: (left) traditional trial and error approach, and (right) data-driven inverse design strategies, optimizing materials target functionalities via high-throughput, machine learning, or evolutionary techniques. Reprinted with permission from [9]. Copyright 2019 American Chemical Society. (b) Machine learning applied to materials simulations in different stages: i) pre-Schrödinger equation, for instance, learning the DFT electronic density [4]; ii) replacing- [10] or accelerating-Schrödinger equation, e.g. Bayesian optimization for geometry relaxation, nudged-elastic band (NEB) calculations, or global optimization [11]; or iii) post-Schrödinger equation, such as directly learning materials properties [5].</figcaption></figure>

Current and Future Challenges

Two main materials science goals are (A) materials design/optimization (local search or exploitation), and (B) materials discovery (global search or exploration). The first aims to find among all known materials, the best candidates for each interest property or application, and then how to improve promising and interesting systems further. The second aims to discover the maximum number of novel materials, exploring simultaneously the atomic (elements), compositional (stoichiometries), and configurational (geometries/structures) spaces. As such, the number of degrees of freedom is immense, and the search (sampling) strategy in this space is of paramount importance.

Currently, both goals present major challenges. The traditional direct approach (A) includes ever-
increasing complex properties such as magnetism, topology, superconductivity, and many others
where quantum and many-body effects are of extreme importance. As such, on the technical side,
more complex calculations demand progressively more computational power, also allowing for larger
realistic systems to be treated. Challenges present for implementing sufficiently higher-level theories
needed to account for quasiparticle and many-body interactions, simultaneously with computational
efficiency for required system sizes. Fundamental challenges also present in developing DFT
functionals better-describing correlation effects. In certain cases, the development of the
fundamental theoretical framework can itself be a challenge, such as the aforementioned
superconductivity description.

In the inverse approach (B), i.e. the search for materials presenting certain desired functionalities,
exploration of the large materials space is fundamental. Therefore, high-throughput, evolutionary,
and machine learning techniques are increasingly important to efficiently sample the materials space,
resulting in screened cases for in-depth investigation. High-throughput approaches present challenges
regarding the construction of consistent databases for expensive calculations, unfeasible to be
performed exhaustively [12]. Although successful, machine learning approaches require four
interdependent components (Fig. 2a), each demanding careful examination and presenting its own
challenges: 1) problem definition, 2) data (inputs), 3) representation, and 4) algorithms and model
training.
(a) Machine Learning
Components

<figure><img src="image_40.png" /></figure>

<figure><img src="image_41.png" /></figure>

<figure><img src="image_42.png" /><figcaption>Figure 2. Machine learning for quantum materials: (a) 4 components. (b) ML representations: essential information should be added to the feature space, balancing the accuracy with efficiency. Adapted with permission from [9]. Copyright 2019 American Chemical Society. (c) Current ML areas for materials science. Atomistic ML allows structural exploration (global search) and properties, while neglecting quantum effects. Model Hamiltonian ML allows exploration of quantum and many-body effects, while hardly applicable to real material systems. Materials or DFT ML must bridge the gap between those areas, possible steps forward are extracting realistic Hamiltonians from DFT calculations [13] and generating wavefunctions from atomistic ML [10].</figcaption></figure>

Advances in Science and Technology to Meet Challenges

Advances in both materials science goals—materials discovery and design—mutually benefit. Beyond
traditional manual developments, artificial intelligence and machine learning techniques can be
broadly employed, improving every problem-solving steps. This includes estimating the computational
cost [14], optimizing simulation and parallelization parameters for faster and numerically efficient
calculations, creating optimized density functionals, obtaining information for use or directly
bypassing simulations, and directly obtaining results and materials properties.

Toward this, machine learning advances in each of its components are needed. First, quantum
materials ML problem definition might be supervised, i.e., regression of a numerical property or
classification among different categories. If on the other hand, an exploratory and unbiased approach
is more suitable, unsupervised tasks such as dimensionality reduction and clustering techniques can
be preferred. Regarding input data, quantum materials generally allow for the creation of smaller
databases, in contrast to big-data datasets. Accordingly, the representation used and the algorithm to
be employed must also be carefully selected. Merging experimental and literature-extracted data can
be valuable [15], with the additional benefit of improving agreement to experiments, as also using
multi-fidelity, data augmentation, and transfer learning techniques. Certainly, the representation, also
called descriptor or fingerprint, is the central element in ML for quantum materials, where nonlocal,
collective, and many-body effects need to be accounted for. The challenge is distilling the fundamental
information from the systems into efficient numerical descriptions (Fig 2b). Insights can be taken from
two more mature ML areas (despite having larger datasets): atomistic potential energy surface (PES)
ML, and many-body Hamiltonian ML (see Roadmap section “Machine Learning Quantum Material
Data”). The first usually uses additive descriptions based on local atomic environments, while the
latter succeeded by translating Hamiltonians into image-like vectors or tensors commonly used with
deep neural networks. Developments for quantum materials lie between these two areas, i.e.
describing quantum phenomena in realistic systems, with ML as a driver for advances (Fig 2c). Lastly,
the algorithm choice must suit the problem data size and representation, for instance, deep learning
can have hundreds to millions of trainable coefficients which is unreasonable for small datasets.

Finally, exciting advances are expected in the area of distilling the data-driven correlation models for
driving theoretical understanding of studied problems. Interpretability of generated ML models is a
first step in this direction, but progress toward physics-enhanced surrogate models [5,16], and the
broader goal of machine-assisted discovery of underlying fundamental physics [17] is essential. In
practice, it is crucial to incorporate machine learning strategies into routine workflows, whether by direct integration into used software or as interoperable standalone software libraries.

## Concluding Remarks

Computational physics has traditionally been at the core of many fundamental advances related to quantum systems and materials. With the development of computational methodologies and high-performance computing and storage technologies, this progress has enabled more and realistic problems to be treated with ever-increasing levels of theory. State-of-the-art DFT has driven the field forward, but it faces challenges to overcome. The expansion in information-generation capability is ideal for data-driven approaches such as machine and deep learning, which can accelerate or make possible knowledge production for the different complex problems related to quantum materials. Challenges exist related to problem definition, data acquisition, efficient and complete representations, and suitable algorithms to be used. The field is still in its beginning, and many opportunities lie ahead for future developments.

## Acknowledgements

GRS and AF acknowledge financial support from the Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), project numbers 2017/18139-6 and 17/02317-2.

## ORCID iDs

Gabriel R. Schleder: https://orcid.org/0000-0003-3129-8682

Adalberto Fazzio: https://orcid.org/0000-0001-5384-7676

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