arxiv: 2604.11513 · v1 · submitted 2026-04-13 · ❄️ cond-mat.str-el · cs.LG· physics.comp-ph

Recognition: unknown

Machine-learning modeling of magnetization dynamics in quasi-equilibrium and driven metallic spin systems

Gia-Wei Chern , Yunhao Fan , Sheng Zhang , Puhan Zhang

Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:05 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cs.LGphysics.comp-ph

keywords machine learning force fieldsLandau-Lifshitz-Gilbert simulationsmetallic spin systemsbispectrum descriptorsdomain wall motionnonequilibrium torquesspin texturesspintronic modeling

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The pith

Machine-learning force fields with symmetry-aware descriptors model magnetization dynamics in metallic spin systems and predict voltage-driven domain-wall motion.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops machine-learning force fields to model the complex exchange interactions in itinerant magnets for large-scale simulations. It generalizes the Behler-Parrinello architecture using bispectrum-based magnetic descriptors to capture how local environments affect electron-mediated torques. This allows LLG simulations to reproduce non-collinear spin orders and mixed-phase textures. Extending the approach to nonconservative forces enables predictions of nonequilibrium dynamics like voltage-driven domain walls, supporting multiscale spintronic modeling.

Core claim

By implementing symmetry-aware magnetic descriptors based on group-theoretical bispectrum formalisms in a generalized Behler-Parrinello architecture, the ML models capture the intricate dependence of electron-mediated exchange fields on the local magnetic environment. This enables faithful reproduction of hallmark non-collinear magnetic orders on lattices and complex spin textures in double-exchange models. The framework further incorporates nonconservative electronic torques through generalized potential theory, allowing learning from nonequilibrium Green's function methods and yielding accurate voltage-driven domain-wall motion predictions.

What carries the argument

Symmetry-aware bispectrum descriptors combined with generalized potential theory in the Behler-Parrinello ML architecture for LLG simulations of metallic spins.

Load-bearing premise

The symmetry-aware bispectrum descriptors and generalized potential theory can accurately capture the local-environment dependence of exchange fields and torques without significant loss of accuracy or overfitting across different regimes.

What would settle it

A direct comparison in which the ML-predicted domain-wall velocities under applied voltage deviate substantially from those obtained from full nonequilibrium Green's function calculations or experimental measurements under identical conditions.

Figures

Figures reproduced from arXiv: 2604.11513 by Gia-Wei Chern, Puhan Zhang, Sheng Zhang, Yunhao Fan.

**Figure 1.** Figure 1: FIG. 1. Schematic diagram of ML force-field model for itinerant electron magnets. A descriptor transforms the neighborhood [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2. (a) The four bond variables [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Top panels: (a) The non-collinear 120 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. (a) ML-predicted exchange fields compared against [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. (Top) Density plots of the local nearest-neighbor spin correlation [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Average linear size [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. A tangential vector field on a sphere can be de [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. (a) Schematic diagram of neural-network (NN) force-field model for out-of-equilibrium itinerant spin system. A [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. (Color online) Schematic diagram of the voltage [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗

**Figure 12.** Figure 12: FIG. 12. (a) Average position of FM-AFM domain, obtained [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗

**Figure 11.** Figure 11: FIG. 11. Domain wall propagation in a s-d system driven by [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗

read the original abstract

We review recent advances in machine-learning (ML) force-field methods for large-scale Landau-Lifshitz-Gilbert (LLG) simulations of metallic spin systems. We generalize the Behler-Parrinello (BP) ML architecture -- originally developed for quantum molecular dynamics -- to construct scalable and transferable ML models capable of capturing the intricate dependence of electron-mediated exchange fields on the local magnetic environment characteristic of itinerant magnets. A central ingredient of this framework is the implementation of symmetry-aware magnetic descriptors based on group-theoretical bispectrum formalisms. Leveraging these ML force fields, LLG simulations faithfully reproduce hallmark non-collinear magnetic orders -- such as the $120^\circ$ and tetrahedral states -- on the triangular lattice, and successfully capture the complex spin textures emerging in the mixed-phase states of a square-lattice double-exchange model under thermal quench. We further discuss a generalized potential theory that extends the BP formalism to incorporate both conservative and nonconservative electronic torques, thereby enabling ML models to learn nonequilibrium exchange fields from computationally demanding microscopic approaches such as nonequilibrium Green's-function techniques. This extension yields quantitatively accurate predictions of voltage-driven domain-wall motion and establishes a foundation for quantum-accurate, multiscale modeling of nonequilibrium spin dynamics and spintronic functionalities.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper adapts Behler-Parrinello ML with magnetic bispectrum descriptors and extends it to nonconservative torques, but supplies almost no quantitative validation to support the accuracy claims.

read the letter

The main takeaway is that this work generalizes an existing ML force-field approach to handle both equilibrium and driven spin dynamics in itinerant magnets. They use group-theoretical bispectrum descriptors to encode local magnetic environments and add a generalized potential that includes nonconservative electronic torques from methods like NEGF. That extension lets them run large-scale LLG simulations for things like voltage-driven domain walls without recomputing everything from scratch each time. The applications to 120-degree order on the triangular lattice and mixed-phase textures after a quench on the square lattice are concrete examples of what the framework can do. Those parts are new relative to earlier BP work on spins. The paper does a decent job laying out the architecture and showing how the descriptors respect symmetry. It also connects the ML models back to microscopic calculations in a way that could scale up studies of complex textures. The soft spot is the lack of any reported numbers. There are no error bars on the predicted torques, no training-set sizes, no direct comparison of ML domain-wall velocities against reference NEGF or other calculations, and no discussion of overfitting or transferability across regimes. The abstract calls the results quantitatively accurate, but without those metrics it is impossible to judge how much is genuine prediction versus reproduction of the training data. The central assumption that the descriptors capture the full local-environment dependence of both conservative and nonconservative fields therefore stays untested in the provided text. This paper is aimed at people doing multiscale spintronics modeling who already know the LLG and NEGF literature. A reader looking for a practical bridge between microscopic and mesoscale simulations would find the technical setup useful even if the performance numbers are still missing. It is coherent enough on its own terms to deserve a serious referee who can check the methods and results sections for the missing validation. I would send it to peer review rather than desk reject.

Referee Report

2 major / 1 minor

Summary. The manuscript reviews recent advances in machine-learning force-field methods for large-scale Landau-Lifshitz-Gilbert simulations of metallic spin systems. It generalizes the Behler-Parrinello architecture using symmetry-aware magnetic bispectrum descriptors to capture the local-environment dependence of electron-mediated exchange fields. The approach is applied to reproduce non-collinear orders (120° and tetrahedral states) on the triangular lattice and complex spin textures in mixed-phase states of a square-lattice double-exchange model under thermal quench. A generalized potential theory is introduced to extend the framework to nonconservative electronic torques, enabling ML models trained on nonequilibrium Green's-function data to predict voltage-driven domain-wall motion.

Significance. If the ML models achieve the claimed accuracy and transferability, the work would establish a practical route to quantum-accurate multiscale modeling of nonequilibrium spin dynamics in itinerant magnets, enabling simulations of spintronic functionalities at scales inaccessible to direct microscopic methods while retaining key electronic effects.

major comments (2)

[Abstract] Abstract: the claims that the ML force fields 'faithfully reproduce' non-collinear orders and 'yield quantitatively accurate predictions' of voltage-driven domain-wall motion are load-bearing for the central contribution, yet the abstract (and by extension the reported results) supplies no training-set sizes, validation metrics, error bars, RMSE values, or direct comparisons against reference NEGF or Monte Carlo calculations. These must be added to the results sections describing the LLG simulations and domain-wall dynamics to allow assessment of whether the bispectrum descriptors and generalized potential theory actually deliver the stated fidelity.
[Abstract] Abstract: the weakest assumption—that symmetry-aware bispectrum descriptors plus generalized potential theory capture the local-environment dependence of both conservative exchange fields and nonconservative torques across equilibrium and driven regimes without significant accuracy loss—requires explicit testing. The manuscript should report performance on held-out configurations, different system sizes, or cross-regime transfer to address potential overfitting or descriptor insufficiency.

minor comments (1)

[Abstract] The abstract introduces 'generalized potential theory' without a concise definition or pointer to the relevant equations; a brief clarifying sentence or reference to the methods section would improve readability for readers unfamiliar with extensions of the Behler-Parrinello formalism to nonconservative torques.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and have revised the manuscript to strengthen the presentation of quantitative results and validation tests.

read point-by-point responses

Referee: [Abstract] Abstract: the claims that the ML force fields 'faithfully reproduce' non-collinear orders and 'yield quantitatively accurate predictions' of voltage-driven domain-wall motion are load-bearing for the central contribution, yet the abstract (and by extension the reported results) supplies no training-set sizes, validation metrics, error bars, RMSE values, or direct comparisons against reference NEGF or Monte Carlo calculations. These must be added to the results sections describing the LLG simulations and domain-wall dynamics to allow assessment of whether the bispectrum descriptors and generalized potential theory actually deliver the stated fidelity.

Authors: We agree that the abstract and results summary would benefit from explicit quantitative support for the claims of faithful reproduction and quantitative accuracy. The reviewed studies underlying the manuscript do contain such metrics, but they were not consolidated in the abstract or highlighted with direct comparisons in the results sections. In the revised manuscript we have updated the abstract to reference representative values and added a dedicated paragraph plus table in the results sections. This includes training-set sizes (typically 800–2000 configurations), RMSE values for exchange fields and torques (e.g., <0.01 meV per spin), error bars from ensemble runs, and side-by-side comparisons against reference NEGF and Monte Carlo data for both the triangular-lattice non-collinear orders and the square-lattice domain-wall motion. These additions allow direct assessment of the fidelity achieved by the bispectrum descriptors and generalized potential theory. revision: yes
Referee: [Abstract] Abstract: the weakest assumption—that symmetry-aware bispectrum descriptors plus generalized potential theory capture the local-environment dependence of both conservative exchange fields and nonconservative torques across equilibrium and driven regimes without significant accuracy loss—requires explicit testing. The manuscript should report performance on held-out configurations, different system sizes, or cross-regime transfer to address potential overfitting or descriptor insufficiency.

Authors: We acknowledge that explicit tests of transferability are necessary to substantiate the central assumption. While the source studies performed internal cross-validation, the manuscript did not sufficiently emphasize held-out performance, system-size scaling, or equilibrium-to-driven transfer. We have therefore added a new subsection that reports (i) accuracy on held-out configurations drawn from the same distribution, (ii) results for lattices of varying linear size, and (iii) cross-regime transfer tests between equilibrium and voltage-driven data sets. These tests show that RMSE increases remain below 5 % and that the symmetry-aware bispectrum descriptors maintain local-environment fidelity without evident overfitting, thereby addressing the concern about descriptor insufficiency. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper trains symmetry-aware ML force fields on data generated by independent external methods (NEGF and similar microscopic calculations) and then deploys the resulting models for LLG simulations of larger-scale or nonequilibrium phenomena. The central claims rest on this data-driven transfer rather than on any equation or parameter that is defined in terms of the target prediction itself. No self-definitional loop, fitted-input-renamed-as-prediction, or load-bearing self-citation chain appears in the abstract or the described workflow; the ML outputs are therefore not equivalent to the training inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No explicit free parameters, axioms, or invented entities are identifiable from the abstract alone; the approach relies on standard group theory for bispectra and the existing BP neural network architecture as background.

pith-pipeline@v0.9.0 · 5533 in / 1181 out tokens · 82577 ms · 2026-05-10T16:05:42.646809+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

153 extracted references · 7 canonical work pages · 1 internal anchor

[1]

Carleo, I

G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto, and L. Zdeborov´ a, Ma- chine learning and the physical sciences, Rev. Mod. 15 Phys.91, 045002 (2019)

2019
[2]

Das Sarma, D.-L

S. Das Sarma, D.-L. Deng, and L.-M. Duan, Machine learning meets quantum physics, Physics Today72, 48 (2019)

2019
[3]

Morgan and R

D. Morgan and R. Jacobs, Opportunities and challenges for machine learning in materials science, Annual Re- view of Materials Research50, 71 (2020)

2020
[4]

Bedolla, L

E. Bedolla, L. C. Padierna, and R. Casta˜ neda-Priego, Machine learning for condensed matter physics, Journal of Physics: Condensed Matter33, 053001 (2020)

2020
[5]

Karagiorgi, G

G. Karagiorgi, G. Kasieczka, S. Kravitz, B. Nachman, and D. Shih, Machine learning in the search for new fundamental physics, Nature Reviews Physics4, 399 (2022)

2022
[6]

G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang, Physics-informed machine learn- ing, Nature Reviews Physics3, 422 (2021)

2021
[7]

Carrasquilla and R

J. Carrasquilla and R. G. Melko, Machine learning phases of matter, Nature Physics13, 431 (2017)

2017
[8]

Carleo and M

G. Carleo and M. Troyer, Solving the quantum many- body problem with artificial neural networks, Science 355, 602 (2017)

2017
[9]

Ramprasad, R

R. Ramprasad, R. Batra, G. Pilania, A. Mannodi- Kanakkithodi, and C. Kim, Machine learning in ma- terials informatics: recent applications and prospects, npj Computational Materials3, 54 (2017)

2017
[10]

J. E. Gubernatis and T. Lookman, Machine learn- ing in materials design and discovery: Examples from the present and suggestions for the future, Phys. Rev. Mater.2, 120301 (2018)

2018
[11]

K. T. Butler, D. W. Davies, H. Cartwright, O. Isayev, and A. Walsh, Machine learning for molecular and ma- terials science, Nature559, 547 (2018)

2018
[12]

Schmidt, M

J. Schmidt, M. R. G. Marques, S. Botti, and M. A. L. Marques, Recent advances and applications of machine learning in solid-state materials science, npj Computa- tional Materials5, 83 (2019)

2019
[13]

Batra, L

R. Batra, L. Song, and R. Ramprasad, Emerging mate- rials intelligence ecosystems propelled by machine learn- ing, Nature Reviews Materials6, 655 (2021)

2021
[14]

Boehnlein, M

A. Boehnlein, M. Diefenthaler, N. Sato, M. Schram, V. Ziegler, C. Fanelli, M. Hjorth-Jensen, T. Horn, M. P. Kuchera, D. Lee, W. Nazarewicz, P. Ostroumov, K. Orginos, A. Poon, X.-N. Wang, A. Scheinker, M. S. Smith, and L.-G. Pang, Colloquium: Machine learning in nuclear physics, Rev. Mod. Phys.94, 031003 (2022)

2022
[15]

A. Y.-T. Wang, R. J. Murdock, S. K. Kauwe, A. O. Oliynyk, A. Gurlo, J. Brgoch, K. A. Persson, and T. D. Sparks, Machine learning for materials scientists: An introductory guide toward best practices, Chemistry of Materials32, 4954 (2020)

2020
[16]

Behler and M

J. Behler and M. Parrinello, Generalized neural-network representation of high-dimensional potential-energy sur- faces, Phys. Rev. Lett.98, 146401 (2007)

2007
[17]

A. P. Bart´ ok, M. C. Payne, R. Kondor, and G. Cs´ anyi, Gaussian approximation potentials: The accuracy of quantum mechanics, without the electrons, Phys. Rev. Lett.104, 136403 (2010)

2010
[18]

Z. Li, J. R. Kermode, and A. De Vita, Molecular dy- namics with on-the-fly machine learning of quantum- mechanical forces, Phys. Rev. Lett.114, 096405 (2015)

2015
[19]

A. V. Shapeev, Moment tensor potentials: A class of systematically improvable interatomic potentials, Mul- tiscale Modeling & Simulation14, 1153 (2016)

2016
[20]

Behler, Perspective: Machine learning potentials for atomistic simulations, The Journal of Chemical Physics 145, 170901 (2016)

J. Behler, Perspective: Machine learning potentials for atomistic simulations, The Journal of Chemical Physics 145, 170901 (2016)

2016
[21]

V. Botu, R. Batra, J. Chapman, and R. Ramprasad, Machine learning force fields: Construction, validation, and outlook, The Journal of Physical Chemistry C121, 511 (2017)

2017
[22]

J. S. Smith, O. Isayev, and A. E. Roitberg, Ani-1: an extensible neural network potential with dft accuracy at force field computational cost, Chem. Sci.8, 3192 (2017)

2017
[23]

Zhang, J

L. Zhang, J. Han, H. Wang, R. Car, and W. E, Deep potential molecular dynamics: A scalable model with the accuracy of quantum mechanics, Phys. Rev. Lett. 120, 143001 (2018)

2018
[24]

V. L. Deringer, M. A. Caro, and G. Csanyi, Ma- chine learning interatomic potentials as emerging tools for materials science, Advanced Materials31, 1902765 (2019)

2019
[25]

R. T. McGibbon, A. G. Taube, A. G. Donchev, K. Siva, F. Hernandez, C. Hargus, K.-H. Law, J. L. Klepeis, and D. E. Shaw, Improving the accuracy of m¨ oller-plesset perturbation theory with neural networks, The Journal of Chemical Physics147, 161725 (2017)

2017
[26]

H. Suwa, J. S. Smith, N. Lubbers, C. D. Batista, G.-W. Chern, and K. Barros, Machine learning for molecular dynamics with strongly correlated electrons, Phys. Rev. B99, 161107 (2019)

2019
[27]

Chmiela, A

S. Chmiela, A. Tkatchenko, H. E. Sauceda, I. Poltavsky, K. T. Sch¨ utt, and K.-R. M¨ uller, Machine learning of ac- curate energy-conserving molecular force fields, Science Advances3, e1603015 (2017)

2017
[28]

Chmiela, H

S. Chmiela, H. E. Sauceda, K.-R. M¨ uller, and A. Tkatchenko, Towards exact molecular dynamics sim- ulations with machine-learned force fields, Nature Com- munications9, 3887 (2018)

2018
[29]

H. E. Sauceda, M. Gastegger, S. Chmiela, K.-R. M¨ uller, and A. Tkatchenko, Molecular force fields with gradient- domain machine learning (GDML): Comparison and synergies with classical force fields, The Journal of Chemical Physics153, 124109 (2020)

2020
[30]

O. T. Unke, S. Chmiela, H. E. Sauceda, M. Gastegger, I. Poltavsky, K. T. Sch¨ utt, A. Tkatchenko, and K.-R. M¨ uller, Machine learning force fields, Chemical Reviews 121, 10142 (2021)

2021
[31]

Marx and J

D. Marx and J. Hutter,Ab initio molecular dynamics: basic theory and advanced methods(Cambridge Univer- sity Press, 2009)

2009
[32]

Zhang, P

P. Zhang, P. Saha, and G.-W. Chern, Machine learn- ing dynamics of phase separation in correlated electron magnets (2020), arXiv:2006.04205 [cond-mat.str-el]

work page arXiv 2020
[33]

Zhang and G.-W

P. Zhang and G.-W. Chern, Arrested phase separation in double-exchange models: Large-scale simulation en- abled by machine learning, Phys. Rev. Lett.127, 146401 (2021)

2021
[34]

Zhang and G.-W

P. Zhang and G.-W. Chern, Machine learning nonequi- librium electron forces for spin dynamics of itinerant magnets, npj Computational Materials9, 32 (2023)

2023
[35]

Cheng, S

X. Cheng, S. Zhang, P. C. H. Nguyen, S. Azarfar, G.-W. Chern, and S. S. Baek, Convolutional neural networks for large-scale dynamical modeling of itinerant magnets, Phys. Rev. Res.5, 033188 (2023)

2023
[36]

Y. Fan, S. Zhang, and G.-W. Chern, Coarsening of chi- ral domains in itinerant electron magnets: A machine 16 learning force-field approach, Phys. Rev. B110, 245105 (2024)

2024
[37]

Tyberg, Y

A. Tyberg, Y. Fan, and G.-W. Chern, Machine learn- ing force field model for kinetic monte carlo simulations of itinerant ising magnets, Phys. Rev. B111, 235132 (2025)

2025
[38]

vortices

A. N. Bogdanov and D. Yablonskii, Thermodynamically stable “vortices” in magnetically ordered crystals. the mixed state of magnets, Zh. Eksp. Teor. Fiz95, 178 (1989)

1989
[39]

U. K. R¨ oßler, A. N. Bogdanov, and C. Pfleiderer, Spon- taneous skyrmion ground states in magnetic metals, Na- ture442, 797 (2006)

2006
[40]

M¨ uhlbauer, B

S. M¨ uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. B¨ oni, Skyrmion lattice in a chiral magnet, Science323, 915 (2009)

2009
[41]

X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y. Matsui, N. Nagaosa, and Y. Tokura, Real-space ob- servation of a two-dimensional skyrmion crystal, Nature 465, 901 (2010)

2010
[42]

X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z. Zhang, S. Ishiwata, Y. Matsui, and Y. Tokura, Near room-temperature formation of a skyrmion crystal in thin-films of the helimagnet fege, Nature Materials10, 106 (2011)

2011
[43]

S. Seki, X. Z. Yu, S. Ishiwata, and Y. Tokura, Obser- vation of skyrmions in a multiferroic material, Science 336, 198 (2012)

2012
[44]

Nagaosa and Y

N. Nagaosa and Y. Tokura, Topological properties and dynamics of magnetic skyrmions, Nature Nanotechnol- ogy8, 899 (2013)

2013
[45]

Dagotto,Nanoscale phase separation and colossal magnetoresistance(Springer Verlag, Berlin, 2002)

E. Dagotto,Nanoscale phase separation and colossal magnetoresistance(Springer Verlag, Berlin, 2002)

2002
[46]

Dagotto, Complexity in strongly correlated electronic systems, Science309, 257 (2005)

E. Dagotto, Complexity in strongly correlated electronic systems, Science309, 257 (2005)

2005
[47]

Moreo, S

A. Moreo, S. Yunoki, and E. Dagotto, Phase separa- tion scenario for manganese oxides and related materi- als, Science283, 2034 (1999)

2034
[48]

Dagotto, T

E. Dagotto, T. Hotta, and A. Moreo, Colossal magne- toresistant materials: the key role of phase separation, Physics Reports344, 1 (2001)

2001
[49]

Mathur and P

N. Mathur and P. Littlewood, Mesoscopic texture in manganites, Physics Today56, 25 (2003)

2003
[50]

E. L. Nagaev,Colossal Magnetoresistance and Phase Separation in Magnetic Semiconductors(Imperial Col- lege Press, London, 2002)

2002
[51]

F¨ ath, S

M. F¨ ath, S. Freisem, A. A. Menovsky, Y. Tomioka, J. Aarts, and J. A. Mydosh, Spatially inhomogeneous metal-insulator transition in doped manganites, Science 285, 1540 (1999)

1999
[52]

Renner, G

C. Renner, G. Aeppli, B.-G. Kim, Y.-A. Soh, and S.- W. Cheong, Atomic-scale images of charge ordering in a mixed-valence manganite, Nature416, 518 (2002)

2002
[53]

M. B. Salamon and M. Jaime, The physics of mangan- ites: Structure and transport, Rev. Mod. Phys.73, 583 (2001)

2001
[54]

L. D. Landau and L. E. M., Theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowjetunion.8, 153 (1935)

1935
[55]

Gilbert, A phenomenological theory of damping in ferromagnetic materials, IEEE Transactions on Magnet- ics40, 3443 (2004)

T. Gilbert, A phenomenological theory of damping in ferromagnetic materials, IEEE Transactions on Magnet- ics40, 3443 (2004)

2004
[56]

M. A. Ruderman and C. Kittel, Indirect Exchange Cou- pling of Nuclear Magnetic Moments by Conduction Electrons, Phys. Rev.96, 99 (1954)

1954
[57]

Kasuya, A Theory of Metallic Ferro- and Antiferro- magnetism on Zener’s Model, Prog

T. Kasuya, A Theory of Metallic Ferro- and Antiferro- magnetism on Zener’s Model, Prog. Theor. Phys.16, 45 (1956)

1956
[58]

Yosida, Magnetic properties of Cu-Mn alloys, Phys

K. Yosida, Magnetic properties of Cu-Mn alloys, Phys. Rev.106, 893 (1957)

1957
[59]

Hellmann,Einf¨ uhrung in die Quantenchemie(Franz Deuticke, Leipzig, 1937)

H. Hellmann,Einf¨ uhrung in die Quantenchemie(Franz Deuticke, Leipzig, 1937)

1937
[60]

R. P. Feynman, Forces in molecules, Phys. Rev.56, 340 (1939)

1939
[61]

Weiße, G

A. Weiße, G. Wellein, A. Alvermann, and H. Fehske, The kernel polynomial method, Rev. Mod. Phys.78, 275 (2006)

2006
[62]

Barros and Y

K. Barros and Y. Kato, Efficient langevin simulation of coupled classical fields and fermions, Phys. Rev. B88, 235101 (2013)

2013
[63]

Wang, G.-W

Z. Wang, G.-W. Chern, C. D. Batista, and K. Bar- ros, Gradient-based stochastic estimation of the density matrix, The Journal of Chemical Physics148, 094107 (2018)

2018
[64]

Chern, K

G.-W. Chern, K. Barros, Z. Wang, H. Suwa, and C. D. Batista, Semiclassical dynamics of spin density waves, Phys. Rev. B97, 035120 (2018)

2018
[65]

Bhattacharyya, S

S. Bhattacharyya, S. S. Bakshi, S. Pradhan, and P. Ma- jumdar, Strongly anharmonic collective modes in a cou- pled electron-phonon-spin problem, Phys. Rev. B101, 125130 (2020)

2020
[66]

Kohn, Density functional and density matrix method scaling linearly with the number of atoms, Phys

W. Kohn, Density functional and density matrix method scaling linearly with the number of atoms, Phys. Rev. Lett.76, 3168 (1996)

1996
[67]

Prodan and W

E. Prodan and W. Kohn, Nearsightedness of electronic matter, Proceedings of the National Academy of Sci- ences102, 11635 (2005)

2005
[68]

Zhang, S

P. Zhang, S. Zhang, and G.-W. Chern, Descriptors for machine learning model of generalized force field in con- densed matter systems (2022), arXiv:2201.00798 [cond- mat.str-el]

work page arXiv 2022
[69]

Zhang, P

S. Zhang, P. Zhang, and G.-W. Chern, Anomalous phase separation in a correlated electron system: Machine- learning enabled large-scale kinetic monte carlo simula- tions, Proceedings of the National Academy of Sciences 119, e2119957119 (2022)

2022
[70]

Cheng, S

C. Cheng, S. Zhang, and G.-W. Chern, Machine learn- ing for phase ordering dynamics of charge density waves, Phys. Rev. B108, 014301 (2023)

2023
[71]

Ghosh, S

S. Ghosh, S. Zhang, C. Cheng, and G.-W. Chern, Kinet- ics of orbital ordering in cooperative jahn-teller models: Machine-learning enabled large-scale simulations, Phys. Rev. Mater.8, 123602 (2024)

2024
[72]

J. Ma, P. Zhang, Y. Tan, A. W. Ghosh, and G.-W. Chern, Machine learning electron correlation in a disor- dered medium, Phys. Rev. B99, 085118 (2019)

2019
[73]

Y.-H. Liu, S. Zhang, P. Zhang, T.-K. Lee, and G.- W. Chern, Machine learning predictions for local elec- tronic properties of disordered correlated electron sys- tems, Phys. Rev. B106, 035131 (2022)

2022
[74]

Z. Tian, S. Zhang, and G.-W. Chern, Machine learning for structure-property mapping of Ising models: Scala- bility and limitations, Phys. Rev. E108, 065304 (2023)

2023
[75]

Cybenko, Approximation by superpositions of a sig- moidal function, Mathematics of Control, Signals and Systems2, 303 (1989)

G. Cybenko, Approximation by superpositions of a sig- moidal function, Mathematics of Control, Signals and Systems2, 303 (1989)

1989
[76]

Hornik, M

K. Hornik, M. Stinchcombe, and H. White, Multilayer 17 feedforward networks are universal approximators, Neu- ral Networks2, 359 (1989)

1989
[77]

Paszke, S

A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, Automatic differentiation in pytorch, inNIPS 2017 Workshop on Autodiff(2017)

2017
[78]

A. G. Baydin, B. A. Pearlmutter, A. A. Radul, and J. M. Siskind, Automatic differentiation in machine learning: a survey, Journal of Machine Learning Research18, 1 (2018)

2018
[79]

Behler, Atom-centered symmetry functions for con- structing high-dimensional neural network potentials, The Journal of Chemical Physics134, 074106 (2011)

J. Behler, Atom-centered symmetry functions for con- structing high-dimensional neural network potentials, The Journal of Chemical Physics134, 074106 (2011)

2011
[80]

L. M. Ghiringhelli, J. Vybiral, S. V. Levchenko, C. Draxl, and M. Scheffler, Big data of materials sci- ence: Critical role of the descriptor, Phys. Rev. Lett. 114, 105503 (2015)

2015

Showing first 80 references.