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Hamiltonian learning quantum magnets with dynamical impurity tomography

Netta Karjalainen1,2, Greta Lupi2, Rouven Koch3, Adolfo O. Fumega2, and Jose L. Lado2,*

  • *Contact author: jose.lado@aalto.fi
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Phys. Rev. Research 8, 033281 – Published 8 September, 2026

DOI: https://doi.org/10.1103/cw27-2qqd

Abstract

Nanoscale engineered spin systems, ranging from spins on surfaces to nanographenes, provide flexible platforms to realize entangled quantum magnets from a bottom-up approach. However, assessing the quantum many-body Hamiltonian realized in a specific experiment remains an exceptional open challenge, due to the difficulty of disentangling competing terms accounting for the many-body excitations. Here, we demonstrate a machine learning strategy to learn a quantum many-body spin Hamiltonian from scanning spectroscopy measurements of spin excitations. Our methodology leverages the spatially resolved reconstruction of the many-body excitations induced by depositing quantum impurities next to the quantum magnet. We demonstrate that our algorithm allows us to predict long-range Heisenberg exchange interactions, anisotropic exchange, and antisymmetric Dzyaloshinskii-Moriya interaction, including in the presence of sizable noise. Our methodology establishes defect-induced spatially resolved dynamical excitations in quantum magnets as a powerful strategy to understand the nature of quantum spin many-body models.

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References (103)

  1. L. Savary and L. Balents, Quantum spin liquids: A review, Rep. Prog. Phys. 80, 016502 (2017).
  2. C. Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R. Norman, and T. Senthil, Quantum spin liquids, Science 367, eaay0668 (2020).
  3. H. Wang, P. Fan, J. Chen, L. Jiang, H.-J. Gao, J. L. Lado, and K. Yang, Construction of topological quantum magnets from atomic spins on surfaces, Nat. Nanotechnol. 19, 1782 (2024).
  4. A. Hirohata, K. Yamada, Y. Nakatani, I.-L. Prejbeanu, B. Diény, P. Pirro, and B. Hillebrands, Review on spintronics: Principles and device applications, J. Magn. Magn. Mater. 509, 166711 (2020).
  5. M. Fukami, J. C. Marcks, D. R. Candido, L. R. Weiss, B. Soloway, S. E. Sullivan, N. Delegan, F. J. Heremans, M. E. Flatté, and D. D. Awschalom, Magnon-mediated qubit coupling determined via dissipation measurements, Proc. Natl. Acad. Sci. USA 121, e2313754120 (2024).
  6. Y. Wang, Y. Chen, H. T. Bui, C. Wolf, M. Haze, C. Mier, J. Kim, D.-J. Choi, C. P. Lutz, Y. Bae, S.-h. Phark, and A. J. Heinrich, An atomic-scale multi-qubit platform, Science 382, 87 (2023).
  7. D. M. Eigler and E. K. Schweizer, Positioning single atoms with a scanning tunnelling microscope, Nature (London) 344, 524 (1990).
  8. K. Yang, W. Paul, S.-H. Phark, P. Willke, Y. Bae, T. Choi, T. Esat, A. Ardavan, A. J. Heinrich, and C. P. Lutz, Coherent spin manipulation of individual atoms on a surface, Science 366, 509 (2019).
  9. A. Spinelli, B. Bryant, F. Delgado, J. Fernández-Rossier, and A. F. Otte, Imaging of spin waves in atomically designed nanomagnets, Nat. Mater. 13, 782 (2014).
  10. R. Drost, S. Kezilebieke, J. L. Lado, and P. Liljeroth, Real-space imaging of triplon excitations in engineered quantum magnets, Phys. Rev. Lett. 131, 086701 (2023).
  11. A. J. Heinrich, J. A. Gupta, C. P. Lutz, and D. M. Eigler, Single-atom spin-flip spectroscopy, Science 306, 466 (2004).
  12. S. Baumann, W. Paul, T. Choi, C. P. Lutz, A. Ardavan, and A. J. Heinrich, Electron paramagnetic resonance of individual atoms on a surface, Science 350, 417 (2015).
  13. W. Paul, S. Baumann, C. P. Lutz, and A. J. Heinrich, Generation of constant-amplitude radio-frequency sweeps at a tunnel junction for spin resonance STM, Rev. Sci. Instrum. 87, 074703 (2016).
  14. S. Phark, H. T. Bui, A. Ferrón, J. Fernández‐Rossier, J. Reina‐Gálvez, C. Wolf, Y. Wang, K. Yang, A. J. Heinrich, and C. P. Lutz, Electric‐field‐driven spin resonance by on‐surface exchange coupling to a single‐atom magnet, Adv. Sci. 10, 2302033 (2023).
  15. J. L. Lado, A. Ferrón, and J. Fernández-Rossier, Exchange mechanism for electron paramagnetic resonance of individual adatoms, Phys. Rev. B 96, 205420 (2017).
  16. D.-J. Choi, N. Lorente, J. Wiebe, K. von Bergmann, A. F. Otte, and A. J. Heinrich, Colloquium: Atomic spin chains on surfaces, Rev. Mod. Phys. 91, 041001 (2019).
  17. K. Sun, N. Cao, O. J. Silveira, A. O. Fumega, F. Hanindita, S. Ito, J. L. Lado, P. Liljeroth, A. S. Foster, and S. Kawai, On-surface synthesis of Heisenberg spin-1/2 antiferromagnetic molecular chains, Sci. Adv. 11, eads1641 (2025).
  18. C. Zhao, G. Catarina, J.-J. Zhang, J. C. G. Henriques, L. Yang, J. Ma, X. Feng, O. Gröning, P. Ruffieux, J. Fernández-Rossier, and R. Fasel, Tunable topological phases in nanographene-based spin-1/2 alternating-exchange Heisenberg chains, Nat. Nanotechnol. 19, 1789 (2024).
  19. K. Yang, Y. Bae, W. Paul, F. D. Natterer, P. Willke, J. L. Lado, A. Ferrón, T. Choi, J. Fernández-Rossier, A. J. Heinrich, and C. P. Lutz, Engineering the eigenstates of coupled spin-1/2 atoms on a surface, Phys. Rev. Lett. 119, 227206 (2017).
  20. R. Kawaguchi, K. Hashimoto, T. Kakudate, K. Katoh, M. Yamashita, and T. Komeda, Spatially resolving electron spin resonance of π-radical in single-molecule magnet, Nano Lett. 23, 213 (2023).
  21. J. Friedel, XIV. The distribution of electrons round impurities in monovalent metals, London Edinburgh Dublin Philos. Mag. J. Sci. 43, 153 (1952).
  22. M. F. Crommie, C. P. Lutz, and D. M. Eigler, Imaging standing waves in a two-dimensional electron gas, Nature (London) 363, 524 (1993).
  23. G. M. Rutter, J. N. Crain, N. P. Guisinger, T. Li, P. N. First, and J. A. Stroscio, Scattering and interference in epitaxial graphene, Science 317, 219 (2007).
  24. N. Avraham, J. Reiner, A. Kumar‐Nayak, N. Morali, R. Batabyal, B. Yan, and H. Beidenkopf, Quasiparticle interference studies of quantum materials, Adv. Mater. 30, 1707628 (2018).
  25. Y. Guan, C. Dutreix, H. González-Herrero, M. M. Ugeda, I. Brihuega, M. I. Katsnelson, O. V. Yazyev, and V. T. Renard, Observation of Kekulé vortices around hydrogen adatoms in graphene, Nat. Commun. 15, 2927 (2024).
  26. P. Roushan, J. Seo, C. V. Parker, Y. S. Hor, D. Hsieh, D. Qian, A. Richardella, M. Z. Hasan, R. J. Cava, and A. Yazdani, Topological surface states protected from backscattering by chiral spin texture, Nature (London) 460, 1106 (2009).
  27. Z. Alpichshev, J. G. Analytis, J.-H. Chu, I. R. Fisher, Y. L. Chen, Z. X. Shen, A. Fang, and A. Kapitulnik, STM imaging of electronic waves on the surface of bi2te3: Topologically protected surface states and hexagonal warping effects, Phys. Rev. Lett. 104, 016401 (2010).
  28. H. Zheng and M. Zahid Hasan, Quasiparticle interference on type-I and type-II Weyl semimetal surfaces: A review, Adv. Phys.: X 3, 1466661 (2018).
  29. C. Dutreix, H. González-Herrero, I. Brihuega, M. I. Katsnelson, C. Chapelier, and V. T. Renard, Measuring the Berry phase of graphene from wavefront dislocations in Friedel oscillations, Nature (London) 574, 219 (2019).
  30. G. Chen and J. L. Lado, Impurity-induced resonant spinon zero modes in Dirac quantum spin liquids, Phys. Rev. Res. 2, 033466 (2020).
  31. Y. Chen, W.-Y. He, W. Ruan, J. Hwang, S. Tang, R. L. Lee, M. Wu, T. Zhu, C. Zhang, H. Ryu, F. Wang, S. G. Louie, Z.-X. Shen, S.-K. Mo, P. A. Lee, and M. F. Crommie, Evidence for a spinon Kondo effect in cobalt atoms on single-layer 1T−TaSe2, Nat. Phys. 18, 1335 (2022).
  32. W.-Y. He and P. A. Lee, Magnetic impurity as a local probe of the u(1) quantum spin liquid with spinon Fermi surface, Phys. Rev. B 105, 195156 (2022).
  33. M. O. Takahashi, W.-H. Kao, S. Fujimoto, and N. B. Perkins, Z2 flux binding to higher-spin impurities in the Kitaev spin liquid, npj Quantum Mater. 10, 14 (2025).
  34. E. J. König, M. T. Randeria, and B. Jäck, Tunneling spectroscopy of quantum spin liquids, Phys. Rev. Lett. 125, 267206 (2020).
  35. W. Ruan, Y. Chen, S. Tang, J. Hwang, H.-Z. Tsai, R. L. Lee, M. Wu, H. Ryu, S. Kahn, F. Liou, C. Jia, A. Aikawa, C. Hwang, F. Wang, Y. Choi, S. G. Louie, P. A. Lee, Z.-X. Shen, S.-K. Mo, and M. F. Crommie, Evidence for quantum spin liquid behaviour in single-layer 1T−TaSe2 from scanning tunnelling microscopy, Nat. Phys. 17, 1154 (2021).
  36. S. C. Ganguli, M. Aapro, S. Kezilebieke, M. Amini, J. L. Lado, and P. Liljeroth, Visualization of moiré magnons in monolayer ferromagnet, Nano Lett. 23, 3412 (2023).
  37. H. Deng, T. Yang, G. Liu, L. Liu, L. Zhao, W. Wang, T. Li, W. Song, T. Neupert, X.-R. Liu, et al., Local excitation of kagome spin ice magnetism seen by scanning tunneling microscopy, Phys. Rev. Lett. 133, 046503 (2024).
  38. H. Wang, M. Madami, J. Chen, H. Jia, Y. Zhang, R. Yuan, Y. Wang, W. He, L. Sheng, Y. Zhang, J. Wang, S. Liu, K. Shen, G. Yu, X. Han, D. Yu, J.-P. Ansermet, G. Gubbiotti, and H. Yu, Observation of spin-wave moiré edge and cavity modes in twisted magnetic lattices, Phys. Rev. X 13, 021016 (2023).
  39. A. Mitra, A. Corticelli, P. Ribeiro, and P. A. McClarty, Magnon interference tunneling spectroscopy as a probe of 2D magnetism, Phys. Rev. Lett. 130, 066701 (2023).
  40. R. Koch, R. Drost, P. Liljeroth, and J. L. Lado, Hamiltonian learning of triplon excitations in an artificial nanoscale molecular quantum magnet, Nano Lett. 25, 13435 (2025).
  41. E. P. L. van Nieuwenburg, Y.-H. Liu, and S. D. Huber, Learning phase transitions by confusion, Nat. Phys. 13, 435 (2017).
  42. J. Carrasquilla and R. G. Melko, Machine learning phases of matter, Nat. Phys. 13, 431 (2017).
  43. J. F. Rodriguez-Nieva and M. S. Scheurer, Identifying topological order through unsupervised machine learning, Nat. Phys. 15, 790 (2019).
  44. N. L. Holanda and M. A. R. Griffith, Machine learning topological phases in real space, Phys. Rev. B 102, 054107 (2020).
  45. M. S. Scheurer and R.-J. Slager, Unsupervised machine learning and band topology, Phys. Rev. Lett. 124, 226401 (2020).
  46. L.-F. Zhang, L.-Z. Tang, Z.-H. Huang, G.-Q. Zhang, W. Huang, and D.-W. Zhang, Machine learning topological invariants of non-Hermitian systems, Phys. Rev. A 103, 012419 (2021).
  47. N. Käming, A. Dawid, K. Kottmann, M. Lewenstein, K. Sengstock, A. Dauphin, and C. Weitenberg, Unsupervised machine learning of topological phase transitions from experimental data, Mach. Learn.: Sci. Technol. 2, 035037 (2021).
  48. D. Carvalho, N. A. García-Martínez, J. L. Lado, and J. Fernández-Rossier, Real-space mapping of topological invariants using artificial neural networks, Phys. Rev. B 97, 115453 (2018).
  49. V. Dunjko and H. J. Briegel, Machine learning and artificial intelligence in the quantum domain: A review of recent progress, Rep. Prog. Phys. 81, 074001 (2018).
  50. G. Torlai, G. Mazzola, J. Carrasquilla, M. Troyer, R. Melko, and G. Carleo, Neural-network quantum state tomography, Nat. Phys. 14, 447 (2018).
  51. W. Hu, R. R. P. Singh, and R. T. Scalettar, Discovering phases, phase transitions, and crossovers through unsupervised machine learning: A critical examination, Phys. Rev. E 95, 062122 (2017).
  52. Y. Zhang, R. Xie, T. Long, D. Günzing, H. Wende, K. J. Ollefs, and H. Zhang, Autonomous atomic Hamiltonian construction and active sampling of X-ray absorption spectroscopy by adversarial Bayesian optimization, npj Comput. Mater. 9, 46 (2023).
  53. A. A. Melnikov, H. Poulsen Nautrup, M. Krenn, V. Dunjko, M. Tiersch, A. Zeilinger, and H. J. Briegel, Active learning machine learns to create new quantum experiments, Proc. Natl. Acad. Sci. USA 115, 1221 (2018).
  54. P. Zhang, H. Shen, and H. Zhai, Machine learning topological invariants with neural networks, Phys. Rev. Lett. 120, 066401 (2018).
  55. M. Krenn, J. Landgraf, T. Foesel, and F. Marquardt, Artificial intelligence and machine learning for quantum technologies, Phys. Rev. A 107, 010101 (2023).
  56. J. Carrasquilla, Machine learning for quantum matter, Adv. Phys.: X 5, 1797528 (2020).
  57. N. Karjalainen, Z. Lippo, G. Chen, R. Koch, A. O. Fumega, and J. L. Lado, Hamiltonian inference from dynamical excitations in confined quantum magnets, Phys. Rev. Appl. 20, 024054 (2023).
  58. A. Valenti, G. Jin, J. Léonard, S. D. Huber, and E. Greplova, Scalable Hamiltonian learning for large-scale out-of-equilibrium quantum dynamics, Phys. Rev. A 105, 023302 (2022).
  59. O. Simard, A. Dawid, J. Tindall, M. Ferrero, A. M. Sengupta, and A. Georges, Learning interactions between Rydberg atoms, PRX Quantum 6, 030324 (2025).
  60. R. Koch and J. L. Lado, Designing quantum many-body matter with conditional generative adversarial networks, Phys. Rev. Res. 4, 033223 (2022).
  61. K. Tucker, A. K. Rege, C. Smith, C. Monteleoni, and T. Albash, Hamiltonian learning using machine-learning models trained with continuous measurements, Phys. Rev. Appl. 22, 044080 (2024).
  62. A. M. Samarakoon, P. Laurell, C. Balz, A. Banerjee, P. Lampen-Kelley, D. Mandrus, S. E. Nagler, S. Okamoto, and D. A. Tennant, Extraction of interaction parameters for α−RuCl3 from neutron data using machine learning, Phys. Rev. Res. 4, L022061 (2022).
  63. T. Heightman, E. Jiang, and A. Acín, Solving the quantum many-body Hamiltonian learning problem with neural differential equations, arXiv:2408.08639.
  64. R. Koch, D. van Driel, A. Bordin, J. L. Lado, and E. Greplova, Adversarial Hamiltonian learning of quantum dots in a minimal Kitaev chain, Phys. Rev. Appl. 20, 044081 (2023).
  65. N. Abuawwad, Y. Zhang, S. Lounis, and H. Zhang, Kalman filter enhanced active learning sampling for inelastic neutron scattering: The case of CrSBr, Phys. Rev. B 111, 054404 (2025).
  66. G. Lupi, A. O. Fumega, M. Amini, R. Drost, P. Liljeroth, and J. L. Lado, Molecular Hamiltonian learning from setpoint-dependent scanning tunneling spectroscopy, arXiv:2601.19371.
  67. M. Khosravian, R. Koch, and J. L. Lado, Hamiltonian learning with real-space impurity tomography in topological moiré superconductors, J. Phys.: Mater. 7, 015012 (2024).
  68. F. Aikebaier, T. Ojanen, and J. L. Lado, Machine learning the Kondo entanglement cloud from local measurements, Phys. Rev. B 109, 195125 (2024).
  69. D. Liu, A. B. Watson, M. Hott, S. Carr, and M. Luskin, Learning the local density of states of a bilayer moire material, Multiscale Model. Simul. 23, 1481 (2025).
  70. F. Noronha, A. Canabarro, R. Chaves, and R. G. Pereira, Predicting topological invariants and unconventional superconducting pairing from density of states and machine learning, Phys. Rev. B 111, 014501 (2025).
  71. D. Liu, M. Luskin, and S. Carr, Seeing moiré: Convolutional network learning applied to twistronics, Phys. Rev. Res. 4, 043224 (2022).
  72. J. A. Sobral, S. Obernauer, S. Turkel, A. N. Pasupathy, and M. S. Scheurer, Machine learning the microscopic form of nematic order in twisted double-bilayer graphene, Nat. Commun. 14, 5012 (2023).
  73. G. Lupi and J. L. Lado, Hamiltonian-learning quantum magnets with nonlocal impurity tomography, Phys. Rev. Appl. 23, 054077 (2025).
  74. F. Nigmatulin, G. Lupi, J. L. Lado, and Z. Sun, Hamiltonian learning for spin-spiral moiré magnets from electronic magnetotransport, arXiv:2604.02959.
  75. P. Willke, Y. Bae, K. Yang, J. L. Lado, A. Ferrón, T. Choi, A. Ardavan, J. Fernández-Rossier, A. J. Heinrich, and C. P. Lutz, Hyperfine interaction of individual atoms on a surface, Science 362, 336 (2018).
  76. K. Yang, P. Willke, Y. Bae, A. Ferrón, J. L. Lado, A. Ardavan, J. Fernández-Rossier, A. J. Heinrich, and C. P. Lutz, Electrically controlled nuclear polarization of individual atoms, Nat. Nanotechnol. 13, 1120 (2018).
  77. M. Bode, M. Heide, K. von Bergmann, P. Ferriani, S. Heinze, G. Bihlmayer, A. Kubetzka, O. Pietzsch, S. Blügel, and R. Wiesendanger, Chiral magnetic order at surfaces driven by inversion asymmetry, Nature (London) 447, 190 (2007).
  78. J. Lee, C. Jang, B. Min, S. Lee, K. Lee, and J. Chang, All-electrical measurement of interfacial Dzyaloshinskii-Moriya interaction using collective spin-wave dynamics, Nano Lett. 16, 62 (2016).
  79. R. D. Johnson, S. C. Williams, A. A. Haghighirad, J. Singleton, V. Zapf, P. Manuel, I. I. Mazin, Y. Li, H. O. Jeschke, R. Valentí, and R. Coldea, Monoclinic crystal structure of α−RuCl3 and the zigzag antiferromagnetic ground state, Phys. Rev. B 92, 235119 (2015).
  80. R. E. Camley and K. L. Livesey, Consequences of the Dzyaloshinskii-Moriya interaction, Surf. Sci. Rep. 78, 100605 (2023).
  81. A. A. Khajetoorians, M. Steinbrecher, M. Ternes, M. Bouhassoune, M. dos Santos Dias, S. Lounis, J. Wiebe, and R. Wiesendanger, Tailoring the chiral magnetic interaction between two individual atoms, Nat. Commun. 7, 10620 (2016).
  82. M. Kuepferling, A. Casiraghi, G. Soares, G. Durin, F. Garcia-Sanchez, L. Chen, C. Back, C. Marrows, S. Tacchi, and G. Carlotti, Measuring interfacial Dzyaloshinskii-Moriya interaction in ultrathin magnetic films, Rev. Mod. Phys. 95, 015003 (2023).
  83. A. Holzner, A. Weichselbaum, I. P. McCulloch, U. Schollwöck, and J. von Delft, Chebyshev matrix product state approach for spectral functions, Phys. Rev. B 83, 195115 (2011).
  84. A. Weiße, G. Wellein, A. Alvermann, and H. Fehske, The kernel polynomial method, Rev. Mod. Phys. 78, 275 (2006).
  85. J. L. Lado and O. Zilberberg, Topological spin excitations in Harper-Heisenberg spin chains, Phys. Rev. Res. 1, 033009 (2019).
  86. J. L. Lado and M. Sigrist, Solitonic in-gap modes in a superconductor-quantum antiferromagnet interface, Phys. Rev. Res. 2, 023347 (2020).
  87. M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor software library for tensor network calculations, SciPost Phys. Codebases, 4 (2022).
  88. J. L. Lado, dmrgpy library (2026), https://github.com/joselado/dmrgpy.
  89. C. Zhao, L. Yang, J. C. G. Henriques, M. Ferri-Cortés, G. Catarina, C. A. Pignedoli, J. Ma, X. Feng, P. Ruffieux, J. Fernández-Rossier, and R. Fasel, Spin excitations in nanographene-based antiferromagnetic spin-1/2 Heisenberg chains, Nat. Mater. 24, 722 (2025).
  90. T. S. Seifert, S. Kovarik, P. Gambardella, and S. Stepanow, Accurate measurement of atomic magnetic moments by minimizing the tip magnetic field in STM-based electron paramagnetic resonance, Phys. Rev. Res. 3, 043185 (2021).
  91. J. Kügel, P.-J. Hsu, M. Böhme, K. Schneider, J. Senkpiel, D. Serrate, M. Bode, and N. Lorente, Jahn-Teller splitting in single adsorbed molecules revealed by isospin-flip excitations, Phys. Rev. Lett. 121, 226402 (2018).
  92. H. Gawronski, M. Mehlhorn, and K. Morgenstern, Imaging phonon excitation with atomic resolution, Science 319, 930 (2008).
  93. N. Lorente and J.-P. Gauyacq, Efficient spin transitions in inelastic electron tunneling spectroscopy, Phys. Rev. Lett. 103, 176601 (2009).
  94. N. Lorente, R. Rurali, and H. Tang, Single-molecule manipulation and chemistry with the STM, J. Phys.: Condens. Matter 17, S1049 (2005).
  95. A. F. Otte, M. Ternes, K. von Bergmann, S. Loth, H. Brune, C. P. Lutz, C. F. Hirjibehedin, and A. J. Heinrich, The role of magnetic anisotropy in the Kondo effect, Nat. Phys. 4, 847 (2008).
  96. M. Ternes, Spin excitations and correlations in scanning tunneling spectroscopy, New J. Phys. 17, 063016 (2015).
  97. S. Loth, K. von Bergmann, M. Ternes, A. F. Otte, C. P. Lutz, and A. J. Heinrich, Controlling the state of quantum spins with electric currents, Nat. Phys. 6, 340 (2010).
  98. F. Delgado and J. Fernández-Rossier, Spin dynamics of current-driven single magnetic adatoms and molecules, Phys. Rev. B 82, 134414 (2010).
  99. D. van Driel, R. Koch, V. P. M. Sietses, S. L. D. ten Haaf, C.-X. Liu, F. Zatelli, B. Roovers, A. Bordin, N. van Loo, G. Wang, et al. Cross-platform autonomous control of minimal Kitaev chains, PRX Intelligence 1, 013005 (2026).
  100. J. Hwang, D. Krylov, R. Elbertse, S. Yoon, T. Ahn, J. Oh, L. Fang, W.-j. Jang, F. H. Cho, A. J. Heinrich, and Y. Bae, Development of a scanning tunneling microscope for variable temperature electron spin resonance, Rev. Sci. Instrum. 93, 093703 (2022).
  101. M. A. Reed, Inelastic electron tunneling spectroscopy, Mater. Today 11, 46 (2008).
  102. N. Karjalainen, Source code Hamiltonian learning quantum magnets with dynamical impurity tomography (2026), https://github.com/nettakarjalainen/hamiltonian-learning-quantum-magnets-with-dynamical-impurity-tomography.git.
  103. N. Karjalainen, dIdV datasets and PCAs for dynamical impurity tomography [Dataset], Zenodo, 2025, https://doi.org/10.5281/zenodo.17378918.

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Phys. Rev. Research 8, 033281– Published 8 September, 2026
  • Received 3 November 2025
  • Accepted 21 July 2026
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