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Many-body neural network wavefunction for a non-Hermitian Ising chain

Lavoisier Wah1,2,*, Remmy Zen1,†, and Flore K. Kunst1,2,‡

  • *Contact author: lavoisier.wahkenounouh@mpl.mpg.de
  • †Contact author: remmy.zen@mpl.mpg.de
  • ‡Contact author: flore.kunst@mpl.mpg.de
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Phys. Rev. Research 7, 043291 – Published 15 December, 2025

DOI: https://doi.org/10.1103/q8p7-k7ms

Abstract

Non-Hermitian (NH) quantum systems have emerged as a powerful framework for describing open quantum systems, nonequilibrium dynamics, and engineered quantum optical materials. However, solving the ground-state properties of NH systems is challenging due to the exponential scaling of the Hilbert space and exotic phenomena such as the emergence of exceptional points. Another challenge arises from the limitations of traditional methods like exact diagonalization (ED). For the past decade, neural networks (NNs) have shown promise in approximating many-body wavefunctions, yet their application to NH systems remains largely unexplored. In this paper, we explore different NN architectures to investigate the ground-state properties of a parity-time-symmetric, one-dimensional NH, transverse field Ising model with a complex spectrum by employing a recurrent neural network (RNN), a restricted Boltzmann machine (RBM), and a multilayer perceptron (MLP). We construct the NN-based many-body wavefunctions and validate our approach by recovering the ground-state properties of the model for small system sizes, finding excellent agreement with ED. Furthermore, for larger system sizes, we demonstrate that the RNN outperforms both the RBM and MLP. However, we show that the accuracy of the RBM and MLP can be significantly improved through transfer learning, allowing them to perform comparably to the RNN for larger system sizes. These results highlight the potential of neural-network-based approaches—particularly for accurately capturing the low-energy physics of NH quantum systems in case of both weak and strong non-Hermiticity.

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References (71)

  1. Y. Ashida, Z. Gong, and M. Ueda, Non-Hermitian physics, Adv. Phys. 69, 249 (2020).
  2. E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Exceptional topology of non-Hermitian systems, Rev. Mod. Phys. 93, 015005 (2021).
  3. C. M. Bender and S. Boettcher, Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80, 5243 (1998).
  4. R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Non-Hermitian physics and PT symmetry, Nat. Phys. 14, 11 (2018).
  5. M. Ateuafack, L. Wah, M. Jipdi, J. N. Kuetche, L. Temdie, J. Diffo, and L. Fai, Probing nonadiabatic transition dynamics in 2D topological superfluid Fermi gas with spin orbit coupling, Phys. Lett. A 533, 130216 (2025).
  6. Y. Zhang and Z. Wei, Non-Hermitian skin effect in non-Hermitian optical systems, Laser Photonics Rev. 19, 2400099 (2025).
  7. Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, Parity–time symmetry and exceptional points in photonics, Nat. Mater. 18, 783 (2019).
  8. J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, and L. Luo, Observation of parity-time symmetry breaking transitions in a dissipative Floquet system of ultracold atoms, Nat. Commun. 10, 855 (2019).
  9. Z. Ren, D. Liu, E. Zhao, C. He, K. K. Pak, J. Li, and G.-B. Jo, Chiral control of quantum states in non-Hermitian spin–orbit-coupled fermions, Nat. Phys. 18, 385 (2022).
  10. L. Ding, K. Shi, Q. Zhang, D. Shen, X. Zhang, and W. Zhang, Experimental determination of PT-symmetric exceptional points in a single trapped ion, Phys. Rev. Lett. 126, 083604 (2021).
  11. J. T. Gohsrich, A. Banerjee, and F. K. Kunst, The non-Hermitian skin effect: A perspective, Europhys. Lett. 150, 60001 (2025).
  12. D. J. Luitz and F. Piazza, Exceptional points and the topology of quantum many-body spectra, Phys. Rev. Res. 1, 033051 (2019).
  13. D.-W. Zhang, Y.-L. Chen, G.-Q. Zhang, L.-J. Lang, Z. Li, and S.-L. Zhu, Skin superfluid, topological Mott insulators, and asymmetric dynamics in an interacting non-Hermitian Aubry-André-Harper model, Phys. Rev. B 101, 235150 (2020).
  14. C.-Z. Lu, X. Deng, S.-P. Kou, and G. Sun, Many-body phase transitions in a non-Hermitian Ising chain, Phys. Rev. B 110, 014441 (2024).
  15. G. Sun, J.-C. Tang, and S.-P. Kou, Biorthogonal quantum criticality in non-Hermitian many-body systems, Front. Phys. 17, 33502 (2022).
  16. L. Lenke, M. Mühlhauser, and K. P. Schmidt, High-order series expansion of non-Hermitian quantum spin models, Phys. Rev. B 104, 195137 (2021).
  17. C. Li, G. Zhang, X. Z. Zhang, and Z. Song, Conventional quantum phase transition driven by a complex parameter in a non-Hermitian PT-symmetric Ising model, Phys. Rev. A 90, 012103 (2014).
  18. E. M. Guerra, I. Gornyi, and Y. Gefen, Correlations and Krylov spread for a non-Hermitian Hamiltonian: Ising chain with a complex-valued transverse magnetic field, Phys. Rev. B 111, 174207 (2025).
  19. M. Krenn, J. Landgraf, T. Foesel, and F. Marquardt, Artificial intelligence and machine learning for quantum technologies, Phys. Rev. A 107, 010101 (2023).
  20. G. Carleo, I. Cirac, K. Cranmer, L. Daudet, M. Schuld, N. Tishby, L. Vogt-Maranto, and L. Zdeborová, Machine learning and the physical sciences, Rev. Mod. Phys. 91, 045002 (2019).
  21. J. Carrasquilla, Machine learning for quantum matter, Adv. Phys.: X 5, 1797528 (2020).
  22. G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science 355, 602 (2017).
  23. M. Hibat-Allah, M. Ganahl, L. E. Hayward, R. G. Melko, and J. Carrasquilla, Recurrent neural network wave functions, Phys. Rev. Res. 2, 023358 (2020).
  24. S. Pilati, E. M. Inack, and P. Pieri, Self-learning projective quantum Monte Carlo simulations guided by restricted Boltzmann machines, Phys. Rev. E 100, 043301 (2019).
  25. T. Bayram, S. Akkoyun, and S. O. Kara, A study on ground-state energies of nuclei by using neural networks, Ann. Nucl. Energy 63, 172 (2014).
  26. M. Bukov, M. Schmitt, and M. Dupont, Learning the ground state of a non-stoquastic quantum Hamiltonian in a rugged neural network landscape, SciPost Phys. 10, 147 (2021).
  27. T. Westerhout, N. Astrakhantsev, K. S. Tikhonov, M. I. Katsnelson, and A. A. Bagrov, Generalization properties of neural network approximations to frustrated magnet ground states, Nat. Commun. 11, 1593 (2020).
  28. M. Schmitt and M. Heyl, Simulating dynamics of correlated matter with neural quantum states, arXiv:2506.03124.
  29. R. Rende, L. L. Viteritti, F. Becca, A. Scardicchio, A. Laio, and G. Carleo, Foundation neural-networks quantum states as a unified ansatz for multiple Hamiltonians, Nat. Commun. 16, 7213 (2025).
  30. O. Sharir, Y. Levine, N. Wies, G. Carleo, and A. Shashua, Deep autoregressive models for the efficient variational simulation of many-body quantum systems, Phys. Rev. Lett. 124, 020503 (2020).
  31. A. Sinibaldi, A. F. Mello, M. Collura, and G. Carleo, Non-stabilizerness of neural quantum states, arXiv:2502.09725.
  32. E. Ibarra-García-Padilla, H. Lange, R. G. Melko, R. T. Scalettar, J. Carrasquilla, A. Bohrdt, and E. Khatami, Autoregressive neural quantum states of Fermi Hubbard models, Phys. Rev. Res. 7, 013122 (2025).
  33. G. A. Starkov, M. V. Fistul, and I. M. Eremin, Quantum phase transitions in non-Hermitian PT-symmetric transverse-field Ising spin chains, Ann. Phys. 456, 169268 (2023).
  34. See Supplemental Material at http://link.aps.org/supplemental/10.1103/q8p7-k7ms for more details on the different types of RNNs, their nature, and how they can be used for two-dimensional systems, which includes Refs. [58, 59, 60, 61, 69].
  35. D. Yu and F. Vollmer, Spontaneous PT-symmetry breaking in lasing dynamics, Commun. Phys. 4, 77 (2021).
  36. F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Biorthogonal bulk-boundary correspondence in non-Hermitian systems, Phys. Rev. Lett. 121, 026808 (2018).
  37. D. C. Brody, Biorthogonal quantum mechanics, J. Phys. A: Math. Theor. 47, 035305 (2014).
  38. X.-J. Yu, Z. Pan, L. Xu, and Z.-X. Li, Non-Hermitian strongly interacting Dirac fermions, Phys. Rev. Lett. 132, 116503 (2024).
  39. Y.-Q. Chen, S.-X. Zhang, C.-Y. Hsieh, and S. Zhang, Non-Hermitian ground-state-searching algorithm enhanced by a variational toolbox, Phys. Rev. A 107, 042418 (2023).
  40. X. Luo, T. Ohtsuki, and R. Shindou, Transfer matrix study of the Anderson transition in non-Hermitian systems, Phys. Rev. B 104, 104203 (2021).
  41. F. K. Kunst and V. Dwivedi, Non-Hermitian systems and topology: A transfer-matrix perspective, Phys. Rev. B 99, 245116 (2019).
  42. K. Yokomizo and S. Murakami, Non-Bloch band theory of non-Hermitian systems, Phys. Rev. Lett. 123, 066404 (2019).
  43. J. Carrasquilla and G. Torlai, How to use neural networks to investigate quantum many-body physics, PRX Quantum 2, 040201 (2021).
  44. J. Kim, G. Pescia, B. Fore, J. Nys, G. Carleo, S. Gandolfi, M. Hjorth-Jensen, and A. Lovato, Neural-network quantum states for ultra-cold Fermi gases, Commun. Phys. 7, 148 (2024).
  45. M. J. Hartmann and G. Carleo, Neural-network approach to dissipative quantum many-body dynamics, Phys. Rev. Lett. 122, 250502 (2019).
  46. F. Döschl, F. A. Palm, H. Lange, F. Grusdt, and A. Bohrdt, Neural network quantum states for the interacting Hofstadter model with higher local occupations and long-range interactions, Phys. Rev. B 111, 045408 (2025).
  47. H. Lange, A. Van de Walle, A. Abedinnia, and A. Bohrdt, From architectures to applications: A review of neural quantum states, Quantum Sci. Technol. 9, 040501 (2024).
  48. Z. Cai and J. Liu, Approximating quantum many-body wave functions using artificial neural networks, Phys. Rev. B 97, 035116 (2018).
  49. M. Hibat-Allah, E. Merali, G. Torlai, R. G. Melko, and J. Carrasquilla, Recurrent neural network wave functions for Rydberg atom arrays on kagome lattice, Commun. Phys. 8, 308 (2025).
  50. L. Medsker and L. C. Jain, Recurrent Neural Networks: Design and Applications (CRC Press, Boca Raton, FL, 1999).
  51. R. Salakhutdinov, A. Mnih, and G. Hinton, Restricted Boltzmann machines for collaborative filtering, in Proceedings of the 24th International Conference on Machine Learning (ACM, New York, 2007), pp. 791–798.
  52. R. G. Melko, G. Carleo, J. Carrasquilla, and J. I. Cirac, Restricted Boltzmann machines in quantum physics, Nat. Phys. 15, 887 (2019).
  53. G. E. Hinton, A practical guide to training restricted Boltzmann machines, in Neural Networks: Tricks of the Trade, edited by G. Montavon, G. B. Orr, and K. R. Müller, 2nd ed. (Springer, Berlin, Heidelberg, 2012), pp. 599–619.
  54. F. Murtagh, Multilayer perceptrons for classification and regression, Neurocomputing 2, 183 (1991).
  55. L. B. Almeida, Multilayer perceptrons, in Handbook of Neural Computation, edited by F. Emile and B. Russell (CRC Press, Boca Raton, FL, 2020), pp. C1–C2.
  56. R. Kruse, S. Mostaghim, C. Borgelt, C. Braune, and M. Steinbrecher, Multi-layer perceptrons, in Computational Intelligence: A Methodological Introduction (Springer International Publishing, Cham, 2022), pp. 53–124.
  57. M. Reh, M. Schmitt, and M. Gärttner, Optimizing design choices for neural quantum states, Phys. Rev. B 107, 195115 (2023).
  58. L. R. Medsker and L. Jain, Recurrent neural networks, Des. Appl. 5, 2 (2001).
  59. Y. Yu, X. Si, C. Hu, and J. Zhang, A review of recurrent neural networks: LSTM cells and network architectures, Neural Comput. 31, 1235 (2019).
  60. K. Cho, B. van Merriënboer, D. Bahdanau, and Y. Bengio, On the properties of neural machine translation: Encoder–decoder approaches, in Proceedings of SSST-8, Eighth Workshop on Syntax, Semantics and Structure in Statistical Translation, edited by D. Wu, M. Carpuat, X. Carreras, and E. M. Vecchi (Association for Computational Linguistics, Doha, Qatar, 2014), pp. 103–111.
  61. S. Hochreiter and J. Schmidhuber, Long short-term memory, Neural Comput. 9, 1735 (1997).
  62. L. Torrey and J. Shavlik, Transfer learning, in Handbook of Research on Machine Learning Applications and Trends: Algorithms, Methods, and Techniques, edited by E. Soria, J. Martin, R. Magdalena, M. Martinez, and A. Serrano (IGI Global, 2010), pp. 242–264.
  63. R. Zen, L. My, R. Tan, F. Hébert, M. Gattobigio, C. Miniatura, D. Poletti, and S. Bressan, Transfer learning for scalability of neural-network quantum states, Phys. Rev. E 101, 053301 (2020).
  64. R. Zen, L. My, R. Tan, F. Hébert, M. Gattobigio, C. Miniatura, D. Poletti, and S. Bressan, Finding quantum critical points with neural-network quantum states, in ECAI 2020 (IOS Press, Spain, 2020), pp. 1962–1969.
  65. R. Rende, S. Goldt, F. Becca, and L. L. Viteritti, Fine-tuning neural network quantum states, Phys. Rev. Res. 6, 043280 (2024).
  66. P. Pfeuty, The one-dimensional Ising model with a transverse field, Ann. Phys. 57, 79 (1970).
  67. I. Mandal and E. J. Bergholtz, Symmetry and higher-order exceptional points, Phys. Rev. Lett. 127, 186601 (2021).
  68. A. Montag and F. K. Kunst, Symmetry-induced higher-order exceptional points in two dimensions, Phys. Rev. Res. 6, 023205 (2024).
  69. J.-Q. Wang, H.-Q. Wu, R.-Q. He, and Z.-Y. Lu, Variational optimization of the amplitude of neural-network quantum many-body ground states, Phys. Rev. B 109, 245120 (2024).
  70. M. Solinas, B. Barton, Y. Zhang, J. Nys, and J. Carrasquilla, Biorthogonal neural network approach to two-dimensional non-Hermitian systems, arXiv:2508.01072.
  71. L. Wah, R. Zen, and F. K. Kunst, Many-body neural network wavefunction for a non-Hermitian Ising chain, GitHub (2025), https://github.com/Kenounouh/Many-Body-Neural-Network-Wavefunction-for-a-Non-Hermitian-Ising-Chain.

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Phys. Rev. Research 7, 043291– Published 15 December, 2025
  • Received 12 June 2025
  • Accepted 31 October 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by Max Planck Society.

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