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Black Hole States in Quantum Spin Chains

Charlotte Kristjansen1 and Konstantin Zarembo1,2

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Phys. Rev. Lett. 137, 021602 – Published 7 July, 2026

DOI: https://doi.org/10.1103/5myp-jksn

Abstract

We define a black hole state in a spin chain by studying thermal correlators in holography. Focusing on the Heisenberg model we investigate the thermal and complexity properties of the black hole state by evaluating its entanglement entropy, emptiness formation probability, and Krylov complexity. The entanglement entropy grows logarithmically with effective central charge c≃5.2. We also find evidence for thermalization at infinite temperature.

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References (53)

  1. E. Witten, Anti-de Sitter space, thermal phase transition, and confinement in gauge theories, Adv. Theor. Math. Phys. 2, 505 (1998).
  2. M. Grinberg and J. Maldacena, Proper time to the black hole singularity from thermal one-point functions, J. High Energy Phys. 03 (2021) 131.
  3. J. A. Minahan and K. Zarembo, The Bethe ansatz for N=4 superYang-Mills, J. High Energy Phys. 03 (2003) 013.
  4. M. de Leeuw, C. Kristjansen, and K. Zarembo, One-point functions in defect CFT and integrability, J. High Energy Phys. 08 (2015) 098.
  5. I. Buhl-Mortensen, M. de Leeuw, C. Kristjansen, and K. Zarembo, One-point functions in AdS/dCFT from matrix product states, J. High Energy Phys. 02 (2016) 052.
  6. C. Kristjansen and K. Zarembo, Integrable holographic defect CFTs, arXiv:2401.17144.
  7. G. Linardopoulos, String theory methods for defect CFTs, arXiv:2501.11985.
  8. R. A. Janik, Ising model as a window on quantum gravity with matter, Phys. Rev. D 111, 106016 (2025).
  9. P. Basteiro, G. Di Giulio, J. Erdmenger, J. Karl, R. Meyer, and Z.-Y. Xian, Towards explicit discrete holography: Aperiodic spin chains from hyperbolic tilings, SciPost Phys. 13, 103 (2022).
  10. P. Basteiro, G. Di Giulio, J. Erdmenger, R. Meyer, and Z.-Y. Xian, Wormhole-induced effective coupling in SYK chains, J. High Energy Phys. 02 (2025) 142.
  11. S. Dey et al., Simulating holographic conformal field theories on hyperbolic lattices, Phys. Rev. Lett. 133, 061603 (2024).
  12. G. Rumer, Zur theorie der spinvalenz, Gottingen Nachr. 1932, 337 (1932).
  13. G. Rumer, E. Teller, and H. Weyl, Eine für die valenztheorie geeignete basis der binären vektorinvarianten, Gottingen Nachr. Tech. 1932, 499 (1932).
  14. L. Hulthén, Über das austauschproblem eines kristalles, Ark. Mat. Astron. Fys. 26A, 1 (1938).
  15. S. Jung, S. Kim, J. Park, and S. Song, Random matrix product state models of gravitationally prepared states, arXiv:2512.11966.
  16. S. Ghoshal and A. B. Zamolodchikov, Boundary S matrix and boundary state in two-dimensional integrable quantum field theory, Int. J. Mod. Phys. A 09, 3841 (1994); Int. J. Mod. Phys. A09, 4353(E) (1994).
  17. L. Piroli, B. Pozsgay, and E. Vernier, What is an integrable quench?, Nucl. Phys. B925, 362 (2017).
  18. M. Brockmann, J. De Nardis, B. Wouters, and J.-S. Caux, A Gaudin-like determinant for overlaps of Néel and XXZ Bethe States, J. Phys. A 47, 145003 (2014).
  19. T. Gombor, Exact overlaps for all integrable two-site boundary states of gl(N) symmetric spin chains, J. High Energy Phys. 05 (2024) 194.
  20. T. Gombor, Exact overlaps for all integrable matrix product states of rational spin chains, Phys. Rev. Lett. 135, 150402 (2025).
  21. B. Pozsgay, Overlaps with arbitrary two-site states in the XXZ spin chain, J. Stat. Mech. (2018) 053103.
  22. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England, 2012).
  23. P. Calabrese and J. L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. (2004) P06002.
  24. N. Laflorencie, Quantum entanglement in condensed matter systems, Phys. Rep. 643, 1 (2016).
  25. J. I. Latorre, E. Rico, and G. Vidal, Ground state entanglement in quantum spin chains, Quantum Inf. Comput. 4, 48 (2004).
  26. V. Korepin, Universality of entropy scaling in one dimensional gapless models, Phys. Rev. Lett. 92, 096402 (2004).
  27. I. Affleck, Universal term in the free energy at a critical point and the conformal anomaly, Phys. Rev. Lett. 56, 746 (1986).
  28. M. Rigol and M. Srednicki, Alternatives to eigenstate thermalization, Phys. Rev. Lett. 108, 110601 (2012).
  29. L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65, 239 (2016).
  30. M. Alishahiha and M. J. Vasli, Eigenstate thermalization hypothesis: A short review, arXiv:2501.07243.
  31. M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature (London) 452, 854 (2008).
  32. V. E. Korepin, A. G. Izergin, F. H. L. Essler, and D. B. Uglov, Correlation function of the spin 1/2 XXX antiferromagnet, Phys. Lett. A 190, 182 (1994).
  33. H. E. Boos and V. E. Korepin, Quantum spin chains and Riemann zeta function with odd arguments, J. Phys. A 34, 5311 (2001).
  34. H. E. Boos, V. E. Korepin, Y. Nishiyama, and M. Shiroishi, Quantum correlations and number theory, J. Phys. A 35, 4443 (2002).
  35. A. G. Abanov and V. E. Korepin, On the probability of ferromagnetic strings in antiferromagnetic spin chains, Nucl. Phys. B 647, 565 (2002).
  36. D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, A universal operator growth hypothesis, Phys. Rev. X 9, 041017 (2019).
  37. V. Balasubramanian, P. Caputa, J. M. Magan, and Q. Wu, Quantum chaos and the complexity of spread of states, Phys. Rev. D 106, 046007 (2022).
  38. P. Nandy, A. S. Matsoukas-Roubeas, P. Martínez-Azcona, A. Dymarsky, and A. del Campo, Quantum dynamics in Krylov space: Methods and applications, Phys. Rep. 1125–1128, 1 (2025).
  39. E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, Krylov complexity, arXiv:2507.06286.
  40. E. Rabinovici, A. Sánchez-Garrido, R. Shir, and J. Sonner, Krylov complexity from integrability to chaos, J. High Energy Phys. 07 (2022) 151.
  41. A. Avdoshkin, A. Dymarsky, and M. Smolkin, Krylov complexity in quantum field theory, and beyond, J. High Energy Phys. 06 (2024) 066.
  42. G. F. Scialchi, A. J. Roncaglia, and D. A. Wisniacki, Integrability-to-chaos transition through the Krylov approach for state evolution, Phys. Rev. E 109, 054209 (2024).
  43. B. Craps, O. Evnin, and G. Pascuzzi, Multiseed Krylov complexity, Phys. Rev. Lett. 134, 050402 (2025).
  44. J. Erdmenger, S.-K. Jian, and Z.-Y. Xian, Universal chaotic dynamics from Krylov space, J. High Energy Phys. 08 (2023) 176.
  45. V. Balasubramanian, P. Caputa, and J. Simón, Variations on a theme of Krylov, J. High Energy Phys. 04 (2026) 172.
  46. R. N. Das, S. Demulder, J. Erdmenger, and C. Northe, Spread complexity for the planar limit of holography, J. High Energy Phys. 06 (2025) 166.
  47. A. Fatemiabhari, H. Nastase, and D. Roychowdhury, Holographic Krylov complexity in N=4 SYM, Phys. Rev. D 113, 106033 (2026).
  48. J. Barrat, D. N. Bozkurt, E. Marchetto, A. Miscioscia, and E. Pomoni, Analytic thermal bootstrap meets holography, J. High Energy Phys. 05 (2026) 180.
  49. K. Alkalaev and S. Mandrygin, One-point thermal conformal blocks from four-point conformal integrals, J. High Energy Phys. 10 (2024) 241.
  50. I. Buric, F. Russo, V. Schomerus, and A. Vichi, Thermal one-point functions and their partial wave decomposition, J. High Energy Phys. 12 (2024) 021.
  51. N. Čeplak, H. Liu, A. Parnachev, and S. Valach, Black hole singularity from OPE, J. High Energy Phys. 10 (2024) 105.
  52. N. Čeplak, H. Liu, A. Parnachev, and S. Valach, Fooling the censor: Going beyond inner horizons with the OPE, J. High Energy Phys. 05 (2026) 001.
  53. I. Burić, I. Gusev, and A. Parnachev, Holographic correlators from thermal bootstrap, J. High Energy Phys. 05 (2026) 059.

Outline

Information

Phys. Rev. Lett. 137, 021602– Published 7 July, 2026
  • Received 9 January 2026
  • Revised 27 March 2026
  • Accepted 4 June 2026
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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. Funded by SCOAP3.

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