- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Hilbert space geometry and quantum chaos

Rustem Sharipov1,*, Anastasiia Tiutiakina2, Alexander Gorsky3, Vladimir Gritsev4, and Anatoli Polkovnikov5

  • 1Physics Department, Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia
  • 2Laboratoire de Physique Theorique et Modelisation, CNRS UMR 8089, CY Cergy Paris Universite, 95302 Cergy-Pontoise Cedex, France
  • 3Laboratory of Complex Networks, Center for Neurophysics and Neuromorphic Technologies, Moscow, Russia
  • 4Institute for Theoretical Physics, University of Amsterdam, Science Park 904, Postbus 94485, 1090 GL Amsterdam, The Netherlands
  • 5Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA

  • *Contact author: rustem.sharipov@fmf.uni-lj.si
PDF

Phys. Rev. Research 8, L022008 – Published 9 April, 2026

DOI: https://doi.org/10.1103/92z8-bqx9

Abstract

The quantum geometric tensor (QGT) characterizes the Hilbert space geometry of the eigenstates of a parameter-dependent Hamiltonian. In recent years, the QGT and related quantities have found extensive theoretical and experimental utility, in particular for quantifying quantum phase transitions both at and out of equilibrium. Here, we consider the symmetric part (quantum Riemannian metric) of the QGT for various multiparametric random matrix Hamiltonians and discuss the possible indication of ergodic or integrable behavior. We found for a two-dimensional parameter space that, while the ergodic phase corresponds to the smooth manifold, the integrable limit marks itself as a singular geometry with a conical defect. Our study thus provides more support for the idea that the landscape of the parameter space yields information on the ergodic-nonergodic transition in complex quantum systems, including the intermediate phase.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (45)

  1. J. P. Provost and G. Vallee, Riemannian structure on manifolds of quantum states, Commun. Math. Phys. 76, 289 (1980).
  2. P. Zanardi and N. Paunković, Ground state overlap and quantum phase transitions, Phys. Rev. E 74, 031123 (2006).
  3. P. Zanardi, P. Giorda, and M. Cozzini, Information-theoretic differential geometry of quantum phase transitions, Phys. Rev. Lett. 99, 100603 (2007).
  4. M. Kolodrubetz, V. Gritsev, and A. Polkovnikov, Classifying and measuring geometry of a quantum ground state manifold, Phys. Rev. B 88, 064304 (2013).
  5. D. Liska and V. Gritsev, Hidden symmetries, the Bianchi classification and geodesics of the quantum geometric ground-state manifolds, SciPost Phys. 10, 020 (2021).
  6. S. Peotta, K.-E. Huhtinen, and P. Törmä, Quantum geometry in superfluidity and superconductivity, in Quantum Mixtures with Ultra-cold Atoms, Proceedings of the International School of Physics “Enrico Fermi”, Vol. 211 (IOS Press, Amsterdam, 2025), pp. 373–404.
  7. A. Alexandrov and A. Gorsky, Information geometry and synchronization phase transition in the Kuramoto model, Phys. Rev. E 107, 044211 (2023).
  8. M. Pandey, P. W. Claeys, D. K. Campbell, A. Polkovnikov, and D. Sels, Adiabatic eigenstate deformations as a sensitive probe for quantum chaos, Phys. Rev. X 10, 041017 (2020).
  9. F. M. Surace and O. Motrunich, Weak integrability breaking perturbations of integrable models, Phys. Rev. Res. 5, 043019 (2023).
  10. S. Vanovac, F. M. Surace, and O. I. Motrunich, Finite-size generators for weak integrability breaking perturbations in the Heisenberg chain, Phys. Rev. B 110, 144309 (2024).
  11. P. Orlov, A. Tiutiakina, R. Sharipov, E. Petrova, V. Gritsev, and D. V. Kurlov, Adiabatic eigenstate deformations and weak integrability breaking of Heisenberg chain, Phys. Rev. B 107, 184312 (2023).
  12. B. Pozsgay, R. Sharipov, A. Tiutiakina, and I. Vona, Adiabatic gauge potential and integrability breaking with free fermions, SciPost Phys. 17, 075 (2024).
  13. D. Sels and A. Polkovnikov, Dynamical obstruction to localization in a disordered spin chain, Phys. Rev. E 104, 054105 (2021).
  14. H. Kim and A. Polkovnikov, Integrability as an attractor of adiabatic flows, Phys. Rev. B 109, 195162 (2024).
  15. V. E. Kravtsov, I. M. Khaymovich, E. Cuevas, and M. Amini, A random matrix model with localization and ergodic transitions, New J. Phys. 17, 122002 (2015).
  16. D. Venturelli, L. F. Cugliandolo, G. Schehr, and M. Tarzia, Replica approach to the generalized Rosenzweig-Porter model, SciPost Phys. 14, 110 (2023).
  17. B. Bhattacharjee and P. Nandy, Krylov fractality and complexity in generic random matrix ensembles, Phys. Rev. B 111, L060202 (2025).
  18. E. Bogomolny and M. Sieber, Eigenfunction distribution for the Rosenzweig-Porter model, Phys. Rev. E 98, 032139 (2018).
  19. W. Tang and I. M. Khaymovich, Non-ergodic delocalized phase with Poisson level statistics, Quantum 6, 733 (2022).
  20. I. M. Khaymovich and V. E. Kravtsov, Dynamical phases in a “multifractal” Rosenzweig-Porter model, SciPost Phys. 11, 045 (2021).
  21. P. von Soosten and S. Warzel, Non-ergodic delocalization in the Rosenzweig–Porter model, Lett. Math. Phys. 109, 905 (2018).
  22. B. Altshuler and V. Kravtsov, Random Cantor sets and mini-bands in local spectrum of quantum systems, Ann. Phys. 456, 169300 (2023).
  23. S. Sugiura, P. W. Claeys, A. Dymarsky, and A. Polkovnikov, Adiabatic landscape and optimal paths in ergodic systems, Phys. Rev. Res. 3, 013102 (2021).
  24. A. I. Larkin and Y. N. Ovchinnikov, Quasiclassical method in the theory of superconductivity, Sov. Phys. JETP 28, 1200 (1969).
  25. D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, A universal operator growth hypothesis, Phys. Rev. X 9, 041017 (2019).
  26. F. Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys. 80, 1355 (2008).
  27. M. Serbyn, Z. Papić, and D. A. Abanin, Criterion for many-body localization-delocalization phase transition, Phys. Rev. X 5, 041047 (2015).
  28. C. Lim, K. Matirko, A. Polkovnikov, and M. O. Flynn, Defining classical and quantum chaos through adiabatic transformations, J. Phys. A: Math. Theor. 59, 075702 (2026).
  29. B. V. Fine, T. A. Elsayed, C. M. Kropf, and A. S. de Wijn, Absence of exponential sensitivity to small perturbations in nonintegrable systems of spins 1/2, Phys. Rev. E 89, 012923 (2014).
  30. I. Kukuljan, S. Grozdanov, and T. Prosen, Weak quantum chaos, Phys. Rev. B 96, 060301(R) (2017).
  31. S. Pappalardi, A. Polkovnikov, and A. Silva, Quantum echo dynamics in the Sherrington-Kirkpatrick model, SciPost Phys. 9, 021 (2020).
  32. M. Berry and P. Shukla, Quantum metric statistics for random-matrix families, J. Phys. A: Math. Theor. 53, 275202 (2020).
  33. P. Sierant, A. Maksymov, M. Kuś, and J. Zakrzewski, Fidelity susceptibility in Gaussian random ensembles, Phys. Rev. E 99, 050102(R) (2019).
  34. A.-G. Penner, F. von Oppen, G. Zaránd, and M. R. Zirnbauer, Hilbert space geometry of random matrix eigenstates, Phys. Rev. Lett. 126, 200604 (2021).
  35. B. Derrida, Random-energy model: Limit of a family of disordered models, Phys. Rev. Lett. 45, 79 (1980).
  36. See Supplemental Material at http://link.aps.org/supplemental/10.1103/92z8-bqx9 for additional details.
  37. M. L. Mehta, Random Matrices (Elsevier, Amsterdam, 2004).
  38. V. R. Motamarri, A. S. Gorsky, and I. M. Khaymovich, Localization and fractality in disordered Russian Doll model, SciPost Phys. 13, 117 (2022).
  39. M. A. Skvortsov, M. Amini, and V. E. Kravtsov, Sensitivity of (multi)fractal eigenstates to a perturbation of the Hamiltonian, Phys. Rev. B 106, 054208 (2022).
  40. L. V. Iliesiu, M. Mezei, and G. Sárosi, The volume of the black hole interior at late times, J. High Energy Phys. 07 (2022) 073.
  41. B. Bhattacharjee, A Lanczos approach to the adiabatic gauge potential, arXiv:2302.07228.
  42. K. Takahashi and A. del Campo, Shortcuts to adiabaticity in Krylov space, Phys. Rev. X 14, 011032 (2024).
  43. I. Dumitriu and A. Edelman, Matrix models for beta ensembles, J. Math. Phys. 43, 5830 (2002).
  44. V. Balasubramanian, J. M. Magan, and Q. Wu, Tridiagonalizing random matrices, Phys. Rev. D 107, 126001 (2023).
  45. P. Saad, S. H. Shenker, and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115.

Outline

Information

Phys. Rev. Research 8, L022008– Published 9 April, 2026
  • Received 4 February 2025
  • Accepted 4 March 2026
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation