- 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
  • Open Access

Undecidability of the spectral gap in rotationally symmetric Hamiltonians

Laura Castilla-Castellano1,* and Angelo Lucia1,2,3,†

  • *Contact author: laurca04@ucm.es
  • †Contact author: angelo.lucia@polimi.it
PDF

Phys. Rev. Research 8, 013033 – Published 14 January, 2026

DOI: https://doi.org/10.1103/5lrn-h3cg

Abstract

The problem of determining the existence of a spectral gap in a lattice quantum spin system was previously shown to be undecidable for one [Phys. Rev. X 10, 031038 (2020)] or more dimensions [Nature (London) 528, 207 (2015)]. In these works, families of nearest-neighbor interactions are constructed whose spectral gap depends on the outcome of a Turing machine halting problem, therefore making it impossible for an algorithm to predict its existence. While these models are translationally invariant, they are not invariant under the other symmetries of the lattice, a property which is commonly found in physically relevant cases. This poses the question of whether the spectral gap problem could be decidable for Hamiltonians with stronger symmetry constraints. We give a negative answer to this question, in the case of models with four-body (plaquette) interactions on the square lattice satisfying rotation, but not reflection, symmetry: rotational symmetry is not enough to make the problem decidable.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (52)

  1. A. Komar, Undecidability of macroscopically distinguishable states in quantum field theory, Phys. Rev. 133, B542 (1964).
  2. M. B. Pour-El and I. Richards, The wave equation with computable initial data such that its unique solution is not computable, Adv. Math. 39, 215 (1981).
  3. C. Moore, Unpredictability and undecidability in dynamical systems, Phys. Rev. Lett. 64, 2354 (1990).
  4. C. H. Bennett, Undecidable dynamics, Nature (London) 346, 606 (1990).
  5. E. Fredkin and T. Toffoli, Conservative logic, Int. J. Theor. Phys. 21, 219 (1982).
  6. E. Domany and W. Kinzel, Equivalence of cellular automata to Ising models and directed percolation, Phys. Rev. Lett. 53, 311 (1984).
  7. S. Omohundro, Modelling cellular automata with partial differential equations, Phys. D 10, 128 (1984).
  8. M. Gu, C. Weedbrook, A. Perales, and M. A. Nielsen, More really is different, Phys. D 238, 835 (2009).
  9. H. Wang, Proving theorems by pattern recognition, Bell Syst. Tech. J. 40, 1 (1961).
  10. R. Berger, The Undecidability of the Domino Problem, Memoirs of the American Mathematical Society, Vol. 66 (American Mathematical Society, Providence, RI, 1966), p. 72.
  11. I. Kanter, Undecidability principle and the uncertainty principle even for classical systems, Phys. Rev. Lett. 64, 332 (1990).
  12. S. Lloyd, Quantum-mechanical computers and uncomputability, Phys. Rev. Lett. 71, 943 (1993).
  13. S. Lloyd, Necessary and sufficient conditions for quantum computation, J. Mod. Opt. 41, 2503 (1994).
  14. J. Eisert, M. P. Müller, and C. Gogolin, Quantum measurement occurrence is undecidable, Phys. Rev. Lett. 108, 260501 (2012).
  15. M. Kliesch, D. Gross, and J. Eisert, Matrix-product operators and states: NP-hardness and undecidability, Phys. Rev. Lett. 113, 160503 (2014).
  16. J. Morton and J. Biamonte, Undecidability in tensor network states, Phys. Rev. A 86, 030301(R) (2012).
  17. G. De las Cuevas, T. S. Cubitt, J. I. Cirac, M. M. Wolf, and D. Pérez-García, Fundamental limitations in the purifications of tensor networks, J. Math. Phys. 57, 071902 (2016).
  18. G. Scarpa, A. Molnár, Y. Ge, J. J. García-Ripoll, N. Schuch, D. Pérez-García, and S. Iblisdir, Projected entangled pair states: Fundamental analytical and numerical limitations, Phys. Rev. Lett. 125, 210504 (2020).
  19. M. Van den Nest and H. J. Briegel, Measurement-based quantum computation and undecidable logic, Found. Phys. 38, 448 (2008).
  20. D. Elkouss and D. Pérez-García, Memory effects can make the transmission capability of a communication channel uncomputable, Nat. Commun. 9, 1149 (2018).
  21. A. Bendersky, G. Senno, G. de la Torre, S. Figueira, and A. Acin, Non-signaling deterministic models for non-local correlations have to be uncomputable, Phys. Rev. Lett. 118, 130401 (2017).
  22. H. Fu, C. A. Miller, and W. Slofstra, The membership problem for constant-sized quantum correlations is undecidable, Commun. Math. Phys. 406, 96 (2025).
  23. D. I. Bondar and A. N. Pechen, Uncomputability and complexity of quantum control, Sci. Rep. 10, 1195 (2020).
  24. Y. Tachikawa, Undecidable problems in quantum field theory, Int. J. Theor. Phys. 62, 199 (2023).
  25. J. M. Agüero Trejo, C. S. Calude, M. J. Dinneen, A. Fedorov, A. Kulikov, R. Navarathna, and K. Svozil, How real is incomputability in physics? Theor. Comput. Sci. 1003, 114632 (2024).
  26. T. Cubitt, D. Perez-Garcia, and M. M. Wolf, Undecidability of the spectral gap, Forum of Mathematics, Pi (Cambridge University Press, 2022), Vol. 10, p. e14.
  27. Á. Perales-Eceiza, T. Cubitt, M. Gu, D. Pérez-García, and M. M. Wolf, Undecidability in physics: A review, Phys. Rep. 1138, 1 (2025).
  28. T. S. Cubitt, D. Perez-Garcia, and M. M. Wolf, Undecidability of the spectral gap, Nature (London) 528, 207 (2015).
  29. J. Bausch, T. S. Cubitt, A. Lucia, and D. Perez-Garcia, Undecidability of the spectral gap in one dimension, Phys. Rev. X 10, 031038 (2020).
  30. A. Young, Quantum spin systems, in Encyclopedia of Mathematical Physics, 2nd ed., edited by M. B. Richard Szabo (Elsevier, 2014).
  31. J. Bausch, T. S. Cubitt, A. Lucia, D. Perez-Garcia, and M. M. Wolf, Size-driven quantum phase transitions, Proc. Natl. Acad. Sci. USA 115, 19 (2018).
  32. J. Bausch, T. S. Cubitt, and J. D. Watson, Uncomputability of phase diagrams, Nat. Commun. 12, 452 (2021).
  33. J. D. Watson and J. Bausch, The complexity of approximating critical points of quantum phase transitions, arXiv:2105.13350.
  34. T. S. Cubitt, A note on the second spectral gap incompleteness theorem, arXiv:2105.09854.
  35. J. D. Watson, E. Onorati, and T. S. Cubitt, Uncomputably complex renormalisation group flows, Nat. Commun. 13, 7618 (2022).
  36. J. D. Watson, J. Bausch, and S. Gharibian, The complexity of translationally invariant problems beyond ground state energies, in 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023), Leibniz International Proceedings in Informatics (LIPIcs) (Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Wadern, Germany, 2023), Vol. 254, pp. 1–21.
  37. D. Gottesman and S. Irani, The quantum and classical complexity of translationally invariant tiling and Hamiltonian problems, Theory Comput. 9, 31 (2013).
  38. L. D. Faddeev, How algebraic Bethe ansatz works for integrable model, in Relativistic Gravitation and Gravitational Radiation: Proceedings of the Les Houches School of Physics, Les Houches, France, 1995, edited by A. Connes et al. (North-Holland, Amsterdam, The Netherlands, 1996), pp. 149–219.
  39. F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Entanglement spectrum of a topological phase in one dimension, Phys. Rev. B 81, 064439 (2010).
  40. D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, Matrix product state representations, Quantum Inf. Comput. 7, 401 (2007).
  41. X. Chen, Z.-C. Gu, and X.-G. Wen, Classification of gapped symmetric phases in one-dimensional spin systems, Phys. Rev. B 83, 035107 (2011).
  42. S. Bravyi and M. Vyalyi, Commutative version of the k-local Hamiltonian problem and common eigenspace problem, Quantum Inf. Comput. 5, 187 (2005).
  43. J. Kempe, A. Kitaev, and O. Regev, The complexity of the local hamiltonian problem, in FSTTCS 2004: Foundations of Software Technology and Theoretical Computer Science, edited by K. Lodaya and M. Mahajan (Springer, Berlin, Heidelberg, 2005), pp. 372–383.
  44. R. M. Robinson, Undecidability and nonperiodicity for tilings of the plane, Invent. Math. 12, 177 (1971).
  45. A. Beekman, L. Rademaker, and J. van Wezel, An introduction to spontaneous symmetry breaking, SciPost Physics Lecture Notes 11 (2019).
  46. L. Landau, J. F. Perez, and W. F. Wreszinski, Energy gap, clustering, and the Goldstone theorem in statistical mechanics, J. Stat. Phys. 26, 755 (1981).
  47. T. Koma and H. Tasaki, Symmetry breaking and finite-size effects in quantum many-body systems, J. Stat. Phys. 76, 745 (1994).
  48. A. M. Turing, On computable numbers, with an application to the Entscheidungsproblem, Proc. Lond. Mathemat. Soc. 2 42, 230 (1936).
  49. E. Bernstein and U. Vazirani, Quantum complexity theory, SIAM J. Comput. 26, 1411 (1997).
  50. R. P. Feynman, Quantum mechanical computers., Found. Phys. 16, 507 (1986).
  51. A. Y. Kitaev, A. Shen, and M. N. Vyalyi, Classical and Quantum Computation (American Mathematical Society, Providence, RI, 2002).
  52. D. Aharonov, D. Gottesman, S. Irani, and J. Kempe, The power of quantum systems on a line, Commun. Math. Phys. 287, 41 (2009).

Outline

Information

Phys. Rev. Research 8, 013033– Published 14 January, 2026
  • Received 29 January 2025
  • Accepted 4 December 2025
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