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Gauging quantum phases: A matrix product state approach

David Blanik1, José Garre-Rubio2,3, and Norbert Schuch1,2

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Phys. Rev. B 112, 115110 – Published 3 September, 2025

DOI: https://doi.org/10.1103/gkh9-lgrk

Abstract

Utilizing the framework of matrix product states, we investigate gauging as a method for exploring quantum phases of matter. Specifically, we describe how symmetry-protected topological (SPT) phases and spontaneous symmetry breaking (SSB) phases in one-dimensional spin systems behave under twisted gauging, a generalization of the well-known gauging procedure for globally symmetric states. Compared to previous order parameter-based approaches, our analysis is not limited to the case of maximally noncommutative (MNC) phases, and we use our findings to propose a generalization of the Kennedy-Tasaki transformation to the non-MNC setting. A key result of our work is that gauging produces configurations characterized by a combination of MNC order and symmetry breaking, when applied to non-MNC SPT phases. More generally, we conjecture a precise correspondence between SSB and non-MNC SPT phases, possibly enabling the detection of such phases using local and string order parameters.

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References (38)

  1. H. A. Kramers and G. H. Wannier, Statistics of the two-dimensional ferromagnet. Part I, Phys. Rev. 60, 252 (1941).
  2. A. Miyake, Quantum computation on the edge of a symmetry-protected topological order, Phys. Rev. Lett. 105, 040501 (2010).
  3. D. V. Else, I. Schwarz, S. D. Bartlett, and A. C. Doherty, Symmetry-protected phases for measurement-based quantum computation, Phys. Rev. Lett. 108, 240505 (2012).
  4. J. Miller and A. Miyake, Resource quality of a symmetry-protected topologically ordered phase for quantum computation, Phys. Rev. Lett. 114, 120506 (2015).
  5. D. T. Stephen, D.-S. Wang, A. Prakash, T.-C. Wei, and R. Raussendorf, Computational power of symmetry-protected topological phases, Phys. Rev. Lett. 119, 010504 (2017).
  6. T. Kennedy and H. Tasaki, Hidden symmetry breaking and the Haldane phase in S = 1 quantum spin chains, Commun. Math. Phys. 147, 431 (1992).
  7. I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Rigorous results on valence-bond ground states in antiferromagnets, Phys. Rev. Lett. 59, 799 (1987).
  8. K. Duivenvoorden and T. Quella, From symmetry-protected topological order to Landau order, Phys. Rev. B 88, 125115 (2013).
  9. D. V. Else, S. D. Bartlett, and A. C. Doherty, Hidden symmetry-breaking picture of symmetry-protected topological order, Phys. Rev. B 88, 085114 (2013).
  10. F. Pollmann and A. M. Turner, Detection of symmetry-protected topological phases in one dimension, Phys. Rev. B 86, 125441 (2012).
  11. L. Li, M. Oshikawa, and Y. Zheng, Noninvertible duality transformation between symmetry-protected topological and spontaneous symmetry breaking phases, Phys. Rev. B 108, 214429 (2023).
  12. D.-C. Lu, Z. Sun, and Y.-Z. You, Realizing triality and p-ality by lattice twisted gauging in (1+1)d quantum spin systems, SciPost Phys. 17, 136 (2024).
  13. M. Oshikawa, Hidden Z2*Z2 symmetry in quantum spin chains with arbitrary integer spin, J. Phys.: Condens. Matter 4, 7469 (1992).
  14. L. Lootens, C. Delcamp, and F. Verstraete, Entanglement and the density matrix renormalisation group in the generalised Landau paradigm, Nat. Phys. (2025), doi: 10.1038/s41567-025-02961-2.
  15. B. V.-D. Cuiper and C. Delcamp, Twisted gauging and topological sectors in (2+1)d Abelian lattice gauge theories, arXiv:2501.16301
  16. M. B. Hastings, An area law for one-dimensional quantum systems, J. Stat. Mech. (2007) P08024.
  17. M. M. Wolf, F. Verstraete, M. B. Hastings, and J. I. Cirac, Area laws in quantum systems: Mutual information and correlations, Phys. Rev. Lett. 100, 070502 (2008).
  18. N. Schuch, D. Perez-Garcia, and I. Cirac, Classifying quantum phases using matrix product states and PEPS, Phys. Rev. B 84, 165139 (2011).
  19. J. Haegeman, K. Van Acoleyen, N. Schuch, J. I. Cirac, and F. Verstraete, Gauging quantum states: From global to local symmetries in many-body systems, Phys. Rev. X 5, 011024 (2015).
  20. D. Perez-Garcia, F. Verstraete, M. Wolf, and J. I. Cirac, Matrix product state representations, Quantum Inf. Comput. 7, 401 (2007).
  21. A. Molnar, J. Garre-Rubio, D. Pérez-García, N. Schuch, and J. I. Cirac, Normal projected entangled pair states generating the same state, New J. Phys. 20, 113017 (2018).
  22. 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).
  23. Y. Ogata, Classification of symmetry protected topological phases in quantum spin chains, Curr. Dev. Math. 2020, 41 (2020).
  24. J. G. Rubio, A. Molnar, and Y. Ogata, Classifying symmetric and symmetry-broken spin chain phases with anomalous group actions, arXiv:2403.18573.
  25. A phase invariant is a property or quantity that is shared by all members of the phase.
  26. J. I. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys. 93, 045003 (2021).
  27. C. de Groot, Topological phases, symmetries and open systems, Ph.D. thesis, Technische Universität München, 2022.
  28. D. J. Williamson, N. Bultinck, M. Mariën, M. B. Şahinoğlu, J. Haegeman, and F. Verstraete, Matrix product operators for symmetry-protected topological phases: Gauging and edge theories, Phys. Rev. B 94, 205150 (2016).
  29. R. Thorngren and Y. Wang, Fusion category symmetry. Part I. Anomaly in-flow and gapped phases, J. High Energy Phys. 04 (2024) 132.
  30. L. Lootens, C. Delcamp, G. Ortiz, and F. Verstraete, Dualities in one-dimensional quantum lattice models: Symmetric Hamiltonians and matrix product operator intertwiners, PRX Quantum 4, 020357 (2023).
  31. J. Garre-Rubio, Emergent (2+1)D topological orders from iterative (1+1)D gauging, Nat. Commun. 15, 7986 (2024).
  32. L. Lootens, C. Delcamp, and F. Verstraete, Dualities in one-dimensional quantum lattice models: Topological sectors, PRX Quantum 5, 010338 (2024).
  33. We use the symbol to denote equivalence up to normalization of the generated MPS.
  34. C. de Groot, D. T. Stephen, A. Molnar, and N. Schuch, Inaccessible entanglement in symmetry protected topological phases, J. Phys. A: Math. Theor. 53, 335302 (2020).
  35. ResHGρ≡ρ|H for ρ∈RepCαG and H≤G.
  36. J. G. Berkovič and E. M. Žmud', Characters of Finite Groups. Part 1, Translations of Mathematical Monographs No. 172 (American Mathematical Society, Providence, RI, 1998).
  37. C. Cheng, A character theory for projective representations of finite groups, Linear Algebra Appl. 469, 230 (2015).
  38. J. I. Cirac, D. Perez-Garcia, N. Schuch, and F. Verstraete, Matrix product density operators: Renormalization fixed points and boundary theories, Ann. Phys. 378, 100 (2017).

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Phys. Rev. B 112, 115110– Published 3 September, 2025
  • Received 15 May 2025
  • Revised 28 July 2025
  • Accepted 30 July 2025
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