Stability of invertible, frustration-free ground states against large perturbations

Sven Bachmann1, Wojciech De Roeck2, Brecht Donvil3,4, and Martin Fraas5

1Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
2Institute of Theoretical Physics, K.U. Leuven, 3001 Leuven, Belgium
3Institute for Complex Quantum Systems and Center for IQST, Ulm University, 89069 Ulm, Germany
4Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland
5Department of Mathematics, University of California, Davis, Davis, CA, 95616, USA

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

A gapped ground state of a quantum spin system has a natural length scale set by the gap. This length scale governs the decay of correlations. A common intuition is that this length scale also controls the spatial relaxation towards the ground state away from impurities or boundaries. The aim of this article is to take a step towards a proof of this intuition. We assume that the ground state is frustration-free and invertible, i.e. it has no long-range entanglement. Moreover, we assume the property that we are aiming to prove for one specific kind of boundary condition; namely open boundary conditions. This assumption is also known as the "local topological quantum order" (LTQO) condition. With these assumptions we can prove stretched exponential decay away from boundaries or impurities, for any of the ground states of the perturbed system. In contrast to most earlier results, we do not assume that the perturbations at the boundary or the impurity are small. In particular, the perturbed system itself can have long-range entanglement.

► BibTeX data

► References

[1] Wojciech De Roeck and Marius Schütz. ``An exponentially local spectral flow for possibly non-self-adjoint perturbations of non-interacting quantum spins, inspired by kam theory''. Letters in Mathematical Physics 107, 505–532 (2017).
https:/​/​doi.org/​10.1007/​s11005-016-0913-z

[2] Simone Del Vecchio, Jürg Fröhlich, Alessandro Pizzo, and Stefano Rossi. ``Lie-schwinger block-diagonalization and gapped quantum chains: analyticity of the ground-state energy''. Journal of Functional Analysis 279, 108703 (2020).
https:/​/​doi.org/​10.1016/​j.jfa.2020.108703

[3] Juerg Froehlich and Alessandro Pizzo. ``Lie–schwinger block-diagonalization and gapped quantum chains''. Communications in Mathematical Physics 375, 2039–2069 (2020).
https:/​/​doi.org/​10.1007/​s00220-019-03613-2

[4] DA Yarotsky. ``Ground states in relatively bounded quantum perturbations of classical lattice systems''. Communications in mathematical physics 261, 799–819 (2006).
https:/​/​doi.org/​10.1007/​s00220-005-1456-9

[5] Nilanjana Datta, Roberto Fernández, and Jürg Fröhlich. ``Low-temperature phase diagrams of quantum lattice systems. i. stability for quantum perturbations of classical systems with finitely-many ground states''. Journal of statistical physics 84, 455–534 (1996).
https:/​/​doi.org/​10.1007/​BF02179651

[6] Christian Borgs, R Koteckỳ, and D Ueltschi. ``Low temperature phase diagrams for quantum perturbations of classical spin systems''. Communications in mathematical physics 181, 409–446 (1996).
https:/​/​doi.org/​10.1007/​BF02101010

[7] Matthew F Lapa and Michael Levin. ``Stability of ground state degeneracy to long-range interactions'' (2021). arXiv:2107.11396.
arXiv:2107.1139

[8] Sergey Bravyi, Matthew B Hastings, and Spyridon Michalakis. ``Topological quantum order: stability under local perturbations''. Journal of mathematical physics 51, 093512 (2010).
https:/​/​doi.org/​10.1063/​1.3490195

[9] Spyridon Michalakis and Justyna P Zwolak. ``Stability of frustration-free hamiltonians''. Communications in Mathematical Physics 322, 277–302 (2013).
https:/​/​doi.org/​10.1007/​s00220-013-1762-6

[10] Bruno Nachtergaele, Robert Sims, and Amanda Young. ``Quasi-locality bounds for quantum lattice systems. part ii. perturbations of frustration-free spin models with gapped ground states''. In Annales Henri Poincaré. Volume 23, pages 393–511. Springer (2022).
https:/​/​doi.org/​10.1007/​s00023-021-01086-5

[11] Bruno Nachtergaele, Robert Sims, and Amanda Young. ``Stability of the bulk gap for frustration-free topologically ordered quantum lattice systems'' (2021). arXiv:2102.07209.
arXiv:2102.0720

[12] Sven Bachmann, Spyridon Michalakis, Bruno Nachtergaele, and Robert Sims. ``Automorphic equivalence within gapped phases of quantum lattice systems''. Communications in Mathematical Physics 309, 835–871 (2012).
https:/​/​doi.org/​10.1007/​s00220-011-1380-0

[13] Wojciech De Roeck and Marius Schütz. ``Local perturbations perturb—exponentially–locally''. Journal of Mathematical Physics 56, 061901 (2015).
https:/​/​doi.org/​10.1063/​1.4922507

[14] Alexei Kitaev. ``Anyons in an exactly solved model and beyond''. Annals of Physics 321, 2–111 (2006).
https:/​/​doi.org/​10.1016/​j.aop.2005.10.005

[15] Alexei Kitaev and Chris Laumann. ``Topological phases and quantum computation''. Exact methods in low-dimensional statistical physics and quantum computing, Lecture Notes of the Les Houches Summer SchoolPages 101–125 (2009). url:.
arXiv:0904.2771

[16] Bruno Nachtergaele and Nicholas E Sherman. ``Dispersive toric code model with fusion and defusion''. Physical Review B 101, 115105 (2020).
https:/​/​doi.org/​10.1103/​PhysRevB.101.115105

[17] Joscha Henheik, Stefan Teufel, and Tom Wessel. ``Local stability of ground states in locally gapped and weakly interacting quantum spin systems''. Letters in Mathematical Physics 112, 1–12 (2022).
https:/​/​doi.org/​10.1007/​s11005-021-01494-y

[18] Matthew B Hastings. ``Quantum belief propagation: An algorithm for thermal quantum systems''. Physical Review B 76, 201102 (2007).
https:/​/​doi.org/​10.1103/​PhysRevB.76.201102

[19] Kohtaro Kato and Fernando GSL Brandao. ``Quantum approximate markov chains are thermal''. Communications in Mathematical Physics 370, 117–149 (2019).
https:/​/​doi.org/​10.1007/​s00220-019-03485-6

[20] Matthew B Hastings and Xiao-Gang Wen. ``Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance''. Physical review b 72, 045141 (2005).
https:/​/​doi.org/​10.1103/​PhysRevB.72.045141

[21] Daniel S Freed. ``Anomalies and invertible field theories''. In Proc. Symp. Pure Math. Volume 88, pages 25–46. (2014). url:.
arXiv:1404.7224

[22] A. Kitaev. ``On the classification of short-range entangled states''. http:/​/​scgp.stonybrook.edu/​video_portal/​video.php?id=2010.
http:/​/​scgp.stonybrook.edu/​video_portal/​video.php?id=2010

[23] Zheng-Cheng Gu and Xiao-Gang Wen. ``Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order''. Physical Review B 80, 155131 (2009).
https:/​/​doi.org/​10.1103/​PhysRevB.80.155131

[24] Anton Kapustin and Nikita Sopenko. ``Hall conductance and the statistics of flux insertions in gapped interacting lattice systems''. Journal of Mathematical Physics 61, 101901 (2020).
https:/​/​doi.org/​10.1063/​5.0022944

[25] E.H. Lieb and D.W. Robinson. ``The finite group velocity of quantum spin systems''. Commun. Math. Phys. 28, 251–257 (1972).
https:/​/​doi.org/​10.1007/​978-3-662-10018-9_25

[26] Bruno Nachtergaele, Robert Sims, and Amanda Young. ``Quasi-locality bounds for quantum lattice systems. i. lieb-robinson bounds, quasi-local maps, and spectral flow automorphisms''. Journal of Mathematical Physics 60, 061101 (2019).
https:/​/​doi.org/​10.1063/​1.5095769

[27] A. Bruckner. ``Minimal superadditive extensions of superadditive functions''. Pacific J. Math. 10, 1155–1162 (1960). url: msp.org/​pjm/​1960/​10-4/​pjm-v10-n4-s.pdf#page=51.
https:/​/​msp.org/​pjm/​1960/​10-4/​pjm-v10-n4-s.pdf#page=51

Cited by

[1] Joscha Henheik, Stefan Teufel, and Tom Wessel, "Local stability of ground states in locally gapped and weakly interacting quantum spin systems", Letters in Mathematical Physics 112 1, 9 (2022).

[2] Joscha Henheik and Tom Wessel, "On adiabatic theory for extended fermionic lattice systems", arXiv:2208.12220.

[3] Angelo Lucia, Alvin Moon, and Amanda Young, "Stability of the spectral gap and ground state indistinguishability for a decorated AKLT model", arXiv:2209.01141.

The above citations are from SAO/NASA ADS (last updated successfully 2022-10-01 13:38:55). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2022-10-01 13:38:53).