Communications in Mathematical Physics

, Volume 366, Issue 3, pp 895–926 | Cite as

Locality at the Boundary Implies Gap in the Bulk for 2D PEPS

  • Michael J. Kastoryano
  • Angelo LuciaEmail author
  • David Perez-Garcia


Proving that the parent Hamiltonian of a Projected Entangled Pair State (PEPS) is gapped remains an important open problem. We take a step forward in solving this problem by showing two results: first, we identify an approximate factorization condition on the boundary state of rectangular subregions that is sufficient to prove that the parent Hamiltonian of the bulk 2D PEPS has a constant gap in the thermodynamic limit; second, we then show that Gibbs state of a local, finite-range Hamiltonian satisfy such condition. The proof applies to the case of injective and MPO-injective PEPS, employs the martingale method of nearly commuting projectors, and exploits a result of Araki (Commun Math Phys 14(2):120–157, 1969) on the robustness of one dimensional Gibbs states. Our result provides one of the first rigorous connections between boundary theories and dynamical properties in an interacting many body system.


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We thank Albert Werner and Wojciech De Roeck for fruitful discussions. M. J. K. was supported by the VILLUM FONDEN Young Investigator Program. A. L. acknowledges financial support from the European Research Council (ERC Grant Agreement No. 337603), the Danish Council for Independent Research (Sapere Aude), VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059), the Walter Burke Institute for Theoretical Physics in the form of the Sherman Fairchild Fellowship as well as support from the Institute for Quantum Information and Matter (IQIM), an NSF Physics Frontiers Center (NFS Grant PHY-1733907). D. P. G. acknowledges support from MINECO (Grant MTM2014-54240-P), Comunidad de Madrid (Grant QUITEMAD+-CM, ref. S2013/ICE-2801), and Severo Ochoa project SEV-2015-556. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 648913).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.NBIA, Niels Bohr InstituteUniversity of CopenhagenCopenhagenDenmark
  2. 2.Institute for Theoretical PhysicsUniversity of CologneCologneGermany
  3. 3.QMATH, Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark
  4. 4.Walter Burke Institute for Theoretical Physics and Institute for Quantum Information & MatterCalifornia Institute of TechnologyPasadenaUSA
  5. 5.Departamento de Análisis MatemáticoUniversidad Complutense de MadridMadridSpain
  6. 6.ICMAT, C/ Nicolás CabreraMadridSpain

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