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Gapless Hamiltonians for the Toric Code Using the Projected Entangled Pair State Formalism

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dc.contributor.authorFernández González, Carlos
dc.contributor.authorSchuch, Norbert
dc.contributor.authorWolf, Michael M.
dc.contributor.authorCirac, J.I.
dc.contributor.authorPérez García, David
dc.date.accessioned2023-06-20T00:25:06Z
dc.date.available2023-06-20T00:25:06Z
dc.date.issued2012-12
dc.description.abstractWe study Hamiltonians which have Kitaev's toric code as a ground state, and show how to construct a Hamiltonian which shares the ground space of the toric code, but which has gapless excitations with a continuous spectrum in the thermodynamic limit. Our construction is based on the framework of projected entangled pair states, and can be applied to a large class of two-dimensional systems to obtain gapless "uncle Hamiltonians."
dc.description.departmentDepto. de Análisis Matemático y Matemática Aplicada
dc.description.facultyFac. de Ciencias Matemáticas
dc.description.facultyInstituto de Matemática Interdisciplinar (IMI)
dc.description.refereedTRUE
dc.description.sponsorshipUnión Europea. FP7
dc.description.sponsorshipComunidad de Madrid
dc.description.sponsorshipMinisterio de Educación
dc.description.sponsorshipMinisterio de Economía y Competitividad (MINECO)
dc.description.sponsorshipGordon and Betty Moore Foundation through Caltech's Center for the Physics of Information
dc.description.sponsorshipNSF
dc.description.sponsorshipARO
dc.description.statuspub
dc.eprint.idhttps://eprints.ucm.es/id/eprint/19863
dc.identifier.doi10.1103/PhysRevLett.109.26040
dc.identifier.issn0031-9007
dc.identifier.officialurlhttp://prl.aps.org/pdf/PRL/v109/i26/e260401
dc.identifier.relatedurlhttp://www.aps.org/
dc.identifier.urihttps://hdl.handle.net/20.500.14352/42540
dc.issue.number26
dc.journal.titlePhysical Review Letters
dc.language.isoeng
dc.publisherAmerican Physical Society
dc.relation.projectIDQUEVADIS (233859)
dc.relation.projectIDQUITEMAD-CM (S2009/ESP-1594)
dc.relation.projectIDI-MATH
dc.relation.projectID(MTM2008-01366)
dc.relation.projectID(PHY-0803371)
dc.relation.projectID(W911NF-09-1-0442)
dc.rights.accessRightsopen access
dc.subject.cdu51
dc.subject.keywordToric code
dc.subject.ucmMatemáticas (Matemáticas)
dc.subject.unesco12 Matemáticas
dc.titleGapless Hamiltonians for the Toric Code Using the Projected Entangled Pair State Formalism
dc.typejournal article
dc.volume.number109
dcterms.references[1] A. Kitaev, Ann. Phys. (Amsterdam) 303, 2 (2003). [2] S. Bravyi, M. Hastings, and S. Michalakis, J. Math. Phys. (N.Y.) 51, 093512 (2010). [3] S. Bravyi and M. B. Hastings, arXiv:1001.4363. [4] F. Verstraete and J. I. Cirac, Phys. Rev. A 70, 060302 (2004); F. Verstraete and J. I. Cirac, arXiv:cond-mat/0407066. [5] F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac, Phys. Rev. Lett. 96, 220601 (2006). [6] M. Fannes, B. Nachtergaele, and R. F. Werner, Commun.Math. Phys. 144, 443 (1992). [7] D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, Quantum Inf. Comput. 7, 401 (2007). [8] M. B. Hastings, Phys. Rev. B 76, 035114 (2007). [9] C.V. Kraus, N. Schuch, F. Verstraete, and J. I. Cirac, Phys. Rev. A 81, 052338 (2010). [10] N. Schuch, I. Cirac, and D. Pérez-García, Ann. Phys. (Amsterdam) 325, 2153 (2010). [11] M.A. Levin and X.-G.Wen, Phys. Rev. B 71, 045110 (2005). [12] A. Affleck, T. Kennedy, E.H. Lieb, and H. Tasaki, Commun. Math. Phys. 115, 477 (1988). [13] S. Michalakis and J. Pytel, arXiv:1109.1588. [14] X. Chen, B. Zeng, Z. C. Gu, I. L. Chuang, and X. G. Wen, Phys. Rev. B 82, 165119 (2010). [15] M.Levin and X.-G.Wen, Phys.Rev.Lett. 96, 110405 (2006). [16] A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006). [17] The parent Hamiltonian is usually obtained by considering local projectors on just two sites, but we need three sites for our construction in this specific case. A discussion on the need of either two or three sites and its relationship with bond and physical dimensions can be found in Ref. [18]. [18] C. Fernández-González, N. Schuch, M. M. Wolf, J. I. Cirac, and D. Pérez-García, arXiv:1210.6613. [19] For periodic boundary conditions, domain walls come in pairs, i.e., for every │0 + 1> (‘‘up’’) domain wall there is a │1 + 0> (‘‘down’’) domain wall. Since both domain walls are in a momentum eigenstate, there is a nonzero probability that they are at adjacent sites, leading to a configuration │010> which is penalized by the Hamiltonian. [20] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.109.260401 for a detailed proof of the structure of the kernel of the uncle Hamiltonian for the toric code (Part A), and the proof that the spectrum of the uncle Hamiltonian for the toric code is[0, ∞) (Part B).
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relation.isAuthorOfPublication5edb2da8-669b-42d1-867d-8fe3144eb216
relation.isAuthorOfPublication.latestForDiscovery5edb2da8-669b-42d1-867d-8fe3144eb216

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