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Classifying quantum phases using matrix product states and projected entangled pair states

Schuch, Norbert and Pérez-García, David and Cirac, Ignacio (2011) Classifying quantum phases using matrix product states and projected entangled pair states. Physical Review B, 84 (16). Art. No. 165139. ISSN 1098-0121.

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We give a classification of gapped quantum phases of one-dimensional systems in the framework of matrix product states (MPS) and their associated parent Hamiltonians, for systems with unique as well as degenerate ground states and in both the absence and the presence of symmetries. We find that without symmetries, all systems are in the same phase, up to accidental ground-state degeneracies. If symmetries are imposed, phases without symmetry breaking (i.e., with unique ground states) are classified by the cohomology classes of the symmetry group, that is, the equivalence classes of its projective representations, a result first derived by Chen, Gu, and Wen [ Phys. Rev. B 83 035107 (2011)]. For phases with symmetry breaking (i.e., degenerate ground states), we find that the symmetry consists of two parts, one of which acts by permuting the ground states, while the other acts on individual ground states, and phases are labeled by both the permutation action of the former and the cohomology class of the latter. Using projected entangled pair states (PEPS), we subsequently extend our framework to the classification of two-dimensional phases in the neighborhood of a number of important cases, in particular, systems with unique ground states, degenerate ground states with a local order parameter, and topological order. We also show that in two dimensions, imposing symmetries does not constrain the phase diagram in the same way it does in one dimension. As a central tool, we introduce the isometric form, a normal form for MPS and PEPS, which is a renormalization fixed point. Transforming a state to its isometric form does not change the phase, and thus we can focus on to the classification of isometric forms.

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Additional Information:© 2011 American Physical Society. Received 15 March 2011; published 31 October 2011. We acknowledge helpful discussions with Salman Beigi, Oliver Buerschaper, Steve Flammia, Stephen Jordan, Alexei Kitaev, Robert König, Spiros Michalakis, John Preskill, Volkher Scholz, Frank Verstraete, and Michael Wolf. This work has been supported by the Gordon and Betty Moore Foundation through Caltech’s Center for the Physics of Information, NSF Grant No. PHY-0803371, ARO Grant No. W911NF-09-1-0442, Spanish Grants No. I-MATH, No. MTM2008-01366, and No. S2009/ESP-1594, the European project QUEVADIS, and the DFG (Forschergruppe 635).
Funding AgencyGrant Number
NSF Caltech's Center for the Physics of Information Gordon and Betty Moore FoundationPHY-0803371
Army Research Office (ARO)W911NF-09-1-0442
Spanish GrantsI-MATH
Spanish GrantsMTM2008-01366
Spanish GrantsS2009/ESP-1594
Deutsche Forschungsgemeinschaft (DFG)Forschergruppe 635
Issue or Number:16
Classification Code:PACS: 71.10.Hf, 03.65.Ud
Record Number:CaltechAUTHORS:20111205-151056277
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:28303
Deposited By: Tony Diaz
Deposited On:06 Dec 2011 23:00
Last Modified:03 Oct 2019 03:31

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