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Hilbert Space Fragmentation and Commutant Algebras

Moudgalya, Sanjay and Motrunich, Olexei I. (2022) Hilbert Space Fragmentation and Commutant Algebras. Physical Review X, 12 (1). Art. No. 011050. ISSN 2160-3308. doi:10.1103/PhysRevX.12.011050.

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We study the phenomenon of Hilbert space fragmentation in isolated Hamiltonian and Floquet quantum systems using the language of commutant algebras, the algebra of all operators that commute with each local term that appears in the Hamiltonian or each local gate of the circuit. We provide a precise definition of Hilbert space fragmentation in this formalism as the case where the dimension of the commutant algebra grows exponentially with the system size. Fragmentation can, hence, be distinguished from systems with conventional symmetries such as U(1) or SU(2), where the dimension of the commutant algebra grows polynomially with the system size. Furthermore, the commutant algebra language also helps distinguish between “classical” and “quantum” Hilbert space fragmentation, where the former refers to fragmentation in the product state basis. We explicitly construct the commutant algebra in several systems exhibiting classical fragmentation, including the t−J_z model and the spin-1 dipole-conserving model, and we illustrate the connection to previously studied “statistically localized integrals of motion.” We also revisit the Temperley-Lieb spin chains, including the spin-1 biquadratic chain widely studied in the literature, and show that they exhibit quantum Hilbert space fragmentation. Finally, we study the contribution of the full commutant algebra to the Mazur bounds in various cases. In fragmented systems, we use expressions for the commutant to analytically obtain new or improved Mazur bounds for autocorrelation functions of local operators that agree with previous numerical results. In addition, we are able to rigorously show the localization of the on-site spin operator in the spin-1 dipole-conserving model.

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Motrunich, Olexei I.0000-0001-8031-0022
Additional Information:© 2022. Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Received 11 October 2021; revised 18 January 2022; accepted 21 January 2022; published 16 March 2022. We thank Berislav Buca, Anushya Chandran, Arpit Dua, Duncan Haldane, Paul Fendley, Chris Laumann, Cheng-Ju Lin, Daniel Mark, Alan Morningstar, Tibor Rakovszky, Pablo Sala, and Shivaji Sondhi for useful discussions. S. M. thanks Abhinav Prem, Rahul Nandkishore, Nicolas Regnault, and B. Andrei Bernevig for related previous collaborations. This work was supported by the Walter Burke Institute for Theoretical Physics at Caltech; the Institute for Quantum Information and Matter, a National Science Foundation (NSF) Physics Frontiers Center (NSF Grant No. PHY-1733907), and the National Science Foundation through Grant No. DMR-2001186. A part of this work was completed at the Aspen Center for Physics, which is supported by National Science Foundation Grant No. PHY-1607611. This work was partially supported by a grant from the Simons Foundation.
Group:Walter Burke Institute for Theoretical Physics, Institute for Quantum Information and Matter
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Walter Burke Institute for Theoretical Physics, CaltechUNSPECIFIED
Institute for Quantum Information and Matter (IQIM)UNSPECIFIED
Simons FoundationUNSPECIFIED
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Record Number:CaltechAUTHORS:20220113-182211896
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:112869
Deposited By: George Porter
Deposited On:13 Jan 2022 21:20
Last Modified:25 Mar 2022 21:24

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