The `operator entanglement' of a quantum operator O is a useful indicator of its complexity, and, in one-dimension, of its approximability by matrix product operators. Here we focus on spin chains with a global U(1) conservation law, and on operators O with a well-defined U(1) charge, for which it is possible to resolve the operator entanglement of O according to the U(1) symmetry. We employ the notion of symmetry resolved operator entanglement (SROE) introduced in [PRX Quantum 4, 010318 (2023)] and extend the results of the latter paper in several directions. Using a combination of conformal field theory and of exact analytical and numerical calculations in critical free fermionic chains, we study the SROE of the thermal density matrix ρ_β = e^(−βH) and of charged local operators evolving in Heisenberg picture O = e^(itH)Oe^(−itH). Our main results are: i) the SROE of ρ_β obeys the operator area law; ii) for free fermions, local operators in Heisenberg picture can have a SROE that grows logarithmically in time or saturates to a constant value; iii) there is equipartition of the entanglement among all the charge sectors except for a pair of fermionic creation and annihilation operators.
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Abstract
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Acknowledgement
We thank Filiberto Ares, Giuseppe Di Giulio, Aniket Rath and Vittorio Vitale for useful discussions. We especially thank Gilles Parez for interesting comments about the draft. PC acknowledges support from ERC under Consolidator grant number 771536 (NEMO). JD acknowledges support from the Agence Nationale de la Recherche through ANR-20-CE30-0017- 01 project ‘QUADY’ and ANR-22-CE30-0004-01 project ‘UNIOPEN’. SM thanks the Caltech Institute for Quantum Information and Matter and the Walter Burke Institute for Theoretical Physics at Caltech. All authors acknowledge the hospitality of the Simons Center for Geometry and Physics in Stony Brook during the program ‘Fluctuations, Entanglement, and Chaos: Exact Results’ where this work was completed.
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Additional details
- ISSN
- 1751-8121
- European Research Council
- 771536
- Agence Nationale de la Recherche
- ANR-20-CE30-0017-01
- Agence Nationale de la Recherche
- ANR-22-CE30-0004-01
- California Institute of Technology
- Institute for Quantum Information and Matter
- California Institute of Technology
- Walter Burke Institute for Theoretical Physics
- Caltech groups
- Walter Burke Institute for Theoretical Physics, Institute for Quantum Information and Matter