A Converse to Lieb–Robinson Bounds in One Dimension Using Index Theory
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23_2022_Article_1193.pdf
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1.29 MB
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Author(s)
Ranard, Daniel
Walter, Michael
Witteveen, Freek
Date Issued
July 26, 2022
Publisher
Springer International Publishing
Citation
Ranard, Daniel, Walter, Michael and Witteveen, Freek. 2022. "A Converse to Lieb–Robinson Bounds in One Dimension Using Index Theory."
Version
Final published version
Abstract
Abstract
Unitary dynamics with a strict causal cone (or “light cone”) have been studied extensively, under the name of quantum cellular automata (QCAs). In particular, QCAs in one dimension have been completely classified by an index theory. Physical systems often exhibit only approximate causal cones; Hamiltonian evolutions on the lattice satisfy Lieb–Robinson bounds rather than strict locality. This motivates us to study approximately locality preserving unitaries (ALPUs). We show that the index theory is robust and completely extends to one-dimensional ALPUs. As a consequence, we achieve a converse to the Lieb–Robinson bounds: any ALPU of index zero can be exactly generated by some time-dependent, quasi-local Hamiltonian in constant time. For the special case of finite chains with open boundaries, any unitary satisfying the Lieb–Robinson bound may be generated by such a Hamiltonian. We also discuss some results on the stability of operator algebras which may be of independent interest.
MIT Department
Massachusetts Institute of Technology. Center for Theoretical Physics
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DOI of Published Version
https://doi.org/10.1007/s00023-022-01193-x
