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arXiv:2205.10460 · EMS Press eBooks

From Lieb–Robinson bounds to automorphic equivalence

Bruno Nachtergaele

Published Jun 29, 2022• 1 citation
arXivHTMLPDF

Abstract

I review the role of Lieb–Robinson bounds in characterizing and utilizing the locality properties of the Heisenberg dynamics of quantum lattice systems. In particular, I discuss two definitions of gapped ground state phases and show that they are essentially equivalent.
DOI
10.4171/90-2/33

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We review recent progress on state-dependent Lieb--Robinson bounds for Bose--Hubbard Hamiltonians. In particular, Kuwahara, Vu, and Saito established that, for general bounded-density initial states, the Lieb--Robinson velocity is bounded by td−1t^{d-1}td−1 for large times, where ddd denotes the lattice dimension. We present a shorter proof of the weaker, but still polynomial velocity bound td+εt^{d+ε}td+ε.
We review the mathematical speed limits on quantum information processing in many-body systems. After the proof of the Lieb-Robinson Theorem in 1972, the past two decades have seen substantial developments in its application to other questions, such as the simulatability of quantum systems on classical or quantum computers, the generation of entanglement, and even the properties of ground states of gapped systems. Moreover, Lieb-Robinson bounds have been extended in non-trivial ways, to demonstrate speed limits in systems with power-law interactions or interacting bosons, and even to prove notions of locality that arise in cartoon models for quantum gravity with all-to-all interactions. We overview the progress which has occurred, highlight the most promising results and techniques, and discuss some central outstanding questions which remain open. To help bring newcomers to the field up to speed, we provide self-contained proofs of the field's most essential results.
Abstract We prove a Lieb–Robinson bound for lattice fermion models with polynomially decaying interactions, which can be used to show the locality of the quasi-local inverse Liouvillian. This allows us to prove automorphic equivalence and the local perturbations perturb locally (LPPL) principle for these systems. The proof of the Lieb–Robinson bound is based on the work of Else et al. [5], and our results also apply to spin systems. We explain why some newer Lieb–Robinson bounds for long-range spin systems cannot be used to prove the locality of the quasi-local inverse Liouvillian, and in some cases may not even hold for fermionic systems.