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arXiv:1912.12552 · Annales Henri Poincaré

Lieb–Robinson Bounds and Strongly Continuous Dynamics for a Class of Many-Body Fermion Systems in $${\mathbb {R}}^d$$

Martin Gebert, Bruno Nachtergaele, Jake Reschke, Robert Sims

Published Sep 24, 2020• 16 citations
arXivHTMLPDF
DOI
10.1007/s00023-020-00959-5

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On Lieb–Robinson Bounds for a Class of Continuum Fermions

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Abstract We consider the quantum dynamics of a many-fermion system in $${{\mathbb {R}}}^d$$ R d with an ultraviolet regularized pair interaction as previously studied in Gebert et al. (Ann Henri Poincaré 21(11):3609–3637, 2020). We provide a Lieb–Robinson bound under substantially relaxed assumptions on the potentials. We also improve the associated one-body Lieb–Robinson bound on $$L^2$$ L 2 -overlaps to an almost ballistic one (i.e., an almost linear light cone) under the same relaxed assumptions. Applications include the existence of the infinite-volume dynamics and clustering of ground states in the presence of a spectral gap. We also develop a fermionic continuum notion of conditional expectation and use it to approximate time-evolved fermionic observables by local ones, which opens the door to other applications of the Lieb–Robinson bounds.

arXiv:2310.17736•Annales Henri Poincaré•Jul 12, 2024•5 citations
Lieb–Robinson Bounds and Strongly Continuous Dynamics for a Class of Many-Body Fermion Systems in Rd{\mathbb {R}}^dRd
This dissertation presents new results on two problems concerning the dynamics of certainclasses of interacting many-body systems in quantum mechanics. Chapter 1 is an introduction,\nwhich includes mathematical preliminaries and a summary of the results.\nIn Chapter 2 we introduce the notion of transmission time to study the dynamics of disordered\nquantum spin chains and prove results relating its behavior to many-body localization properties.\nWe also study two versions of the so-called Local Integrals of Motion (LIOM) representation of\nspin chain Hamiltonians and their relation to dynamical many-body localization. We prove that\nuniform-in-time dynamical localization expressed by a zero-velocity Lieb-Robinson bound implies\nthe existence of a LIOM representation of the dynamics as well as a weak converse of this statement.\nWe also prove that for a class of spin chains satisfying a form of exponential dynamical localization,\nsparse perturbations result in a dynamics in which transmission times diverge at least as a power\nlaw of distance, with a power for which we provide lower bound that diverges with increasing\nsparseness of the perturbation.\nIn Chapter 3 we introduce a class of UV-regularized two-body interactions for fermions in Rd\nand prove a Lieb-Robinson estimate for the dynamics of this class of many-body systems. As a\nstep toward this result, we also prove a propagation bound of Lieb-Robinson type for Schrodinger\noperators. We apply the propagation bound to prove the existence of infnite-volume dynamics as\na strongly continuous group of automorphisms on the CAR algebra.
Abstract We consider the quantum dynamics of a many-fermion system in Rd{{\mathbb {R}}}^dRd R d with an ultraviolet regularized pair interaction as previously studied in Gebert et al. (Ann Henri Poincaré 21(11):3609–3637, 2020). We provide a Lieb–Robinson bound under substantially relaxed assumptions on the potentials. We also improve the associated one-body Lieb–Robinson bound on L2L^2L2 L 2 -overlaps to an almost ballistic one (i.e., an almost linear light cone) under the same relaxed assumptions. Applications include the existence of the infinite-volume dynamics and clustering of ground states in the presence of a spectral gap. We also develop a fermionic continuum notion of conditional expectation and use it to approximate time-evolved fermionic observables by local ones, which opens the door to other applications of the Lieb–Robinson bounds.