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On the dynamics of lattice systems with unbounded on-site terms in the\n Hamiltonian

Bruno Nachtergaele, Robert Sims

Published Oct 29, 2014• 3 citations
OpenAlex

Abstract

We supply the mathematical arguments required to complete the proofs of two\n previously published results: Lieb-Robinson bounds for the dynamics of quantum lattice\n systems with unbounded on-site terms in the Hamiltonian and the existence of the\n thermodynamic limit of the dynamics of such systems.

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On the dynamics of lattice systems with unbounded on-site terms in the Hamiltonian

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We supply the mathematical arguments required to complete the proofs of two previously published results: Lieb-Robinson bounds for the dynamics of quantum lattice systems with unbounded on-site terms in the Hamiltonian and the existence of the thermodynamic limit of the dynamics of such systems.

arXiv:1410.8174•Oct 29, 2014•6 citations

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Carla Rubiliani, Marius Wesle, Tom Wessel

In this note, we study a broad class of time-dependent and irreversible dynamics for quantum spin systems in both finite and infinite volume. We develop a simple and flexible strategy for expanding commutators of the form $[A,τ_{s,t} \, B]$ in terms of self-avoiding paths of sets, where $A$ and $B$ are local operators and $τ_{s,t}$ is the Heisenberg time evolution. This adaptive Dyson expansion requires no backwards time evolution and is thus also applicable to irreversible dynamics. Furthermore, we discuss how such an expansion directly implies Lieb-Robinson bounds. A central feature of our result is that the obtained Lieb-Robinson bounds are independent of any Hamiltonian and dissipative on-site terms of the generator.

math-phCross-listedarXiv:2610.12731•Oct 12, 2026•0 citations
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.
In this note, we study a broad class of time-dependent and irreversible dynamics for quantum spin systems in both finite and infinite volume. We develop a simple and flexible strategy for expanding commutators of the form [A,τs,t B][A,τ_{s,t} \, B][A,τs,t​B] in terms of self-avoiding paths of sets, where AAA and BBB are local operators and τs,tτ_{s,t}τs,t​ is the Heisenberg time evolution. This adaptive Dyson expansion requires no backwards time evolution and is thus also applicable to irreversible dynamics. Furthermore, we discuss how such an expansion directly implies Lieb-Robinson bounds. A central feature of our result is that the obtained Lieb-Robinson bounds are independent of any Hamiltonian and dissipative on-site terms of the generator.