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arXiv:2510.27495

Lieb-Robinson bounds in classical oscillating lattice systems

Ian Koot, Christiaan J. F. van de Ven

Published Oct 31, 2025• 0 citations
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Abstract

The aim of this paper is two-fold. First, we prove the existence of Lieb-Robinson bounds for classical particle systems describing harmonic oscillators interacting with arbitrarily many neighbors, both on lattices and on more general structures. Second, we prove the existence of a global dynamical system on the commutative resolvent algebra, a C*-algebra of bounded continuous functions on an infinite dimensional vector space, which serves as the classical analog of the Buchholz--Grundling resolvent algebra.

DOI
10.48550/arxiv.2510.27495

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The aim of this paper is two-fold. First, we prove the existence of Lieb-Robinson bounds for classical particle systems describing harmonic oscillators interacting with arbitrarily many neighbors, both on lattices and on more general structures. Second, we prove the existence of a global dynamical system on the commutative resolvent algebra, a C*-algebra of bounded continuous functions on an infinite dimensional vector space, which serves as the classical analog of the Buchholz--Grundling resolvent algebra.
We consider the dynamics of systems of lattice bosons with infinitely many degrees of freedom. We show that their dynamics defines a group of automorphisms on a C∗C^*C∗--algebra introduced by Buchholz, which extends the resolvent algebra of local field operators. For states that admit uniform bounds on moments of the local particle number, we derive propagation bounds of Lieb--Robinson type. Using these bounds, we show that the dynamics of local observables gives rise to a strongly continuous unitary group in the GNS representation. Moreover, accumulation points of finite-volume Gibbs states satisfy the KMS condition with respect to this group. This, in particular, proves the existence of KMS states.
The Lieb-Robinson theorem states that locality is approximately preserved in the dynamics of quantum lattice systems. Whenever one has finite-dimensional constituents, observables evolving in time under a local Hamiltonian will essentially grow linearly in their support, up to exponentially suppressed corrections. In this work, we formulate Lieb-Robinson bounds for general harmonic systems on general lattices, for which the constituents are infinite-dimensional, as systems representing discrete versions of free fields or the harmonic approximation to the Bose-Hubbard model. We consider both local interactions as well as infinite-ranged interactions, showing how corrections to locality are inherited from the locality of the Hamiltonian: Local interactions result in stronger than exponentially suppressed corrections, while non-local algebraic interactions result in algebraic suppression. We derive bounds for canonical operators, Weyl operators and outline generalization to arbitrary operators. As an example, we discuss the Klein-Gordon field, and see how the approximate locality in the lattice model becomes the exact causality in the field limit. We discuss the applicability of these results to quenched lattice systems far from equilibrium, and the dynamics of quantum phase transitions.
We prove global existence and uniqueness of dynamics on the quasi-local algebra A\mathcal{A}A of a quantum lattice system for spatially growing derivations LΦ=∑x[Φx,⋅]\mathcal{L}_Φ= \sum_x [ Φ_x , \cdot ]LΦ​=∑x​[Φx​,⋅]. Existing results assume that the local terms Φx∈AΦ_x\in\mathcal{A}Φx​∈A of the generator are uniformly bounded in space with respect to appropriate weighted norms ∥Φx∥G,x\lVert Φ_x \rVert_{G,x}∥Φx​∥G,x​. Analogous to the global existence result for first order ODEs, we show that global existence and uniqueness persist if the size of the local terms ∥Φx∥G,x\lVert Φ_x \rVert_{G,x}∥Φx​∥G,x​ grows at most linearly in space. This considerably enlarges the class of derivations known to have well-defined dynamics. Moreover, we obtain Lieb-Robinson bounds with exponential light cones for such dynamics. For the proof, we assume Lieb-Robinson bounds with linear light cones for dynamics, whose generators have uniformly bounded local terms. Such bounds are known to hold, for example, if the local terms are of finite range or exponentially localized.
We consider a large class of harmonic systems, each defined as a quasi-free dynamics on the Weyl algebra over ℓ2(Zd)\ell^2(\mathbb{Z}^d)ℓ2(Zd). In contrast to recently obtained, short-time locality estimates, known as Lieb-Robinson bounds, we prove a number of long-time dispersive estimates for these models.