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arXiv:2512.02181 · Journal of Mathematical Physics

The Lieb-Robinson condition and the Fréchet topology

Sven Bachmann, Giuseppe De Nittis, Julián Gómez

Published Jul 1, 2026• 0 citations
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Abstract

We define various notions of locality for *-automorphisms of the algebra of observables for an infinitely extended quantum spin system and study their relationship. In particular, we show that the ubiquitous characterization which arises from the Lieb-Robinson bound implies but is not equivalent to continuity with respect to the natural Fréchet topology of almost local observables, which is a non-commutative analog of the Schwartz space.

DOI
10.1063/5.0315251

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We define various notions of locality for *-automorphisms of the algebra of observables for an infinitely extended quantum spin system and study their relationship. In particular, we show that the ubiquitous characterization which arises from the Lieb-Robinson bound implies but is not equivalent to continuity with respect to the natural Fréchet topology of almost local observables, which is a non-commutative analog of the Schwartz space.
Lieb-Robinson bounds show that the speed of propagation of information under the Heisenberg dynamics in a wide class of nonrelativistic quantum lattice systems is essentially bounded. We review works of the past dozen years that has turned this fundamental result into a powerful tool for analyzing quantum lattice systems. We introduce a unified framework for a wide range of applications by studying quasilocality properties of general classes of maps defined on the algebra of local observables of quantum lattice systems. We also consider a number of generalizations that include systems with an infinite-dimensional Hilbert space at each lattice site and Hamiltonians that may involve unbounded on-site contributions. These generalizations require replacing the operator norm topology with the strong operator topology in a number of basic results for the dynamics of quantum lattice systems. The main results in this paper form the basis for a detailed proof of the stability of gapped ground state phases of frustrationfree models satisfying a local topological quantum order condition, which we present in a sequel to this paper.
We extend the concept of locality to enclose a situation where a tensor-product structure for the Hilbert space is not \textit {a priori} assumed; rather, this locality is related to a given matrix representation of the Hamiltonian associated to the system. As a result, we formulate a Lieb-Robinson-like bound for Hamiltonians local in a given basis. In particular, we employ this bound to obtain alternatively the adiabatic condition, where adiabaticity is naturally ensued from a locality in energy basis and a relatively small Lieb-Robinson bound.
This thesis analyses several locality properties of extended quantum lattice systems, considering both spins and fermions. We begin by assuming locality of the Hamiltonian and then discuss how this translates into locality of the time evolution and the thermal states. Starting from a local Hamiltonian, it is well known that the Heisenberg time evolution satisfies Lieb-Robinson bounds, which quantify its locality. We strengthen these bounds for long-range interacting fermions and for commuting interactions. Furthermore, we establish the first Lieb-Robinson bounds for Hamiltonians whose local terms can grow linearly away from the origin. We also prove stability of Lieb-Robinson bounds with respect to local perturbations in the Hamiltonian. For the thermal states we focus on three types of locality known as decay of correlations, the local perturbations perturb locally (LPPL) principle and local indistinguishability. As a key tool we use spectral filters, which were introduced by Hastings in the 2000s for various applications. It enable us to use Lieb-Robinson bounds to obtain information about locality properties of states. In particular, we use quantum belief propagation to prove an equivalence of decay of correlations, LPPL and local indistinguishability for Gibbs states at any positive temperature. Using the improved Lieb-Robinson bounds mentioned before, we obtain improvements for decay of correlations in the ground state, the locality of the spectral flow and LPPL for long-range interacting systems and commuting Hamiltonians. Furthermore, we adjust the filter functions to achieve exponential decay for certain applications that were previously limited to stretched exponential decay. Spectral filters are also used to rigorously justify response theory, which usually requires a gapped system to begin with. To relax this assumption, we introduce various notions of local gaps, discuss their relations and use one to justify response theory and Kubo’s formula in the absence of a global gap. In another result we use local indistinguishability as an input to prove bulk-edge correspondence at finite temperature, in the sense that the bulk magnetisation is equal to the edge current. This result has mostly been studied for non-interacting models or at zero temperature before. Finally, we present a class of weakly interacting spin systems with an on-site gap and prove that the ground state satisfies a strong form of LPPL. Moreover, we show that these systems satisfy decay of correlations and local indistinguishability uniformly in temperature.
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.