Asymptotic quasi--local structure induced by Magnetic field
Abstract
We introduce a magnetic coherent--state frame over the Landau levels of the two--dimensional Landau Hamiltonian, indexed by a discrete metric space combining spatial and energy degrees of freedom. Although overcomplete, the frame is almost--orthogonal, and we use it to endow the CAR C*--algebra over the one--particle Hilbert space with an \emph{asymptotic quasi--local} (AQL) structure, in which graded commutators of observables localized on disjoint index sets decay exponentially rather than vanish exactly. By means of a family of conditional expectations onto the resulting local subalgebras, we construct a Fréchet *--subalgebra of almost--local observables, invariant under the discrete group of lattice translations, which act on it by continuous automorphisms. For the full continuous group of magnetic translations we establish a quantitative control on how local observables spread through this almost--local algebra, and we show that this control suffices to extend graded asymptotic abelianness from the lattice to the entire continuous translation group. The results provide a discrete--continuum dictionary intended as the starting point for extending Lieb--Robinson--type techniques from lattice systems to interacting continuum electrons in a magnetic field.
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BibTeX
@misc{nittis2026asymptotic,
title = {{Asymptotic quasi--local structure induced by Magnetic field}},
author = {Giuseppe De Nittis},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26987},
eprint = {2609.26987},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26987},
}
RIS
TY - UNPB
TI - Asymptotic quasi--local structure induced by Magnetic field
AU - Giuseppe De Nittis
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26987
UR - https://arxiv.org/abs/2609.26987
AB - We introduce a magnetic coherent--state frame over the Landau levels of the two--dimensional Landau Hamiltonian, indexed by a discrete metric space combining spatial and energy degrees of freedom. Although overcomplete, the frame is almost--orthogonal, and we use it to endow the CAR C^*--algebra over the one--particle Hilbert space with an \emph{asymptotic quasi--local} (AQL) structure, in which graded commutators of observables localized on disjoint index sets decay exponentially rather than vanish exactly. By means of a family of conditional expectations onto the resulting local subalgebras, we construct a Fréchet *--subalgebra of almost--local observables, invariant under the discrete group of lattice translations, which act on it by continuous automorphisms. For the full continuous group of magnetic translations we establish a quantitative control on how local observables spread through this almost--local algebra, and we show that this control suffices to extend graded asymptotic abelianness from the lattice to the entire continuous translation group. The results provide a discrete--continuum dictionary intended as the starting point for extending Lieb--Robinson--type techniques from lattice systems to interacting continuum electrons in a magnetic field.
ER -