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.