On Lieb–Robinson Bounds for a Class of Continuum Fermions

We consider the quantum dynamics of a many-fermion system in \({{\mathbb {R}}}^d\) with an ultraviolet regularized pair interaction as previously studied in Gebert et al. (Ann Henri Poincaré 21(11):3609–3637, 2020). We provide a Lieb–Robinson bound under substantially relaxed assumptions on the potentials. We also improve the associated one-body Lieb–Robinson bound on \(L^2\)-overlaps to an almost ballistic one (i.e., an almost linear light cone) under the same relaxed assumptions. Applications include the existence of the infinite-volume dynamics and clustering of ground states in the presence of a spectral gap. We also develop a fermionic continuum notion of conditional expectation and use it to approximate time-evolved fermionic observables by local ones, which opens the door to other applications of the Lieb–Robinson bounds.