Long-Range Free Fermions: Lieb-Robinson Bound, Clustering Properties, and Topological Phases

TL;DR

This paper investigates long-range free fermion systems with power-law decaying hopping amplitudes, establishing optimal bounds, clustering properties, and implications for topological phase classifications in arbitrary dimensions.

Contribution

It derives an optimal Lieb-Robinson bound, confirms clustering properties, and extends topological phase classification to long-range free fermion systems with decay powers exceeding the spatial dimension.

Findings

Derived an optimal Lieb-Robinson bound for long-range fermions.

Established clustering properties for Green's functions outside the energy spectrum.

Extended topological phase classification to systems with decay power larger than the dimension.

Abstract

We consider free fermions living on lattices in arbitrary dimensions, where hopping amplitudes follow a power-law decay with respect to the distance. We focus on the regime where this power is larger than the spatial dimension (i.e., where the single particle energies are guaranteed to be bounded) for which we provide a comprehensive series of fundamental constraints on their equilibrium and nonequilibrium properties. First we derive a Lieb-Robinson bound which is optimal in the spatial tail. This bound then implies a clustering property with essentially the same power law for the Green's function, whenever its variable lies outside the energy spectrum. The widely believed (but yet unproven in this regime) clustering property for the ground-state correlation function follows as a corollary among other implications. Finally, we discuss the impact of these results on topological phases in long-range free-fermion systems: they justify the equivalence between Hamiltonian and state-based definitions and the extension of the short-range phase classification to systems with decay power larger than the spatial dimension. Additionally, we argue that all the short-range topological phases are unified whenever this power is allowed to be smaller.