Lieb-Robinson Bound and Adiabatic Evolution

TL;DR

This paper generalizes Lieb-Robinson bounds to systems without an initial tensor-product structure by relating locality to Hamiltonian matrix representations, and uses this to derive an adiabatic condition based on energy basis locality.

Contribution

It introduces a Lieb-Robinson-like bound applicable to Hamiltonians defined in a specific basis, extending the concept of locality beyond tensor-product structures.

Findings

Established a basis-dependent Lieb-Robinson bound

Derived an adiabatic condition from locality in energy basis

Showed adiabaticity follows from a small Lieb-Robinson bound

Abstract

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.