A ballistic upper bound on the accumulation of bosonic on-site energies

Abstract

In this note, we study transport properties of the dynamics generated by translation-invariant and possibly long-ranged Hamiltonians of Bose-Hubbard type. For translation-invariant initial states with controlled boson density, we improve the known bound on the local repulsive energy at time $t$ from $\langle n^2_x\rangle_t\lesssim t^{2d}$ to $\langle n^2_x\rangle_t\lesssim t^d$. This shows that bosonic on-site energies accumulate at most ballistically. Extending the result to higher moments would have powerful implications for bosonic Lieb-Robinson bounds. While previous approaches focused on controlling particle transport, our proof develops novel ASTLOs (adiabatic space-time localization observables) that are able to track the growth of local boson-boson correlations.