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

TL;DR

This work establishes a ballistic-cap on the growth of local bosonic energy in translation-invariant Bose-Hubbard-type systems with unbounded interactions. By introducing quartic adiabatic space-time localization observables (ASTLOs) that track local particle-particle correlations, the authors derive a bound showing $\langle n_x^2\rangle_t \lesssim t^d$ for bounded-density initial states, improving previous $t^{2d}$ scalings that arose from solely controlling particle transport. The key technical advance is the correlation-focused ASTLO framework, which closes under commutator dynamics and admits a recursive bootstrapping argument, aided by a recent higher-moments bound. This result sharpens our understanding of energy accumulation and has potential implications for higher-moment bounds and tighter bosonic Lieb-Robinson bounds in long-range, unbounded-interaction lattice systems.

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.

Theorems & Definitions (12)

• Theorem 2.1: Bound on correlations
• Remark 2.2
• Remark 2.3
• Proposition 3.1: Geometric properties of the ASTLO
• proof
• Proposition 4.1: Differential inequality
• Remark 4.2
• Proposition 4.3: Bootstrapping
• Theorem 4.4: lemm2023microscopic*Theorem 2.1
• proof : Proof of Theorem \ref{['prop corr tr inv']}