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Lieb-Robinson bounds for Bose-Hubbard Hamiltonians: A review with a simplified proof

Marius Lemm, Carla Rubiliani

Abstract

We review recent progress on state-dependent Lieb-Robinson bounds for Bose-Hubbard Hamiltonians. In particular, Kuwahara, Vu, and Saito established that, for general bounded-density initial states, the Lieb-Robinson velocity is bounded by $t^{d-1}$ for large times, where $d$ denotes the lattice dimension. We present a shorter proof of the weaker, but still polynomial velocity bound $t^{d+ε}$.

Lieb-Robinson bounds for Bose-Hubbard Hamiltonians: A review with a simplified proof

Abstract

We review recent progress on state-dependent Lieb-Robinson bounds for Bose-Hubbard Hamiltonians. In particular, Kuwahara, Vu, and Saito established that, for general bounded-density initial states, the Lieb-Robinson velocity is bounded by for large times, where denotes the lattice dimension. We present a shorter proof of the weaker, but still polynomial velocity bound .

Paper Structure

This paper contains 26 sections, 15 theorems, 191 equations, 1 figure.

Table of Contents

  1. Setup and main result
  2. Setup and notation
  3. Main result
  4. Particle propagation bounds
  5. The particle propagation bound
  6. ASTLO
  7. Basic lemmas
  8. Proof of Lemma \ref{['prop geom prop']}
  9. Proof of Lemma \ref{['lemm sym taylor']}
  10. Proof of particle propagation bound
  11. First moment of the number operator
  12. Higher moments
  13. Proof of Theorem \ref{['theo final particle']}
  14. Proof of the first moment bound
  15. Proof of Proposition \ref{['prop recurs stru']}
  16. ...and 11 more sections

Key Result

Theorem 1.1

Consider an initial state $\rho\in\mathcal{D}_\eta$ satisfying CD rho for some $\lambda >0$ and $\eta\ge 2(d+1)$. Then, there exists a positive $C=C(d, \eta, J)$, such that for all $R>2$, $t\in\mathbb{R}$, $X\subset\Lambda$ compact, and $A\in\mathcal{A}_{X}^\mathrm{inv}$, the following holds

Figures (1)

  • Figure 1: Comparison between the light cones obtained in Theorem \ref{['teo LRB']}, that scales as $R\sim t^{d+1+\epsilon}$, and Kuwahara2024, with scaling $R\sim t^d$.

Theorems & Definitions (30)

  • Theorem 1.1: Lieb--Robinson bound
  • Remark 1.2
  • Theorem 2.1: Particle propagation bound
  • Lemma 2.2: Geometric properties LRZ2025*Lemma 3.1
  • Lemma 2.3: Symmetrized Taylor expansion, FLS2022*Lem. 2.2
  • Lemma 2.4: Commutator expansion, FLS2022*Lem. A.2
  • proof
  • proof
  • Proposition 3.1: Recursive structure
  • Corollary 3.2: Bootstrapping
  • ...and 20 more