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On the quantum dynamics of long-ranged Bose-Hubbard Hamiltonians

Marius Lemm, Carla Rubiliani, Jingxuan Zhang

TL;DR

The paper tackles the challenge of understanding quantum dynamics in long-range Bose-Hubbard systems with power-law decays, aiming to prove propagation bounds that are uniform in the thermodynamic limit. It develops and applies a multiscale ASTLO (adiabatic space-time localization observables) framework to handle unbounded, long-range hopping and interactions, delivering a sharp uniform particle transport bound for exponents $\\alpha>d+1$ and a thermodynamically stable Lieb-Robinson bound under particle-free shell conditions. A key methodological advance is the logarithmically renormalized ASTLO and a multiscale induction that yields rigorous control over all moments and a light-cone approximation, establishing fixed-speed propagation independent of the total particle number. The results strengthen the locality intuition for long-range, strongly interacting bosonic systems and provide a versatile toolkit for analyzing macroscopic quantum dynamics in high-dimensional, strongly coupled settings. The findings have potential implications for quantum simulations with ultracold atoms in optical lattices, where long-range effects and large system sizes are relevant, and they contribute to the rigorous foundations of dynamical locality in many-body quantum physics.

Abstract

We study the quantum dynamics generated by Bose-Hubbard Hamiltonians with long-ranged (power law) terms. We prove two ballistic propagation bounds for suitable initial states: (i) A bound on all moments of the local particle number for all power law exponents $α>d+1$ in $d$ dimensions, the sharp condition. (ii) The first thermodynamically stable Lieb-Robinson bound (LRB) for these Hamiltonians. To handle the long-ranged and unbounded terms, we further develop the multiscale ASTLO (adiabatic space time localization observables) method introduced in our recent work [arXiv:2310.14896].

On the quantum dynamics of long-ranged Bose-Hubbard Hamiltonians

TL;DR

The paper tackles the challenge of understanding quantum dynamics in long-range Bose-Hubbard systems with power-law decays, aiming to prove propagation bounds that are uniform in the thermodynamic limit. It develops and applies a multiscale ASTLO (adiabatic space-time localization observables) framework to handle unbounded, long-range hopping and interactions, delivering a sharp uniform particle transport bound for exponents and a thermodynamically stable Lieb-Robinson bound under particle-free shell conditions. A key methodological advance is the logarithmically renormalized ASTLO and a multiscale induction that yields rigorous control over all moments and a light-cone approximation, establishing fixed-speed propagation independent of the total particle number. The results strengthen the locality intuition for long-range, strongly interacting bosonic systems and provide a versatile toolkit for analyzing macroscopic quantum dynamics in high-dimensional, strongly coupled settings. The findings have potential implications for quantum simulations with ultracold atoms in optical lattices, where long-range effects and large system sizes are relevant, and they contribute to the rigorous foundations of dynamical locality in many-body quantum physics.

Abstract

We study the quantum dynamics generated by Bose-Hubbard Hamiltonians with long-ranged (power law) terms. We prove two ballistic propagation bounds for suitable initial states: (i) A bound on all moments of the local particle number for all power law exponents in dimensions, the sharp condition. (ii) The first thermodynamically stable Lieb-Robinson bound (LRB) for these Hamiltonians. To handle the long-ranged and unbounded terms, we further develop the multiscale ASTLO (adiabatic space time localization observables) method introduced in our recent work [arXiv:2310.14896].
Paper Structure (42 sections, 28 theorems, 347 equations, 8 figures)

This paper contains 42 sections, 28 theorems, 347 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Setup and Main Results
  3. Power-law decay assumption
  4. Main result 1: uniform particle propagation bound
  5. Lieb-Robinson--type bounds
  6. Proof of Theorem \ref{['thm4']} for p=1
  7. ASTLOs and Their Properties
  8. Proof of Particle Propagation Estimates (Theorem \ref{['thm4']}) for R/r large
  9. Proof of Particle Propagation Estimate (Theorem \ref{['thm4']}) assuming $R/r$ small
  10. Proof of Lower Bound for First Bad Time T1
  11. Completing the proof of Theorem \ref{['thm4']} for ${p=1}$
  12. Deriving integral ASTLO estimate for p=1 from Theorem \ref{['thm4']} for p=1
  13. Proof of Theorem \ref{['thm4']} Assuming ASTLO Estimates
  14. Main ASTLO estimates
  15. Completing the proof of Theorem \ref{['thm4']} assuming main ASTLO Estimates
  16. ...and 27 more sections

Key Result

Theorem 2.1

Let $\alpha>d+1$. Consider an initial state satisfying the bounded-density assumption for some $\infty>\lambda_2>\lambda_1>0$ and $p\ge1$. Then, for any $v>12\kappa$ and $\delta_0>0$, there exist with $C=0$ for $p=1$ and $C>0$ for $p>1,$ such that for all $R>r\ge\rho$ with $R-r>\delta_0r$, there hold

Figures (8)

  • Figure 1: Schematic diagram illustrating the notations above.
  • Figure 2: Schematic diagram illustrating the movement of $f_+$ and $f_-$.
  • Figure 3: Flowchart illustrating the proof of Theorem \ref{['thm4']} for $p=1$ and small $R/r$. The yellow boxes correspond to new ideas that go beyond our prior multiscale induction scheme lemm2023microscopic.
  • Figure 4: Flowchart illustrating the proof structure of Theorem \ref{['thm4']}. The yellow boxes correspond to conceptually new ideas beyond our prior multiscale induction scheme lemm2023microscopic.
  • Figure 5: Schematic diagram illustrating the curved annular region in \ref{['NgxiDef']}.
  • ...and 3 more figures

Theorems & Definitions (47)

  • Theorem 2.1: Main result 1, particle propagation bound
  • Remark 1
  • Theorem 2.2: Main result 2, light-cone approximation
  • Theorem 2.3: Thermodynamically stable Lieb-Robinson bound for long-range bosons
  • proof
  • Remark 2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.4
  • ...and 37 more