arXiv:2505.01786

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

Marius Lemm, Carla Rubiliani, Jingxuan Zhang

Published May 3, 2025• 0 citations

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].

DOI: 10.48550/arxiv.2505.01786

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arXiv:2505.01786

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

Marius Lemm, Carla Rubiliani, Jingxuan Zhang

Published May 3, 2025• 0 citations

arXiv HTML PDF

Abstract

α>d+1

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].

DOI: 10.48550/arxiv.2505.01786

Related papers

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

Tomotaka Kuwahara, Marius Lemm, Carla Rubiliani

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.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.

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Abstract Several recent works have derived Lieb–Robinson bounds (LRBs) for Bose–Hubbard-type Hamiltonians. For certain structured initial states, e.g., vacuum perturbations or near-stationary states, information propagates with velocity $v \le C$ v ≤ C . However, for general bounded-density initial states, it was shown by the first author, Vu, and Saito that the velocity can grow in time as $v \sim t^{D-1}$ v ∼ t D - 1 , where D is the spatial dimension—demonstrating the possibility of accelerated information spreading in bosonic systems. In this work, we introduce a new perspective on this phenomenon: we show that translation invariance combined with local p -body repulsion ( $n^p$ n p with $p > D+1$ p > D + 1 ) qualitatively alters the propagation behavior, leading to a bound of the form $v \sim t^{\frac{D}{p - D - 1}}$ v ∼ t D p - D - 1 for general bounded-energy-density initial states. In particular, this establishes an almost-linear light cone at large p , in stark contrast to the previously found accelerated regimes. Our result identifies symmetry-driven constraints as a new mechanism for suppressing propagation speed in bosonic systems and thereby reframes the scope of what types of LRBs can hold. We further provide matching examples showing that, under the given assumptions, this bound is sharp—no further improvement in the power of t is possible without invoking additional dynamical constraints.Abstract Several recent works have derived Lieb–Robinson bounds (LRBs) for Bose–Hubbard-type Hamiltonians. For certain structured initial states, e.g., vacuum perturbations or near-stationary states, information propagates with velocity $$v \le C$$ v ≤ C . However, for general bounded-density initial states, it was shown by the first author, Vu, and Saito that the velocity can grow in time as $$v \sim t^{D-1}$$ v ∼ t D - 1 , where D is the spatial dimension—demonstrating the possibility of accelerated information spreading in bosonic systems. In this work, we introduce a new perspective on this phenomenon: we show that translation invariance combined with local p -body repulsion ( $$n^p$$ n p with $$p > D+1$$ p > D + 1 ) qualitatively alters the propagation behavior, leading to a bound of the form $$v \sim t^{\frac{D}{p - D - 1}}$$ v ∼ t D p - D - 1 for general bounded-energy-density initial states. In particular, this establishes an almost-linear light cone at large p , in stark contrast to the previously found accelerated regimes. Our result identifies symmetry-driven constraints as a new mechanism for suppressing propagation speed in bosonic systems and thereby reframes the scope of what types of LRBs can hold. We further provide matching examples showing that, under the given assumptions, this bound is sharp—no further improvement in the power of t is possible without invoking additional dynamical constraints.

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We study the quantum many-body dynamics of a Bose-Einstein condensate (BEC) on the lattice in the mean-field regime. We derive a local enhancement of the mean-field approximation: At positive distance $ρ>0$ from the initial BEC, the mean-field approximation error at time $t\leq ρ/v$ is bounded as $ρ^{-n}$ , for arbitrarily large $n\geq 1$ . This is a consequence of new ballistic propagation bounds on the fluctuations around the condensate. To prove this, we develop a variant of the ASTLO (adiabatic spacetime localization observable) method for the particle non-conserving generator of the fluctuation dynamics around Hartree states.We study the quantum many-body dynamics of a Bose-Einstein condensate (BEC) on the lattice in the mean-field regime. We derive a local enhancement of the mean-field approximation: At positive distance $ρ>0$ from the initial BEC, the mean-field approximation error at time $t\leq ρ/v$ is bounded as $ρ^{-n}$, for arbitrarily large $n\geq 1$. This is a consequence of new ballistic propagation bounds on the fluctuations around the condensate. To prove this, we develop a variant of the ASTLO (adiabatic spacetime localization observable) method for the particle non-conserving generator of the fluctuation dynamics around Hartree states.

arXiv:2412.13868Dec 18, 20240 citations

α>d+1

d

t

from

\langle n^2_x\rangle_t\lesssim t^{2d}

\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.

v \le C

v ≤ C . However, for general bounded-density initial states, it was shown by the first author, Vu, and Saito that the velocity can grow in time as

v \sim t^{D-1}

v ∼ t D - 1 , where D is the spatial dimension—demonstrating the possibility of accelerated information spreading in bosonic systems. In this work, we introduce a new perspective on this phenomenon: we show that translation invariance combined with local p -body repulsion (

n^p

n p with

p > D+1

p > D + 1 ) qualitatively alters the propagation behavior, leading to a bound of the form

v \sim t^{\frac{D}{p - D - 1}}

v ∼ t D p - D - 1 for general bounded-energy-density initial states. In particular, this establishes an almost-linear light cone at large p , in stark contrast to the previously found accelerated regimes. Our result identifies symmetry-driven constraints as a new mechanism for suppressing propagation speed in bosonic systems and thereby reframes the scope of what types of LRBs can hold. We further provide matching examples showing that, under the given assumptions, this bound is sharp—no further improvement in the power of t is possible without invoking additional dynamical constraints.

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}

\ell

-photon loss,

\ell>2

. For our main result, we establish an almost-ballistic Lieb-Robinson bound. Dissipation rapidly depletes highly occupied sites, thus regularizing the state on the Sobolev-type scale of local particle moments. The resulting moment bounds diverge as time approaches zero but remain integrable near zero at sufficiently low orders, which is crucial for our argument. We extend these ideas to treat cat-code dissipation and, for initial states in the code space, we prove a local adiabatic approximation uniform in the total volume. Further consequences include local channel approximation, a thermodynamic limit, and efficient hybrid bosonic-qubit simulation. The point is that these applications are now available uniformly in the input state, as for quantum spin systems, but in contrast to closed Bose-Hubbard systems.

ρ>0

from the initial BEC, the mean-field approximation error at time

t\leq ρ/v

is bounded as

ρ^{-n}

, for arbitrarily large

n\geq 1

. This is a consequence of new ballistic propagation bounds on the fluctuations around the condensate. To prove this, we develop a variant of the ASTLO (adiabatic spacetime localization observable) method for the particle non-conserving generator of the fluctuation dynamics around Hartree states.