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arXiv:2405.04672 · Annales Henri Poincaré

Symmetry-Enhanced Lieb–Robinson Bounds for a Class of Bose–Hubbard Type Hamiltonians

Tomotaka Kuwahara, Marius Lemm

Published Jul 28, 2026• 0 citations
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Abstract

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.

DOI
10.1007/s00023-026-01733-9

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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≤Cv \le Cv≤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∼tD−1v \sim t^{D-1}v∼tD−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 ( npn^pnp n p with p>D+1p > D+1p>D+1 p > D + 1 ) qualitatively alters the propagation behavior, leading to a bound of the form v∼tDp−D−1v \sim t^{\frac{D}{p - D - 1}}v∼tp−D−1D​ 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.
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 td−1t^{d-1}td−1 for large times, where ddd denotes the lattice dimension. We present a shorter proof of the weaker, but still polynomial velocity bound td+εt^{d+ε}td+ε.
In this work, we investigate how quickly local perturbations propagate in interacting boson systems with Bose-Hubbard-type Hamiltonians. In general, these systems have unbounded local energies, and arbitrarily fast information propagation may occur. We focus on a specific but experimentally natural situation in which the number of bosons at any one site in the unperturbed initial state is approximately limited. We rigorously prove the existence of an almost-linear information-propagation light cone, thus establishing a Lieb-Robinson bound: the wave front grows at most as t log^{2}(t). We prove the clustering theorem for gapped ground states and study the time complexity of classically simulating one-dimensional quench dynamics, a topic of great practical interest.
Recent works have revealed the intricate effect of long-range interactions on information transport in quantum many-body systems: In DDD spatial dimensions, interactions decaying as a power-law r−αr^{-α}r−α with α>2D+1α> 2 D+1α>2D+1 exhibit a Lieb-Robinson bound (LRB) with a linear light cone and the threshold 2D+12D +12D+1 is sharp in general. Here, we observe that mutually commuting, long-range interactions satisfy an enhanced LRB of the form t r−αt \, r^{-α}tr−α for any α>0α>0α>0, and this scaling is sharp. In particular, the linear light cone occurs at α=1α= 1α=1 in any dimension. Part of our motivation stems from quantum error-correcting codes. As applications, we derive enhanced bounds on ground state correlations and an enhanced local perturbations perturb locally (LPPL) principle for which we adapt a recent subharmonicity argument of Wang-Hazzard. Similar enhancements hold for commuting interactions with stretched exponential decay.
Closed bosonic lattice systems do not generally admit operator norm Lieb-Robinson bounds uniformly in the initial state. The reason is that the information propagation velocity can grow with the local boson occupancy, which can be macroscopically large. Here, we show that local dissipation restores an operator-norm Lieb-Robinson bound. We consider the dissipative Bose-Hubbard model described by a Lindbladian operator with on-site ℓ\ellℓ-photon loss, ℓ>2\ell>2ℓ>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.
The Lieb-Robinson (LR) bound rigorously shows that in quantum systems with short-range interactions, the maximum amount of information that travels beyond an effective "light cone" decays exponentially with distance from the light-cone front, which expands at finite velocity.Despite being a fundamental result, existing bounds are often extremely loose, limiting their applications.We introduce a method that dramatically and qualitatively improves LR bounds in models with finite-range interactions.Most prominently, in systems with a large local Hilbert space dimension D, our method gives a LR velocity that grows much slower than previous bounds with D as D → ∞.For example, in the Heisenberg model with spin S, we find v ≤ const.compared to the previous v ∝ S, which diverges at large S, and in multiorbital Hubbard models with N orbitals, we find v ∝ √ N instead of previous v ∝ N , and similarly in the Nstate truncated Bose-Hubbard model and Wen's quantum rotor model.Our bounds also scale qualitatively better in some systems when the spatial dimension or certain model parameters become large, for example in the d-dimensional quantum Ising model and perturbed toric code models.Even in spin-1/2 Ising and Fermi-Hubbard models, our method improves the LR velocity by an order of magnitude with typical model parameters, and significantly improves the LR bound at large distance and early time.
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+1α>d+1α>d+1 in ddd 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].