Lieb-Robinson bounds with exponential-in-volume tails

TL;DR

This work derives volume-based tails for time-evolved operators in local quantum systems by recasting Lieb-Robinson bounds through an equivalence-class, nested-commutator framework. The central technical result shows that, beyond the light cone, operator support decays as $\exp[-(R-vt)^d/(vt)^{d-1}]$, strengthening locality control and enabling sharper bounds on classical simulability and phase diagnostics. The authors also extend the analysis to quasilocal Hamiltonians with exponential tails, provide explicit simulatability bounds with favorable scaling in error and time, and apply the theory to spontaneous symmetry breaking, proving volume-law suppression of disorder parameters and offering a diagnostic for quantum phases. The results have practical implications for simulating many-body dynamics and for characterizing phases via disorder operators, with potential extensions to bosonic systems and power-law interactions.

Abstract

Lieb-Robinson bounds demonstrate the emergence of locality in many-body quantum systems. Intuitively, Lieb-Robinson bounds state that with local or exponentially decaying interactions, the correlation that can be built up between two sites separated by distance $r$ after a time $t$ decays as $\exp(vt-r)$, where $v$ is the emergent Lieb-Robinson velocity. In many problems, it is important to also capture how much of an operator grows to act on $r^d$ sites in $d$ spatial dimensions. Perturbation theory and cluster expansion methods suggest that at short times, these volume-filling operators are suppressed as $\exp(-r^d)$ at short times. We confirm this intuition, showing that for $r > vt$, the volume-filling operator is suppressed by $\exp(-(r-vt)^d/(vt)^{d-1})$. This closes a conceptual and practical gap between the cluster expansion and the Lieb-Robinson bound. We then present two very different applications of this new bound. Firstly, we obtain improved bounds on the classical computational resources necessary to simulate many-body dynamics with error tolerance $ε$ for any finite time $t$: as $ε$ becomes sufficiently small, only $ε^{-O(t^{d-1})}$ resources are needed. A protocol that likely saturates this bound is given. Secondly, we prove that disorder operators have volume-law suppression near the "solvable (Ising) point" in quantum phases with spontaneous symmetry breaking, which implies a new diagnostic for distinguishing many-body phases of quantum matter.

Figures (8)

• Figure 1: (a) The dominant contribution to the growth of an operator outside of the Lieb-Robinson light cone. Traditional Lieb-Robinson bounds suggest that the fraction of $A(t)$ acting on the red dotted circle to be suppressed by $\exp[-\mathrm{O}(r)]$. We show that it is suppressed by $\exp[-\mathrm{O}(r^d)]$, which highly favors operators with "noodle-shaped" support like the one shown in blue. (b) An illustration of why this problem is so difficult to approach with conventional techniques, which count the number of paths of subsets (illustrated in purple) that connect the support of $A(0)$ to the support of $\mathcal{O}_1, \dots \mathcal{O}_m$. The combinatorics of counting branching paths that grow operators to have large support becomes difficult at higher orders. This makes the equivalence-class method a very valuable tool.
• Figure 2: Example of two sequences of factors that appear in the expansion of the Liouvillian propagator which have the same irreducible path and thus belong to the same equivalence class. The boxes are factors and the circles are vertices. The irreducible path is shown in blue, while the irrelevant terms in the sequence are shown in green.
• Figure 3: Example system showing the advantage to shortening the irreducible paths.
• Figure 4: Illustration of the problem setup. This illustration is schematic, and the underlying vertex set is not shown. The region $R$ is depicted in red, and can be imagined as having infinite extent. The regions $S_1, S_2, S_3$ are depicted in blue, contained within the white regions $B_1, B_2, B_3$ which are disjoint from $R$. The shapes are arbitrary, but for particular applications we will find optimal choices for the shape of these regions.
• Figure 5: Example of a causal forest constructed by the algorithm. The red region again corresponds to $R$ and the two blue regions depict $S_1, S_2$. The blue and green line segments represent factors in the sequence $M$. The blue segments participate in the irreducible paths, and so they form part of the corresponding irreducible skeleton, while the green ones are "reducible" terms. The ordering of the sequence is not depicted.

Theorems & Definitions (57)

• Definition 3.1: Factor Graph
• Definition 3.2: $d$-dimensional system
• Definition 3.3: Many-body Quantum System with Nearest-Neighbor Interactions
• Definition 3.4: Vector space of operators
• Theorem 3.5: Lieb-Robinson bound
• Theorem 3.6: Theorem 3, chen2021operator
• Definition 4.1
• Proposition 4.2
• proof
• Definition 4.3