Lieb-Robinson bounds with exponential-in-volume tails

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 $\epsilon$ for any finite time $t$: as $\epsilon$ becomes sufficiently small, only ${\epsilon }^{-O(t^{d-1})}$ 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.

32 pages, 8 figures; Updated to address reviewer comments; Added Section 4.4

References in corpus (50)

• An Area Law for One Dimensional Quantum Systems
• Operator Spreading in Random Unitary Circuits
• Lieb-Schultz-Mattis in Higher Dimensions
• Lieb-Robinson bounds and the generation of correlations and topological quantum order
• Spectral Gap and Exponential Decay of Correlations
• Operator hydrodynamics, OTOCs, and entanglement growth in systems without conservation laws
• Lieb-Robinson Bounds and the Exponential Clustering Theorem
• Quasi-adiabatic Continuation of Quantum States: The Stability of Topological Ground State Degeneracy and Emergent Gauge Invariance
• Topological quantum order: stability under local perturbations
• Topological Order and Conformal Quantum Critical Points
• General entanglement scaling laws from time evolution
• Stability of Frustration-Free Hamiltonians
• The Dynamics of 1D Quantum Spin Systems Can Be Approximated Efficiently
• Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems
• Quantum algorithm for simulating real time evolution of lattice Hamiltonians
• Locality of temperature
• Lieb-Robinson Bounds for Harmonic and Anharmonic Lattice Systems
• Finite speed of quantum scrambling with long range interactions
• A quantum topological phase transition at the microscopic level
• A Short Introduction to Numerical Linked-Cluster Expansions
• Strictly linear light cones in long-range interacting systems of arbitrary dimensions
• Euclidean operator growth and quantum chaos
• Quantization of Hall Conductance For Interacting Electrons on a Torus
• Hierarchy of linear light cones with long-range interactions
• Simulating adiabatic evolution of gapped spin systems
• Entanglement area laws for long-range interacting systems
• Locality and digital quantum simulation of power-law interactions
• Information propagation for interacting particle systems
• Speed limits and locality in many-body quantum dynamics
• Absence of fast scrambling in thermodynamically stable long-range interacting systems
• Tightening the Lieb-Robinson Bound in Locally-Interacting Systems
• Higher-form symmetry breaking at Ising transitions
• Signaling and Scrambling with Strongly Long-Range Interactions
• Classical simulation of short-time quantum dynamics
• Lieb-Robinson bound and almost-linear light-cone in interacting boson systems
• Lieb-Robinson bounds for spin-boson lattice models and trapped ions
• Optimal State Transfer and Entanglement Generation in Power-law Interacting Systems
• Maximal speed for macroscopic particle transport in the Bose-Hubbard model
• Finite speed of quantum information in models of interacting bosons at finite density
• Optimal learning of quantum Hamiltonians from high-temperature Gibbs states
• On Lieb-Robinson bounds for the Bose-Hubbard model
• Operator growth bounds from graph theory
• Prethermalization and the local robustness of gapped systems
• Effective light cone and digital quantum simulation of interacting bosons
• Low-density parity-check codes as stable phases of quantum matter
• Fast high-fidelity multi-qubit state transfer with long-range interactions
• Theory of metastable states in many-body quantum systems
• Disordered Lieb-Robinson bounds in one dimension
• Heisenberg-limited metrology with perturbing interactions
• Optimal light cone for macroscopic particle transport in long-range systems: A quantum speed limit approach