Concentration of OTOC and Lieb-Robinson velocity in random Hamiltonians

Abstract

The commutator between operators at different space and time has been a diagnostic for locality of unitary evolution. Most existing results are either for specific tractable (random) Hamiltonians(Out-of-Time-Order-Correlators calculations), or for worse case Hamiltonians (Lieb-Robinson-like bounds or OTOC bounds). In this work, we study commutators in typical Hamiltonians. Draw a sample from any zero-mean bounded independent random Hamiltonian ensemble, time-independent or Brownian, we formulate concentration bounds in the spectral norm and for the OTOC with arbitrary non-random state. Our bounds hold with high probability and scale with the sum of interactions squared. Our Brownian bounds are compatible with the Brownian limit while deterministic operator growth bounds must diverge. We evaluate this general framework on short-ranged, 1d power-law interacting, and SYK-like k-local systems and the results match existing lower bounds and conjectures. Our main probabilistic argument employs a robust matrix martingale technique called uniform smoothness and may be applicable in other settings.

References in corpus (24)

• A bound on chaos
• Gapless Spin-Fluid Ground State in a Random Quantum Heisenberg Magnet
• Black holes and the butterfly effect
• Operator Spreading in Random Unitary Circuits
• Lieb-Robinson bounds and the generation of correlations and topological quantum order
• Towards the fast scrambling conjecture
• Solving Gapped Hamiltonians Locally
• Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges
• Bounds on Information Propagation in Disordered Quantum Spin Chains
• Lieb-Robinson Bound and Locality for General Markovian Quantum Dynamics
• Limit Theorem for Continuous-Time Quantum Walk on the Line
• Finite speed of quantum scrambling with long range interactions
• Strictly linear light cones in long-range interacting systems of arbitrary dimensions
• Emergent perturbation independent decay of the Loschmidt echo in a many-spin system studied through scaled dipolar dynamics
• Operator Dynamics in Brownian Quantum Circuit
• Fast scrambling on sparse graphs
• Quantum chaos dynamics in long-range power law interaction systems
• Absence of fast scrambling in thermodynamically stable long-range interacting systems
• The operator Lévy flight: light cones in chaotic long-range interacting systems
• Information propagation through quantum chains with fluctuating disorder
• Scrambling speed of random quantum circuits
• Non-perturbative dynamics of the operator size distribution in the Sachdev-Ye-Kitaev model
• Matrix Concentration for Products
• Freedman's inequality for matrix martingales

Cited by in corpus (5)

• Concentration for random product formulas
• Speed limits and locality in many-body quantum dynamics
• Lieb-Robinson bound and almost-linear light-cone in interacting boson systems
• Finite speed of quantum information in models of interacting bosons at finite density
• Optimal Frobenius light cone in spin chains with power-law interactions