The Lieb-Robinson theorem is a classic result in mathematical physics which proves that in a quantum system with local interactions, the commutators of local operators essentially vanish outside of a “light cone” with an emergent, finite velocity. This result has numerous applications, from bounding classical simulatability of quantum systems to constraining entanglement growth, and many-body operator growth and chaos. In this talk, I will present new frameworks for understanding operator growth and chaos in quantum many-body systems, both with local and without local interactions, which provide qualitative improvements over existing techniques. Using these techniques, I will prove two previously open problems in the community: (1) in spin chains with interactions that fall off with distance faster than 1/r^3, commutators of local operators can be made arbitrarily small outside of a “linear light cone” which grows at a finite velocity, just as in local systems; (2) the scrambling time for an operator to grow large in the Sachdev-Ye-Kitaev model of N fermions grows no slower than log N, when N is large but finite.
