Lieb-Robinson Bounds with Exponential-in-Volume Tails
Journal Article
Overview
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; ; (; v; t; −; 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; ; ); ; . We confirm this intuition, showing that, for; ; r; >; v; t; ; , the volume-filling operator is suppressed by; ; exp; ; (; −; (; r; −; v; t; ; ); d; ; /; (; v; t; ; ); ; 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.;
