Lieb-Robinson Bounds and the Simulation of Time-Evolution of Local Observables in Lattice Systems

Abstract

This is an introductory text reviewing Lieb-Robinson bounds for open and closed quantum many-body systems. We introduce the Heisenberg picture for time-dependent local Liouvillians and state a Lieb-Robinson bound that gives rise to a maximum speed of propagation of correlations in many body systems of locally interacting spins and fermions. Finally, we discuss a number of important consequences concerning the simulation of time evolution and properties of ground states and stationary states.

Holographic thermalization with Lifshitz scaling and hyperscaling violation

“…In figure 6 we show A (3) reg for various dimensions. It seems that a limiting curve is approached as d increases.…”

“…For t < 0 we have A (3) reg = 0 because the background is hvLif. When t > 0, it is possible to identify three regimes: an initial one, when the growth is characterized by a power law, an intermediate regime where the growth is linear and a final regime, when A (3) reg (t) saturates to the thermal value. We report our results for the different regimes in the main text while the details of the computation are described in appendix section B.…”

“…The initial regime is characterized by times that are short compared to the horizon scale In appendix section B.1, following [31], we expand A (3) reg around t = 0 and consider the first non trivial order for an n dimensional spatial region whose boundary Σ has a generic shape. Given the metric (2.11) with (4.13), the final result for this regime is (see (B.9))…”

“…[2]), new techniques have been developed for the theoretical study of time evolution of observables in perturbed quantum lattices (see e.g. [3]),analytical results have been obtained for the time evolution of observables after quenches in conformal field theory [4][5][6] and entanglement entropy has been given a geometric interpretation [7][8][9][10] in the context of the holographic duality of strongly interacting conformal field theory [11]. The present paper follows up on this last direction.…”

“Light-Cone” Dynamics After Quantum Quenches in Spin Chains

“…This gives rise to a causal structure in commutators of local operators at different times, although Schrödinger's equation, unlike relativistic theories, has no built-in speed limit. Recently, the Lieb-Robinson bounds have been refined [19][20][21] and extended to mixed state dynamics in open quantum systems [21,22] as well as creation of topological quantum order [23].A striking consequence of the Lieb-Robinson bound is that the equal-time correlators after a quantum quench feature a "light-cone" effect [23], which is most pronounced for quenches to conformal field theories from initial density matrices with a finite correlation length [24]: connected correlations are initially absent, but exhibit a marked increase after a time t 0 = x/2v. This observation is explained by noting [25,26] that entangled pairs of quasi-particles initially located half-way between the two points of measurement, propagate with the speed of light v and hence induce correlations after a time t 0 .…”