In this thesis we present our results on the propagation of correlations in long-range interacting quantum systems. The dynamics of local observables in these systems cannot be described with the standard methods used in equilibrium statistical physics and completely new methods have to be developed. Several bounds on the time evolution of correlations have been derived for these systems. However the propagation found in experimental and numerical results is completely different and several regimes are present depending on the long-range character of the interactions. Here we present analytical expressions to describe the time evolution of generic observables in systems where the Hamiltonian takes a quadratic form with long- and short-range interactions. These expressions describe the spreading of local observables as the spreading of the fundamental excitations of the system. We apply these expressions to a spin model finding three different propagation regimes. They can be described qualitatively et quantitatively by the divergences in the energy spectrum. The most important result is that the propagation is at most ballistic, but it can be also significantly slower, where the general bounds predict a propagation faster than ballistic. This points out that the bounds are not able to describe properly the propagation, but a more specific approach is needed. We then move to a system of lattice bosons interacting via long-range interactions. In this case we study two different observables finding completely different results for the same interactions: the spreading of two-body correlations is always ballistic while the one of the one-body correlations ranges from faster-than-ballistic to ballistic. Using our general analytic expressions we find that different parts of the spectrum contribute differently to different observables determining the previous differences. This points out that an observable-dependent notion of locality, missing in the general bounds, have to be developed to correctly describe the time evolution.
Understanding the dynamics of many-body systems is crucial for understanding, e.g., thermalization or transmission of information. Nevertheless, little is known in the case of quantum systems with long-range interactions. Here, we analyze the long-range Ising model in a transverse field, where interactions decay as a power-law with distance
∝r−α,
α>0. Using complementary numerical and analytical techniques, we identify three dynamical regimes: short-range-like with an emerging light cone for
α>2; weakly long-range for
1<α<2 without a clear light cone but with a finite propagation speed of excitations; and fully non-local for
α<1 with instantaneous transmission of correlations. This last regime breaks generalized Lieb--Robinson bounds. Numerical calculation of the entanglement spectrum demonstrates that the usual picture of propagating quasi-particles remains valid for long-range interactions. This allows an intuitive interpretation in terms of qualitative changes to the spin-wave dispersion, leading to diverging quasi-particle velocities in the long-range regime. Our results may be tested in state-of-the-art trapped-ion experiments.