**March 2019**. The quantum matter team at Institut Polytechnique de Paris has unveilled the emergence of a twofold light-cone structure for the propagation of information in lattice models with short and long range interactions [1,2]. The two structures may be related to different characteristics of the many-body excitation spectrum and shed new light on Libe-Robinson-like bounds.

When changing abruptly the value of some physical parameter (performing a so-called quench), a system is brought out of equilibrium and some dynamics sets in. In an homogeneous system, no effect is visible on average local values such as the density in a particle system for instance. Correlations, such as density correlations for instance, may, however, develop and spread out throught out the system. In 1972, Lieb and Robinson have shown that for a lattice model with short range interactions the spreading of information is limited by some bound,, hence creating a so-called light cone by analogy with special relativity. Lieb-Robinson bounds have fundamental effects on many effects, such as the propagation of information, thermalization effects, and entanglement areas laws for instance. Whether a light-cone-like effect emerges in the presence of long-range interactions, however, remains an open question. While extended bounds have been proposed, numerical simulations as well as experiments on artificial ion crystals point towards significantly slower spreading in lattice spin models.

To understand how information spreas out in generic lattice models with either short-range or long-range interactions, the team first devised an analytic theory based on quasi-particle expansions [1]. They have shown that most correlation functions can be cast into a simple form, from which a twofold structure emerges. In the case of short-range interactions, it is reminiscent of the spreading of a wave packet in a dispersive media. The signal is a sine-like function modulated by an envelop. In the vicinity of the correlation edge, only the fastest quasi-particles contribute and the velocity of the envelop is determined by the maximum group velocity of the quasi-particles. Moreover, the signal shows strong oscillations inside the envelop, which, in contrast, move at the phase velocity. This creates a twofold signal, characterized by two velocities, which are related to different microscopic properties of the system.

Building upon this first work, the team has developed an exact many-body approach based on tensor-network techniques to test and extend the predictions. In a first work [2], they have studied the spreading of one-body and two-body correlations in the Bose-Hubbard model (short range interactions) spanning the full phase diagram. They found a universal twofold cone structure, except is specific cases discussed in Ref. [2]. In the superfluid mean-field regime and in the Mott insulator phase, the two velocities associated to the correlation spreading are in excellent agreement with the predictions. In the strongly-correlated superfluid regime, there were no analytic predictions, except for harmonic liquids. The results, however, clearly show beyond Luttinger-liquid behavior. In particular, it still show a twofold structure. These results provide useful information on the excitation spectrum beyond the phonon branch.

The team is now extending this study to spin systems with long-range interactions. The theory of Ref. [1] predicts the coexistence of sub-ballistic and super-ballistic signals. Surprisingly enough, the correlation edge may be sub-ballistic as a result of destructive interferences. In contrast the maxima may spread ballistically or super-ballistically depending on whether the model is gapped or not. Testing this theory in exact many-body approaches is the next challenge taken by the team.

These results have fundamental impact on our understading of the spreading of information in short and long range systems. They show that different signals (ballistic, sub-ballistic orsuper-ballistic) may coexist, which spread differently. In practice, this study shows that local maxima can spreading completely differently as the front edge and should not be used to draw conclusions on bounds to the propagation of information.

[1] L. Cevolani, J. Despres, G. Carleo, L. Tagliacozzo, and L. Sanchez-Palencia, “Universal scaling laws for correlation spreading in quantum systems with short- and long-range interactions”, Phys. Rev. B **98**, 024302 (2018)

[2] J. Despres, L. Villa, and L. Sanchez-Palencia, “Twofold correlation spreading in a strongly correlated lattice Bose gas”, Sci. Rep. **9**, 4135 (2019)