Genuine multipartite correlations distribution in the criticality of the Lipkin-Meshkov-Glick model

Correlations play a key role in critical phenomena. Quantities such as the entanglement entropy, for instance, provide key insights into the mechanisms underlying quantum criticality. Almost all of our present knowledge, however, is restricted to bipartite correlations. Some questions still remain unanswered, such as: What parcel of the total correlations are genuinely $k$-partite? With the goal of shedding light on this difficult question, in this paper we put forth a detailed study of the behavior of genuine multipartite correlations (GMC) of arbitrary orders in the Lipkin-Meshkov-Glick model. We find that GMC of all orders serve to signal the second order quantum phase transition presented in the model. Applying finite-size scaling methods, we also find the critical exponents for some orders of correlations.