Entanglement patterns and generalized correlation functions in quantum many body systems

We introduce transition operators that in a given basis of the single-site states of a many-body system have a single non-vanishing matrix element and introduce their correlation functions. We show that they fall into groups that decay with the same rate. The mutual information defined in terms of the von Neumann entropy between two sites is given in terms of these so-called generalized correlation functions. We confirm numerically that the long-distance decay of the mutual information follows the square of that of the most slowly decaying generalized correlation function. The main advantage of our procedure is that, in order to identify the most relevant physical processes, there is no need to know a priori the nature of the ordering in the system, i.e., no need to explicitly construct particular physical correlation functions. We explore the behavior of the mutual information and the generalized correlation functions for conformally invariant models and for the SU(n) Hubbard model with n=2,3,4, and 5, which are, in general, not conformally invariant. In this latter case, we show that for filling f=1/q and q<n, the ground state consists of highly entangled q-site units that are further entangled by single bonds. In addition, we extend the picture of the two-site mutual information and the corresponding generalized correlation functions to the n-site case.