We investigate entanglement properties at quantum phase transitions of an integrable extended Hubbard model in the momentum space representation. Two elementary subsystems are recognized: the single mode of an electron, and the pair of modes (electro
ns coupled through the eta-pairing mechanism). We first detect the two/multi-partite nature of each quantum phase transition by a comparative study of the singularities of Von Neumann entropy and quantum mutual information. We establish the existing relations between the correlations in the momentum representation and those exhibited in the complementary picture: the direct lattice representation. The presence of multipartite entanglement is then investigated in detail through the Q-measure, namely a generalization of the Meyer-Wallach measure of entanglement. Such a measure becomes increasingly sensitive to correlations of a multipartite nature increasing the size of the reduced density matrix. In momentum space, we succeed in obtaining the latter for our system at arbitrary size and we relate its behaviour to the nature of the various QPTs.
The quantum states built with the eta paring mechanism i.e., eta pairing states, were first introduced in the context of high temperature superconductivity where they were recognized as important example of states allowing for off-diagonal long-range
order (ODLRO). In this paper we describe the structure of the correlations present in these states when considered in their momentum representation and we explore the relations between the quantum bipartite/multipartite correlations exhibited in k space and the direct lattice superconducting correlations. In particular, we show how the negativity between paired momentum modes is directly related to the ODLRO. Moreover, we investigate the dependence of the block entanglement on the choice of the modes forming the block and on the ODLRO; consequently we determine the multipartite content of the entanglement through the evaluation of the generalized Meyer Wallach measure in the direct and reciprocal lattice. The determination of the persistency of entanglement shows how the network of correlations depicted exhibits a self-similar structure which is robust with respect to local measurements. Finally, we recognize how a relation between the momentum-space quantum correlations and the ODLRO can be established even in the case of BCS states.
