Structure of quantum correlations in momentum space and off-diagonal long-range order in $\mathrm{\eta }$ pairing and BCS states

The quantum states built with the $\mathrm{\eta }$ pairing mechanism, i.e., $\mathrm{\eta }$ 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 and 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.

Figure 1

(Color online) Left: schematic representation of the correlations of the generic single mode $-k_i$ and of the paired modes $\left(-k_i,k_i\right)$. QC stands for quantum correlations, CC for classical correlations, while QMP stands for quantum multipartite. Right: two opposite ways of grouping the modes in blocks, i.e., paired $\left({\mathcal{A}}_{D_1}\right)$ and unpaired $\left({\mathcal{A}}_{D_2}\right)$ modes.

Figure 2

(Color online) Plot of the measure of multipartite entanglement $Q_{D,4}$ in $k$ space for different sizes of the elementary subsystem; the size varies from bottom to top: $D=1,2,4,8,16$ $\left(L=1000\right)$.

Figure 3

(Color online) Plot of the measure of multipartite entanglement $Q_{D,2}$ in direct lattice picture for different sizes of the elementary subsystem; the size varies form bottom to top: $D=4,8,16,32,100$ $\left(L=1000\right)$.