Momentum-space analysis of multipartite entanglement at quantum phase transitions

We investigate entanglement properties at quantum phase transitions (QPTs) of a one-dimensional 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 (electrons coupled through the $\eta$-pairing mechanism). We first detect the two or multipartite 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 quantum correlations in the momentum representation and the relevant (superconducting) ones exhibited in the complementary picture: the direct-lattice representation. We then investigate the presence of multipartite entanglement in momentum space by means of a generalization of the Meyer-Wallach measure and we relate its behavior to the nature of the various QPTs.

Figure 1

(Color online) $k$-space representation of the ground-state phase diagram of $H$ in the $n\text{−}u$ plane. Dotted lines represent some cases of isocorrelation curves (see Sec. 3B).

Figure 2

(Color online) Plot of ${\partial }_n{\mathcal{S}}_{-k_j,k_j}$ (straight blue line) and ${\partial }_n{\mathcal{I}}_{\left(-k_i,k_i\right),\left(-k_j,k_j\right)}$ (dashed red line) in region II.

Figure 3

(Color online) Plot of ${\partial }_u{\mathcal{S}}_{-k_j,k_j}$ (straight blue line) and ${\partial }_u{\mathcal{I}}_{\left(-k_i,k_i\right),\left(-k_j,k_j\right)}$ (dashed red line) in region II.

Figure 4

(Color online) Plot of the $Q$ measure for a chain of length $L=1000$ at quarter filling; $D=1,2,4,8,16,32$.