Quantum discord and classical correlations in the bond-charge Hubbard model: Quantum phase transitions, off-diagonal long-range order, and violation of the monogamy property for discord

We study the quantum discord (QD) and the classical correlations (CC) in a reference model for strongly correlated electrons, the one-dimensional bond-charge extended Hubbard model. We show that the comparison of QD and CC and of their derivatives in the direct and reciprocal lattice allows one to efficiently inspect the structure of two-points driven quantum phase transitions, discriminating those at which off diagonal long-range order (ODLRO) is involved. Moreover, we observe that QD between pair of sites is a monotonic function of ODLRO, thus establishing a direct relation between the latter and two point quantum correlations that differ from the entanglement. The study of the ground-state properties allows us to show that for a whole class of permutation invariant ($\eta$-pair) states quantum discord can violate the monogamy property, both in presence and in absence of bipartite entanglement. In the thermodynamic limit, due to the presence of ODLRO, the violation for $\eta$-pair states is maximal, while, for the purely fermionic ground state, it is finite. From a general perspective, all our results validate the importance of the concepts of QD and CC for the study of critical condensed-matter systems.

Figure 1

Ground-state phase diagram of $H$. (Left) $n$-$u$ plane; empty circles stand for empty sites, and slashed and full circles stand for singly and doubly occupied sites, respectively. (Right) $\mu$-$u$ plane.

Figure 2

Quantum mutual information $I_{i,j}$ (blue, solid), quantum discord $Q_{i,j}$ (red, solid), classical correlation $C_{i,j}$ (green, solid), and concurrence $K_{i,j}$ (red, dashed) as a function of $\mu$ in region I ($u=4$) for $|i-j|=1$ (top, left), $|i-j|=2$ (top, right), $|i-j|=3$ (bottom, left), and $|i-j|=4$ (bottom, right).

Figure 3

Quantum mutual information $I_{i,j}$ (blue, solid), quantum discord $Q_{i,j}$ (red, solid), and classical correlation $C_{i,j}$ (green, solid) as a function of $|i-j|$ in region I for $\mu =-0.1,u=4$. Upper dashed lines represent the envelope of the respective maxima that exhibits a power-law decay ($\sim {|i-j|}^{-2}$).

Figure 4

Maxima of quantum discord $Q_{i,j}$ (solid lines) and concurrence $K_{i,j}$ (dashed lines) for $|i-j|=16$ (blue), $|i-j|=32$ (red), $|i-j|=64$ (green), and $|i-j|=128$ (black) as a function of $\sqrt{\left|\mu \right|}$ in region I ($u=4$).

Figure 5

Quantum mutual information $I$ (blue), quantum discord $Q$ (red), and classical correlation $C$ (green) as a function of $n_d$ in region III.

Figure 6

Quantum mutual information $I_{i,j}$ (blue), quantum discord $Q_{i,j}$ (red), and classical correlation $C_{i,j}$ (green) as a function of $u$ in region II for $n=1$, $|i-j|=1$ (top, left), $|i-j|=2$ (top, right), $|i-j|=3$ (bottom, left), and $|i-j|=4$ (bottom, right).

Figure 7

Quantum mutual information $I_{i,j}$ (blue), quantum discord $Q_{i,j}$ (red), and classical correlation $C_{i,j}$ (green) as a function of $u$ in region II for $n=1$, $|i-j|=1$ (top, left), $|i-j|=2$ (top, right), $|i-j|=3$ (bottom, left), and $|i-j|=4$ (bottom, right). We note that at the multipartite transition II $\to$ I ($u=0$) all the two-point correlation measures behave in a smooth way.

Figure 8

(Left) Ratio $R$ in phase III for chains of varying length $L$, with $N_d=1/L$ (red) and $N_d=\lfloor L/2\rfloor /L$ (green). (Right) Ratio $R$ in phase I in the TDL.