Correlation energy and quantum correlations in a solvable model

Typically in many-body systems the correlation energy, which is defined as the difference between the exact ground-state energy and the mean-field solution, has been a measure of the system's total correlations. However, under the quantum information context, it is possible to define some quantities in terms of the system's constituents that measure the classical and quantum correlations, such as the entanglement entropy, mutual information, quantum discord, one-body entropy, etc. In this work we apply concepts of quantum information in fermionic systems in order to study traditional correlation measures (the relative correlation energy). Concretely, we analyze the two- and three-level Lipkin models, which are exactly solvable (but nontrivial) models commonly used in the context of the many-body problem. We conclude that the correlation energy is not a good estimator of the total correlation of the system, and information theoretic measures should be considered.

Figure 1

A sketch of the two-level Lipkin model with $N=4$, where the lines represent the different orbitals.

Figure 2

Relative correlation energy as a function of the overall entropy, for the exact ground state of the two-level Lipkin model and different particle number $N$.

Figure 3

Second derivative of the relative correlation energy (Fig. 2) for increasing values of $N$ (from left to right). We observe a discontinuity in $\chi =1$, related to the phase transition of the model

Figure 4

Averaged one-body entropy of the exact ground state as a function of the parameter $\kappa$. As the number of particles increases, the transition between the spherical phase (low correlation regime) and the deformed one (highly correlated regime) is sharper.

Figure 5

Quantum discord between up and down levels of same quantum number $p$ (same degeneration level) for the HF ground state.

Figure 6

Relative correlation energy as a function of the overall entropy for the exact ground state of the three-level Lipkin model.

Figure 7

Second derivative of Fig. 6 for increasing values of $N$ (from left to right). We observe a discontinuity in $\chi =1$ and $\chi =3$ related to the two phase transitions of this model.

Figure 8

Averaged one-body entropy of the exact ground state as a function of the parameter $\chi$.

Figure 9

Quantum discord between levels of the same degeneration number $p$ for the HF ground state. The sum of the quantum discord between the three different configurations is represented with the solid red line.