Quantum correlations, entanglement spectrum, and coherence of the two-particle reduced density matrix in the extended Hubbard model

Abstract

We study the ground state properties of the one-dimensional extended Hubbard model at half filling from the perspective of its particle reduced density matrix. We focus on the reduced density matrix of two fermions and perform an analysis of its quantum correlations and coherence along the different phases of the model. Specifically, we study its (i) entanglement entropy, (ii) ${\ensuremath{\ell}}_{1}$ norm of coherence, (iii) irreducible two-body cumulant matrix, and (iv) entanglement spectrum. Our results show that these different properties are complementary to each other depending on the phase of the system, exhibiting peculiar behaviors such as discontinuities and maximum or minimum values at the quantum phase transitions, thus providing a qualitative view of the phase diagram of the model. In particular, in the superconducting region, we obtain that the entanglement spectrum signals a transition from a dominant singlet (SS) to triplet (TS) pairing ordering in the system. Moreover, from the analysis of the dominant eigenvector in the reduced state, we can relate the SS-TS transition to the spatial separation between the fermion pairs in the two different pairing orderings. The entanglement gap is also able to highlight a transition---at a few-body level---in the ground state wave function, not discussed previously in the literature. While other quantifiers are less sensitive to few-body defects in the wave function, the entanglement gap can work as a magnifying glass for these, capturing such small fluctuations.