We investigate the quantum phase transitions of the extended Hubbard model at half-filling with periodic boundary conditions employing the entanglement of particles, as opposed to the more traditional entanglement of modes. Our results show that the entanglement has either discontinuities or local minima at the critical points. We associate the discontinuities to first-order transitions, and the minima to second-order ones. Thus we show that the entanglement of particles can be used to derive the phase diagram, except for the subtle transitions between the spin-density wave (SDW) and bond-order wave (BOW) phases and the triplet superconducting (TS) and singlet superconducting (SS) phases.
We investigate the quantum phase transitions of the extended Hubbard model at half-filling with periodic boundary conditions employing the entanglement of particles, as opposed to the more traditional entanglement of modes. Our results show that the entanglement has either discontinuities or local minima at the critical points. We associate the discontinuities to first-order transitions, and the minima to second-order ones. Thus we show that the entanglement of particles can be used to derive the phase diagram, except for the subtle transitions between the spin-density wave (SDW) and bond-order wave (BOW) phases and the triplet superconducting (TS) and singlet superconducting (SS) phases.
By using the density matrix renormalization group technique, the phase diagram of the half-filling extended Hubbard model is investigated. The conventional order parameter, the two-site entanglement entropy, and the block-block entanglement entropy are analyzed in detail. According to the numerical results, in the weak coupling region, an intermediate bond-order-wave (BOW) phase is shown to exist indeed between the charge-density-wave (CDW) and the spin-density-wave (SDW) phases. The critical phase transition points are determined by the singularity of the first order derivative of two-site entanglement entropy. In strong coupling region, a direct phase transition occurs from SDW phase to CDW phase, and shows discontinuous (first order) character accompanied with energy level crossing. The numerical results support the phase diagram proposed previously by some authors Sengupta et al., (2002); Sandvik et al., (2004); Zhang, (2004). Therefore, the quantum entanglement is a sensitive tool to describe quantum phase transitions in strongly correlated electron systems.
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.
We examine the phase diagram of the half-filled one-dimensional extended Hubbard model using quantum information entropies within the density-matrix renormalization group. It is well known that there is a charge-density-wave phase at large nearest-neighbor and a small on-site Coulomb repulsion and a spin-density wave at small nearest-neighbor and large on-site Coulomb repulsion. At intermediate Coulomb interaction strength, we find an additional narrow region of a bond-order phase between these two phases. The phase transition line for the transition out of the charge-density-wave phase changes from first order at strong coupling to second order in a parameter regime where all three phases are present. We present evidence that the additional phase-transition line between the spin-density-wave and bond-order phases is infinite order. While these results are in agreement with recent numerical work, our study provides an independent unbiased means of determining the phase boundaries by using quantum information analysis, yields values for the location of some of the phase boundaries that differ from those previously found, and provides insight into the limitations of numerical methods in determining phase boundaries, especially those of infinite-order transitions.
We investigated the fidelity susceptibility in the one-dimension (1D) Hubbard model and the extended Hubbard model at half-filling via the density matrix renormalization group. From the numerical results, we argue that in the 1D Hubbard model, the fidelity susceptibility shows a divergence at two points which is infinitesimally close to the critical point while it is always extensive exactly at the critical point. For the extended Hubbard model, we found that for a properly chosen driving parameter, the fidelity susceptibility is able to reveal the quantum phase transitions between the PS (phase separation)-superconducting, superconducting-CDW (charge-density wave), CDW-SDW(spin-density wave), SDW-PS, CDW-BOW (bond-order wave), and the BOW-SDW phases.
We determine the ground-state phase diagram of the one-dimensional half-filled Hubbard model with on-site (nearest-neighbor) repulsive interaction U (V) and nearest-neighbor hopping t using the density-matrix renormalization group technique. Based on the results of the excitation gaps, Luttinger-liquid exponents, and bond-order-wave (BOW) order parameter, we confirm that the BOW phase appears in a substantial region between the charge-density-wave (CDW) and spin-density-wave phases. Each phase boundary is determined by multiple means and it allows us to make a cross-check on the validity of our estimations. We also find that the BOW-CDW transition changes from continuous to first order at the tricritical point (U(t),V(t)) approximately (5.89 t,3.10 t) and the BOW phase shrinks to zero at the critical end point (U(c),V(c)) approximately (9.25 t,4.76 t).
In this paper, we study block-block entanglement in the ground state of a one-dimensional extended Hubbard model. Our results show that the phase diagram derived from the block-block entanglement manifests a richer structure than that of the local (single-site) entanglement because it comprises nonlocal correlation. Besides phases characterized by the charge-density wave, the spin-density wave, and phase separation, which can be sketched out by the local entanglement, singlet superconductivity phase could be identified on the contour map of the block-block entanglement if the block size is larger than 1. Scaling analysis shows that log2(l) behavior of the block-block entanglement may exist in both the noncritical and critical regions, while some local extremum are induced by the finite-size effect. We also study the block-block entanglement defined in the momentum space and discuss its relation to the phase transition from singlet superconducting state to the charge-density-wave state.