Entanglement Properties of the One-Dimensional Dimerized Fermi-Hubbard Model

Abstract

We study the entanglement properties of the one-dimensional dimerized Fermi-Hubbard model. Using a matrix-product-state approach, we compute the ground state and identify two insulating phases at 1/2- and 3/4-filling, along with a metallic phase, whose mechanisms can be characterized by their entanglement spectra. Our findings indicate that the two insulating phases are distinct, implying that the phase at 1/2-filling has a charge gap arising from the band gap, which is enhanced by repulsive interactions, while the phase at 3/4-filling exhibits a Mott gap resulting from particle interactions. This difference between the two insulating phases is reflected in the scaling properties of the half-chain entanglement entropy and the distribution of the entanglement spectrum.

Figures (5)

• Figure 1: Schematic phase diagram of the one-dimensional dirmerized Fermi-Hubbard model for a dimerization strength of $\delta t=0.5$. At 1/2-filling, a band insulating phase with a band gap $\Delta=4\delta t$ emerges when $U=0$, while at 3/4 (or 1/4)-filling a Mott insulating phase appears.
• Figure 2: Entanglement entropy for (a) a single-site (b) a double-site, and (c) a half-chain system as a function of $\mu$ when $U=2$ and $\delta t=0.5$. The black lines represent the entanglement entropy when the region is coupled to the rest of the system via a $t_+$-bond, while the lower red lines correspond to the case of coupling via a $t_-$-bond. In (a), the entanglement entropy shows overlapping black and red lines. The inset illustrates the density behavior, where the transitions between the insulating phase at fixed density and the metallic phase can be easily identified.
• Figure 3: Half-chain entanglement entropy $S_h(\chi)$ in the insulating phases is plotted as a function of the matrix size $\chi$ for $U=4$ and $\delta t=0.5$. At 3/4-filling, $S_h(\chi)$ exhibits a logarithmic scaling relation indicative of a Mott insulating phase, characterized by a correlation length with the exponent $\kappa=1.344$. On the other hand, at 1/2-filling, $S_h(\chi)$ shows minimal dependence on $\chi$, reflecting the properties of a gapped band insulator with finite correlation lengths. This difference shows the distinct nature of the two insulating phases.
• Figure 4: Typical half-chain entanglement spectrum for $t_+$- and $t_-$-bonds at different phases of the dimerized Fermi-Hubbard model. In the metallic phase, the entanglement spectrum exhibits double degeneracy. This double degeneracy is lifted in the insulating phase for $t_-$-bonds, while $t_+$-bonds retain their double degeneracy, reflecting the spin-1/2 isospin state at 3/4-filling and the spin-singlet state at 1/2-filling.
• Figure 5: The behavior of $n(w)$, the mean number of eigenvalues larger than a given $w$, for different $\chi$ shows a universal scaling behavior for $t_-$-bonds in the 3/4-filled Mott insulating phase. No universal behavior is observed for $t_+$-bonds and in other phases (not shown in the figure). The dotted line represents the theoretical prediction given by $2I_0(\xi_w)-1$ with the global degeneracy $g=2$.