Ground-state properties of the one-dimensional attractive Hubbard model with confinement: A comparative study

We revisit the one-dimensional attractive Hubbard model by using the Bethe-ansatz-based density-functional theory and density-matrix renormalization method. The ground-state properties of this model are discussed in details for different fillings and different confining conditions in weak-to-intermediate coupling regime. We investigate the ground-state energy, energy gap, and pair-binding energy and compare them with those calculated from the canonical Bardeen-Cooper-Schrieffer approximation. We find that the Bethe-ansatz-based density-functional theory is computationally easy and yields an accurate description of the ground-state properties for weak-to-intermediate interaction strength, different fillings, and confinements. In order to characterize the quantum phase transition in the presence of a harmonic confinement, we calculate the thermodynamic stiffness, the density-functional fidelity, and fidelity susceptibility, respectively. It is shown that with the increase in the number of particles or attractive interaction strength, the system can be driven from the Luther-Emery-type phase to the composite phase of Luther-Emery-type in the wings and insulatinglike in the center.

Figure 1

(Color online) Top panel: the exchange-correlation potential (in units of $t$) of the 1D attractive Hubbard model as a function of $n$ for various values of $s$ with $u=-1$. The upper three data are for $V_{\text{xc},\uparrow }^{\text{hom}}$ and the lower three for $V_{\text{xc},\downarrow }^{\text{hom}}$. Obviously there is no charge gap for the attractive Hubbard model. Bottom panel: the same as that in the top panel but for $u=-4$.

Figure 2

(Color online) Top panel: the exchange-correlation potential (in units of $t$) $V_{\text{xc},\uparrow }^{\text{hom}}$ of the attractive Hubbard model as a function of $s$ for various values of $n$ with $u=-4$. The spin channel opens a gap at $s=0$. Bottom panel: the same as that in the top panel but for $V_{\text{xc},\downarrow }^{\text{hom}}$.

Figure 3

(Color online) The xc gap ${\Delta }_{\text{xc},\sigma }\left(n,u\right)$ (in units of $t$) as a function of $\left|u\right|$. In the positive direction, ${\Delta }_{\text{xc},\uparrow }\left(n,u\right)$ are shown and in the negative direction ${\Delta }_{\text{xc},\downarrow }\left(n,u\right)$ are plotted. In the inset, the semilog plot for ${\Delta }_{\text{xc},\uparrow }\left(n,u\right)$ is shown for $\left|u\right|=0.1–0.5$.

Figure 4

(Color online) Top panel: the ground-state energy per site (in units of $t$) of the attractive 1D Hubbard model with periodic boundary condition as a function of particle density $n=N_{\text{f}}/N_{\text{s}}$ for $u=-2$ in a lattice with $N_{\text{s}}=32$ sites. SDFT/BALSDA results (crosses) are compared with the CBCS data (triangles) and the exact results (open circles). The CBCS data are taken from Ref. 48 and the exact data are calculated from the couple Bethe-ansatz equations. Bottom panel: the same as that in the top panel but for $u=-4$ in a lattice with $N_{\text{s}}=16$ sites. The system of even number of particles is implicit with zero magnetization $\left(s=0\right)$. For the system of odd number of particles, we choose $N_{\uparrow }-N_{\downarrow }=1$, that is, $s=1/\left(2N_{\text{s}}\right)$. In this case, SDFT has to be used to obtain an accurate GS energy.

Figure 5

(Color online) The ground-state energy per site (in units of $t$) of the attractive 1D Hubbard model with periodic boundary condition as a function of the coupling strength $u$ with even (left panel) and odd (right panel) number of particles in a lattice of $N_{\text{s}}=8$ (upper frame) and $N_{\text{s}}=64$ (lower frame) sites for various values of fillings. SDFT/BALSDA results $(\times )$ are compared with the exact (solid line) and CBCS (dotted lines) ones. (a) For $N_{\text{s}}=8$ and 64 with even numbers of particles (the magnetization $s=0$). (b) The same as (a), but with odd numbers of particles [the magnetization $s=1/\left(2N_{\text{s}}\right)$, and $N_{\uparrow }-N_{\downarrow }=1$].

Figure 6

(Color online) The energy gap (in units of $t$) of the attractive 1D Hubbard model with periodic boundary condition as a function of the particle density $n=N_{\text{f}}/N_{\text{s}}$ is plotted for various coupling strength in a lattice of sites $N_{\text{s}}=4$ and 16. The SDFT/BALSDA (crosses) results are compared with the exact (open circles) and CBCS data (upper triangle). The upper frame is for $N_{\text{s}}=4$ and the lower frame $N_{\text{s}}=16$. The gap is negative for odd $N_{\text{f}}$ but positive for even $N_{\text{f}}$ reflecting the pairing information induced by the negative interactions. (a) For $u=-1$; (b) $u=-1.5$; and (c) $u=-4$. For the system of even number of particles, the zero magnetization $\left(s=0\right)$ is used. For the system of odd number of particles, we choose $N_{\uparrow }-N_{\downarrow }=1$, that is, $s=1/\left(2N_{\text{s}}\right)$.

Figure 7

(Color online) (a) The ground-state energy per site (in units of $t$) of the attractive 1D Hubbard model in a hard wall as a function of particle density $n=N_{\text{f}}/N_{\text{s}}$ in a lattice of $N_{\text{s}}=32$ sites for $u=-1$ and $-4$.

Figure 8

(Color online) The ground-state energy per site (in units of $t$) of the attractive 1D Hubbard model in a hard wall as a function of the coupling strength $u$ in a lattice of sites $N_{\text{s}}=32$ for various values of fillings $n$.

Figure 9

(Color online) The thermodynamic stiffness $S_{\rho }$ (in units of $t$) as a function of $N_{\text{f}}$ for $u=-1$ and $-2$ with $V_2/t={10}^{-3}$, and $N_{\text{s}}=200$ lattice sites. In the inset we show the density profile in the bottom (top) for $N_{\text{f}}=40\left(N_{\text{f}}=132\right)$ for $u=-2$. Their corresponding positions are indicated by the arrows in the $N_{\text{f}}$ axis. The vertical line indicates another possible quantum phase transition driven by the attractive interaction strength at fixed $N_{\text{f}}=128$, which is checked through the concept of density-functional fidelity and fidelity susceptibility in Fig. 10.

Figure 10

(Color online) Density-functional fidelity (upper panel) and fidelity susceptibility (lower panel) as a function of couple strength $u$ for $N_{\text{s}}=100$, $N_{\text{f}}=128$, and $V_2/t={10}^{-3}$. $\delta u=0.004$ (solid red line) and $\delta u=0.002$ (dotted blue line) are shown in the figures.