Phase separation and pairing regimes in the one-dimensional asymmetric Hubbard model

We address some open questions regarding the phase diagram of the one-dimensional Hubbard model with asymmetric hopping coefficients and balanced species. In the attractive regime we present a numerical study of the passage from on-site pairing dominant correlations at small asymmetries to charge-density waves in the region with markedly different hopping coefficients. In the repulsive regime we exploit two analytical treatments in the strong- and weak-coupling regimes in order to locate the onset of phase separation at small and large asymmetries, respectively.

Figure 1

(Color online) Schematic representation of the phases discussed in the text. Large and small disks represent heavy and light fermions, respectively. The ellipses encircling the pairs in the SS phase denote on-site pairing. The HP state defined in Eq. (2) has not been drawn in real space because it is defined directly in momentum space by filling the two noninteracting bands up to $k_F$.

Figure 2

(Color online) Example of cubic splines interpolation to locate the transition point and the associated error using the DMRG data of Table . The horizontal solid segment at $K_{\rho }=1$ indicates the reported interval.

Figure 3

(Color online) Numerical estimates of the transition lines from SS (on the left) to CDW (on the right) dominant correlations in the attractive regime for the values of filling reported in the legend. Not all the data used to obtain the figure have been reported in Table . The error bars associated to the transition points have been determined by means of splines passing through the upper and lower edges of each interval of possible values of $K_{\rho }$ reported in the table (see the construction of Fig. 2); when the uncertainty in $z$ turned to be smaller than ${10}^{-2}$ we have conventionally set it to this value to account for the approximation introduced by cubic splines interpolation. The lines joining the points are guides for the eye.

Figure 4

Upper panel: plot of the function $\overline{z}\left(n\right)$. Lower panel: optimal value of $\nu$ as a function of $z$ in the whole interval [0,1] for filling $n=1/2$.

Figure 5

(Color online) Energy density of the HP state at $n=1$ for different asymmetry parameters: upper red line $t_{\downarrow }/t_{\uparrow }=0.1$ and lower blue line $t_{\downarrow }/t_{\uparrow }=0.3$.

Figure 6

(Color online) Second-order perturbative contribution at half filling $\vartheta \left(\pi /2,\pi /2,a\right)/{\left(2\pi \right)}^3$ for different values of $a$; solid line is a guide for the eye.

Figure 7

(Color online) Weak-coupling phase diagram with $n=1/3$: white region represents the HP and green regions are different phase-separated states. The phase-transition lines are computed using Eq. (4) (dashed) and Eq. (9) (solid). In this section we have adopted the parameters $U/t_{\uparrow }=u/\left(1+z\right)$ and $a=t_{\downarrow }/t_{\uparrow }=\left(1-z\right)/\left(1+z\right)$ instead of $u$ and $z$ to facilitate the comparison with previous results in the weak-coupling region.

Figure 8

(Color online) Spatial density profiles for an open chain with $z=0.1$, filling $n=0.2$ and $u=100$. The continuous curve labeled SF is the local density for an equivalent chain of spinless fermions [i.e. Eq. (17) with $\ell =L$ and $p=nL$], showing a very good agreement with the spinful numerical data.

Figure 9

(Color online) Spatial density profiles for open chains with $z=0.5$ and $z=0.9$ (offset by $+0.4$ for clarity), filling $n=0.2$ and $u=100$.

Figure 10

(Color online) Transition lines in the $\left(z,u\right)$ plane at filling $n=0.4$ indicating how the ground state changes from SF to PS. In the separated regime above the transition lines, the state can be either PS (left) or TSS (right) (see Sec. 3B) and the edge between the two is marked by a vertical line (analogously to Fig. 3 of Ref. 20). The dashed line corresponds to the bosonization prediction.

Figure 11

(Color online) Same as Fig. 10 but with filling $n=0.9$. Now also the ${\text{PS}}^{\prime }$ (see text) state is relevant for moderate values of $u$ and the corresponding transition lines with SF and PS are marked.