Doped $A{B}_2$ Hubbard chain: Spiral, Nagaoka and resonating-valence-bond states, phase separation, and Luttinger-liquid behavior

We present an extensive numerical study of the Hubbard model on the doped $A{B}_2$ chain, both in the weak coupling and the infinite-$U$ limit. Due to the special unit-cell topology, this system displays a rich variety of phases as a function of hole doping $\left(\delta \right)$ away from half-filling. Near half-filling spiral states develop in the weak-coupling regime, while Nagaoka itinerant ferromagnetism is observed in the infinite-$U$ limit. For higher doping, the system phase-separates before reaching a Mott insulating phase of short-range resonating-valence-bond states at $\delta =1∕3$. Moreover, for $\delta >1∕3$ we observe a crossover, which anticipates the Luttinger-liquid behavior for $\delta >2∕3$.

Figure 1

(Color online) (a) Illustration of the $A{B}_2$ chain showing $A$, $B_1$, and $B_2$ sites. (b) Illustration of the effective linear chain (ELC). (c) Electronic bands of the tight-binding model: two dispersive (continuous line) and one flat (dashed line).

Figure 2

DMRG results for the energy difference between the lowest energy at the symmetry sector $(-{)}^x(+{)}^{N_c-x}$ and the GS energy for $N=100$ $(N_c=33)$ for (a) $U=2$ at $\delta =0.32$ (triangle up), $\delta =0.26$ (diamond), and $\delta =0.18$ (triangle down); (b) $U=\infty$ at $\delta =0.32$ (triangle up), $\delta =0.28$ (triangle up), and $\delta =0.24$ (triangle down), taking 108 states per block (open symbols) and 216 states per block (filled symbols). Average local parity $p$ as a function of $\delta$ for (c) $U=2$ and (d) $U=\infty$. (e) ED results for $\mid z^{\left(q\right)}\mid$. Dashed lines are guides to the eye.

Figure 3

(a) and (b) Magnetic structure factor for $U=2$ and $N=100$ in the underdoped region. (c) Total spin per cell $S∕N_c$ as function of $\delta$ for $U=\infty$.

Figure 4

(Color online) ED results of (a) $N_c$ times the energy gap ${\Delta }_0$ between the saturated ferromagnetism energy $(\Phi ={\Phi }_F)$ and the lowest energy state for an Aharanov-Bohm flux $\Phi$ as a function of $\tilde{\Phi }=\mid \Phi -{\Phi }_F\mid$ at $\delta =1∕6$ and $N_c=$ 4 (dashed-dot line), 6 (dashed line), and 8 (solid line). ED calculation for the $\tilde{\Phi }$-dependent behavior of (b) the spin correlation function $⟨{\mathbf{S}}_{\mathrm{cell}}\left(l_0\right)\bullet {\mathbf{S}}_{\mathrm{cell}}(l_0+l)⟩$ between the cell spins as a function of $l$ and (c) the magnetic structure factor as a function of lattice wave vector $q$ at $\delta =1∕6$. (d) Charge structure factor calculated at the lowest-energy state for any $\tilde{\Phi }$ at $\delta =1∕6$ for $N_c=$ 4 $(●)$, 6 $(▴)$, and 8 $(▿)$.

Figure 5

GS properties at $\delta =0.18$ $(U=2)$ and $\delta =0.28$ $(U=\infty )$ for $N=100$ using DMRG. (a) Spin correlation function $⟨{\mathbf{S}}_1\bullet {\mathbf{S}}_i⟩$ for $U=2$. (b) Expectation value of $S_i^z$ for $U=\infty$ in the sector $S^z=S_g$. Spin correlation function $⟨{\mathbf{S}}_{B_1}\bullet {\mathbf{S}}_{B_2}{⟩}_i$ for (c) $U=2$ and (d) $U=\infty$. $-$ $(+)$ indicates odd (even) local parity. Effective linear chain notation: (●) identifies $A$ sites and (엯) $B_1+B_2$ at the same cell. Dashed lines are guides to the eye.

Figure 6

(Color online) GS properties at $\delta =0.18$ $(U=2)$ and $\delta =0.28$ $(U=\infty )$ for $N=100$ using DMRG. Expectation value of $n_{h,i}$ for (a) $U=2$ and (b) $U=\infty$. Effective linear chain notation: $(●)$ identifies $A$ sites and $(엯)$ $B_1+B_2$ at the same cell. (c) Illustration of the GS for $U=\infty$ in the phase-separated regime: singlet bonds are represented by ellipses and holes by circles. $-(+)$ indicates odd (even) local parity. Dashed lines are guides to the eye.

Figure 7

DMRG results for (a) the size dependence of the charge $\left({\Delta }_c\right)$ and spin $\left({\Delta }_S\right)$ gaps as a function of $1∕N_c$ at $\delta =1∕3$: solid lines are polynomial fittings. ED results for (b) the expectation values of $S_A^z\left(l\right)$, $S_B^z\left(l\right)$ and the correlation function $⟨{\mathbf{S}}_{B_1}\left(l\right)\bullet {\mathbf{S}}_{B_2}\left(l\right)⟩$ at spin sector $S^z=1$ as a function of cell number $l$. The $\pm$ signs below the horizontal axis in (b) indicate the cell parity.

Figure 8

Spin correlation functions (a) $⟨{\mathbf{S}}_A\left(l_c\right)\bullet {\mathbf{S}}_A\left(l\right)⟩$, (b) $⟨{\mathbf{S}}_A\left(l_c\right)\bullet {\mathbf{S}}_B\left(l\right)⟩$, and (c) $⟨{\mathbf{S}}_B\left(l_c\right)\bullet {\mathbf{S}}_B\left(l\right)⟩$ as a function of $l-l_c$ in the ELC at $\delta =1∕3$ for $N_c=33$ using DMRG; in the above expressions, $l_c$ denotes the central cell; $(◇)$ refers to $U=2$ and $(◆)$ to $U=\infty$. Dashed lines are guides to the eye. (d) Illustration of the GS at $\delta =1∕3$; singlet bonds are represented by ellipses and holes by circles.

Figure 9

(a) ED results for the indicated spin correlation functions as a function of cell index $l$, with ${\mathbf{S}}_B\equiv {\mathbf{S}}_{B_1}+{\mathbf{S}}_{B_2}$. (b) Illustration of the GS at $\delta =1∕3$ doped with two holes: singlet bonds are represented by ellipses and holes by circles. (c) ED results for the spin correlation functions between $B$ sites at the same cell $⟨{\mathbf{S}}_{B_1}\left(l_0\right)\bullet {\mathbf{S}}_{B_2}\left(l_0\right)⟩$, electron densities at $A$ sites $⟨n_A\left(l_0\right)⟩$, and at $B\equiv B_1+B_2$ sites $⟨n_B\left(l_0\right)⟩\equiv ⟨n_{B_1}\left(l_0\right)+n_{B_2}\left(l_0\right)⟩$. (d) ED results for the indicated nearest-neighbor spin correlation functions as a function of $\delta$, with ${\mathbf{S}}_B\equiv {\mathbf{S}}_{B_1}+{\mathbf{S}}_{B_2}$. In (a), (c), and (d), $l_0$ denotes an arbitrary cell.

Figure 10

ED results for the charge susceptibility $\chi$, the charge excitation velocity $u_{\rho }$, and the Drude weight $D$, for $U=2$ [(a), (b), and (c)] and $U=\infty$ [(d), (e), and (f)], and $N_c=4(●)$, $6(▵)$, and 8 $(▾)$.

Figure 11

DMRG results for (a) the charge gap ${\Delta }_c$ as a function of $1∕N_c$ at $\delta =2∕3$ using DMRG; the inset presents extrapolated values of the charge gap as a function of $U$. DMRG calculation of (b) the spin gap ${\Delta }_S$ as a function of $1∕N_c$ for $U=2$ $(●)$ and $U=\infty$ $(◆)$. Solid lines are polynomial fittings, except in the inset of (a), where we have used an essential singularity form as explained in the text.

Figure 12

(a) ED results for the ratio $u_{\rho }∕\sqrt{D\chi ∕\pi }$. (b) ED results for $K_{\rho }$ as a function of $\delta$.

Figure 13

Spin correlation functions $C\left(l\right)$ for (a) $U=2$ and (b) $U=\infty$ at $\delta =88∕106$ for $N=106$ using DMRG: solid lines are fittings using Eq. (25).

Figure 14

Energy per unit cell $ϵ$ as a function of the fraction $x=L_{-}∕N_c$ for $\delta =0.28$. The physical region is also shown.