Figure 1

(a) $A(k,\omega )$ for $L=60$ (red solid line) and 80 (blue dotted line) at $k\simeq 0.8\pi$ for $U/t=8$ when $\delta =0.1$. (b) $A(\omega )$ in the LHB (red lines) for (b-1) $U/t=8$, $\delta \simeq 0.133$; (b-2) $U/t=8$, $\delta \simeq 0.033$; and (b-3) $U/t=4$, $\delta \simeq 0.033$. In (b-1), the blue dashed-dotted line denotes the contribution of dominant modes, while the green region shows the contribution of continua except the dominant modes. In (b-2,3), the light blue regions denote the contribution from $\omega >0$. Arrows indicate $ϵ(k=\pi )$ of Eq. (1). (c) Dispersion relations of the upper edge of the $s{h}^{*}$ continuum (red dotted line) and the $h^{*}$ mode with $s(k_F)$ (brown solid line) for $U/t=8$ at $\delta \simeq 0.133$. The green dotted line and the dot at $k=0.5\pi$ show those of $\delta \to 0$. (d) $A(k)=\int d\omega A(k,\omega )$ for $U/t=8$ at $\delta \simeq 0.133$. Blue squares with the light blue region denote the contribution from $\omega <0$, which is the momentum distribution function in the ground state $n(k)$. Green diamonds denote the contribution from $\omega >0$; red crosses and pink open circles denote that of the LHB for $\omega >0$ and that of the UHB. The vertical yellow line indicates $k=k_F$. Brown solid circles denote the total weight at each $k$, which satisfies the sum rule within numerical accuracy. (e) Doping dependence of spectral weight $A$. Solid lines show $A$ of the LHB for $\omega >0$ at $U/t=4$ (red circles) and 8 (blue squares). Red and blue triangles denote $A$ for $\omega <0$ at $U/t=4$ and 8. The green dashed line and orange dashed-dotted line indicate $A=\delta$ and ($1-\delta )/2$, respectively. Open symbols denote data for $L=80$. (f) $A(k,\omega )$ for $U/t=4$ at $\delta \simeq 0.033$.Reuse & Permissions

Figure 2

(a),(b),(d) $A(k,\omega )$ for $U/t=8$ at $\delta \simeq 0.4,0.133,0.033$ (from the left) for (a) overall views, (b) the LHB, and (d) the UHB. (c),(e) Dispersion relations obtained by using the Bethe ansatz in $L=240$, corresponding to (b),(d). Solid lines except those at $\omega =0$ show holon modes and antiholon modes. Blue dashed-dotted lines denote spinon modes. Red (pink) dotted lines show upper (lower) edges of continua of $h{h}^{*}$ with $s(k_F)$ and $h^{*}(2k_F)$, $s{h}^{*}$, $h{h}^{*}$ with $s(k_F)$, and $sh{h}^{*}$ (from above) in (c), and $shd$ in (e). Purple dashed double-dotted lines in (c) denote spinon modes in 2-$\Lambda$-string solutions. Open circles indicate peaks of dominant modes of (b),(d). The inset in (c-3) is the close-up near the gapless points. The pink [yellow] region in (e) denotes the $hd$ continuum with $s(-k_F)$ [$s(k_F)$]. Spinon [holon] modes in (e) have $h(\pm 2k_F)$ and $d\mathbf{(}\pm (\pi -2k_F)\mathbf{)}$ with $q_d{q}_h<0$ [$s(\pm k_F)$ and $d(|q_d|\simeq \pi \delta /2,q_d{q}_h<0)$].Reuse & Permissions