Wide applicability of high-$T_c$ pairing originating from coexisting wide and incipient narrow bands in quasi-one-dimensional systems

We study superconductivity in the Hubbard model on various quasi-one-dimensional lattices with coexisting wide and narrow bands originating from multiple sites within a unit cell, where each site corresponds to a single orbital. The systems studied are the two-leg and three-leg ladders, the diamond chain, and the crisscross ladder. These one-dimensional lattices are weakly coupled to form two-dimensional (quasi-one-dimensional) ones, and the fluctuation exchange approximation is adopted to study spin-fluctuation-mediated superconductivity. When one of the bands is perfectly flat and the Fermi level intersecting the wide band is placed in the vicinity of, but not within, the flat band, superconductivity arising from the interband scattering processes is found to be strongly enhanced owing to the combination of the light electron mass of the wide band and the strong pairing interaction due to the large density of states of the flat band. Even when the narrow band has finite bandwidth, the pairing mechanism still works since the edge of the narrow band, due to its large density of states, plays the role of the flat band. The results indicate the wide applicability of the high-$T_c$ pairing mechanism due to coexisting wide and “incipient” narrow bands in quasi-one-dimensional systems.

Figure 1

Upper panel: Schematic image of coexisting wide and incipient narrow bands. Lower panels: Models in real space. (a) Two-leg ladder, (b) three-leg ladder, (c) diamond chain, (d) crisscross ladder; (e) and (f) show the coupling between the ladders and the diamond chains, respectively.

Figure 2

The bare energy bands of the tight-binding models having flat bands. (a) Two-leg ladder with $t^{\prime }=1$, (b) three-leg ladder with $t^{\prime }=1/\sqrt{2}$, diamond chain with (c) $t^{\prime }=0$ and (d) $t^{\prime }=1$, crisscross ladder with (e) $t_r=1$, (f) $t_r=0.5$, and (g) $t_r=0$.

Figure 3

$\lambda$ against the band filling for (a) the two-leg and three-leg ladders, (b) the diamond chain with $t^{\prime }=0$ or $t^{\prime }=1$, (c) the crisscross ladder with $t_r=1,0.5,0$. The linearized Eliashberg equation could not be solved at the band fillings where $\lambda$ is not plotted; there $\lambda$ is expected to be very small.

Figure 4

$\lambda$ against ${\mu }^{*}$ (see text for definition) for all flat-band cases.

Figure 5

Green's function for the two-leg ladder plotted against $k_x$. (a) $n=2.7$, (b) $n=2.45$, (c) $n=2.2$. Throughout the paper, the bare band structure is superimposed onto the Green's function by the solid lines on the right panels. The Green's function of a certain portion of the band is indicated by those with the same color as the band. The dashed line indicates the position of the bare chemical potential. See the text for the definition of $k_F^{\left(0\right)}$ and $k_F^{\mathrm{eff}}$.

Figure 6

Green's function for the three-leg ladder plotted against $k_x$. (a) $n=3.4$, (b) $n=3.1$.

Figure 7

Schematic image of the effective band structure of the three-leg ladder, where the two bands effective to superconductivity are extracted. (a) The bare band structure when the narrow band is perfectly flat. (b) When $U$ is introduced in the case of (a); here superconductivity is optimized within cases where the bare narrow band is flat (corresponds to $n=3.4$, $t^{\prime }=1/\sqrt{2}$). (c) When the chemical potential is reduced from the case of (b); superconductivity is suppressed compared to (b) (corresponds to $n=3.1$, $t^{\prime }=1/\sqrt{2}$). (d) When $t^{\prime }$ is reduced from the case of (b) (the bare narrow band has finite width); here superconductivity is enhanced compared to (b) (corresponds to $n=3.4$, $t^{\prime }=0.45$).

Figure 8

Green's function for the diamond chain with $t^{\prime }=0$ plotted against $k_x$. (a) $n=1.9$, (b) $n=1.98$.

Figure 9

$\lambda$ plotted against $t^{\prime }$ for the two-leg ladder for various band fillings.

Figure 10

Green's function against $k_x$ for the two-leg ladder with $n=2.45,t^{\prime }=0.25$.

Figure 11

Green's function against $k_x$ for the two-leg ladder with $n=2.7$ and (a) $t^{\prime }=1$, (b) $t^{\prime }=0.3$, or (c) $t^{\prime }=0$.

Figure 12

$\lambda$ plotted against $t^{\prime }$ for the three-leg ladder for various band fillings.

Figure 13

Green's function against $k_x$ for the three-leg ladder with $n=3.4$ and (a) $t^{\prime }=1/\sqrt{2}$ or (b) $t^{\prime }=0.45$.

Figure 14

Evolution of the band structure of the diamond chain with (a) $t^{\prime }=0$, (b) $t^{\prime }=0.5$, (c) $t^{\prime }=1$.

Figure 15

$\lambda$ plotted against the temperature for the diamond chain with $t^{\prime }=0$, $t^{\prime }=0.5$, or $t^{\prime }=1$, along with the results for the three-leg ladder with $t^{\prime }=1/\sqrt{2}$ and those for the crisscross ladder with $t_r=1$ or 0.5.