Superconductivity in a model of two Hubbard chains coupled with ferromagnetic exchange interaction

We study the ground-state properties of the double-chain Hubbard model coupled with ferromagnetic exchange interaction by using the weak-coupling theory, density-matrix renormalization-group technique, and Lanczos exact-diagonalization method. We determine the ground-state phase diagram in the parameter space of the ferromagnetic exchange interaction and band filling. We find that, in high electron-density regime, the spin gap opens and the spin-singlet $d_{xy}$-wave-like pairing correlation is most dominant, whereas in low electron-density regime, the fully-polarized ferromagnetic state is stabilized where the spin-triplet $p_y$-wave-like pairing correlation is most dominant.

Figure 1

Schematic representation of the double-chain Hubbard model coupled with ferromagnetic exchange interaction. No hopping process is allowed between the two chains and no exchange interactions exist along the chains.

Figure 2

Symmetry of the pairing state in the double-chain Hubbard model. The sign of the order parameter is also shown.

Figure 3

(Color online) Calculated values of the total-spin quantum number $S$ as a function of $J$ for $U=20$ (circles), 40 (squares), and $\infty$ (crosses). $L=48$ and $n=0.5$ are assumed.

Figure 4

(Color online) ${\Delta }_c\left(L\right)$ as a function of the inverse system length $1/L$. Lines are the fits with the second-order polynomials.

Figure 5

(Color online) Upper panels: ${\Delta }_s\left(L\right)$ as a function of inverse system size for (a) $n=1$ and (b) $n=0.75$. Solid lines are the polynomial fits to the data for the finite-size scaling analysis. Lower panels: Semilogarithmic plot of the magnitude of the spin-spin correlation function $S\left(\left|r_x-r_x^{\prime }\right|,1,2^{\prime }\right)$ at $J=1$ for (c) $n=1$ and (d) $n=0.75$. The data are fitted with a function $S\left(\left|r_x-r_x^{\prime }\right|,1,2\right)\simeq \text{exp}\left(-\frac{\left|r_x-r_x^{\prime }\right|}{\xi }\right)$.

Figure 6

(Color online) Upper panel: ${\Delta }_B^{\pm }\left(L\right)$ as a function of inverse system length $1/L$ at $n=0.5$. Solid lines are the polynomial fits to the data for finite-size scaling analysis. Solid (empty) symbols denote the binding energy of electrons (holes). Lower panels: Values of ${\Delta }_B^{\pm }$ extrapolated to $1/L\to 0$ plotted as a function of $J$ for (b) $U=4$ and (c) $U=10$.

Figure 7

(Color online) Log-log plot of the pair-correlation functions $D^{\alpha }\left(\left|r_x-r_x^{\prime }\right|\right)$ calculated for $L=128$, $U=10$, and $J=4$.

Figure 8

Left panel: Anomalous Green’s function at $k_y=0$, $k_y^{\prime }=\pi$, $n=3/4$, $U=10$, $J=2$, and $L=8$. Right panel: Same as the left panel but at $k_y=\pi$, $k_y^{\prime }=0$.

Figure 9

(Color online) DMRG results for $K_{\rho +}$ as a function of $J$. The results for $U=10$ and 20 at $n=0.5$ are shown. Dotted line is guide for eyes.

Figure 10

(Color online) Phase diagram of our double-chain Hubbard model obtained from the DRMG calculations. The phase boundary between the fully-polarized ferromagnetic phase and $d_{xy}$-like superconducting (SC) phase is shown for $U=5$, 10, and 20.