Superconductivity in the repulsive Hubbard model on different geometries induced by density-assisted hopping

Abstract

We study the effect of density-assisted hopping on different dimerized lattice geometries, such as bilayers and ladder structures. We show analytically that the density-assisted hopping induces an attractive interaction in the lower (bonding) band of the dimer structure and a repulsion in the upper (anti-bonding) band. Overcoming the onsite repulsion, this can lead to the appearance of superconductivity. The superconductivity depends strongly on the filling, and present a pairing structure more complex than s-wave pairing. Combining numerical and analytical methods such as the matrix product states ansatz, bosonization and perturbative calculations we map out the phase diagram of the two-leg ladder system and identify its superconducting phase. We characterize the transition from the non-density-assisted repulsive regime to the spin-gapped superconducting regime as a Berezinskii-Kosterlitz-Thouless transition.

Figures (6)

• Figure 1: (a) Sketch of the Hubbard ladder with $L$ rungs. The value of the on-site interaction is $U$, hopping amplitudes on different directions are $t_\parallel$ and $t_\perp$, the density-assisted hopping contribution is $t_{n\perp}$. (b) In the weak coupling limit, the density assisted hopping term makes the bonding band attractive; (c) the value of the central charge extracted from MPS simulations in the strong coupling regime. The green (purple) line shows an example of a constant (effective) perpendicular hopping $t_{\perp}$ ($\tilde{t}_{\perp}$) level line. The effective attraction opens a spin gap and induces a superconducting phase. The corresponding simulation parameters are $n=0.9375$, $U=4t_{\parallel}$, and $L=80$ rungs.
• Figure 2: Transition at $U=4t_\parallel$, $n=0.9375$ and $\tilde{t}_\perp=3t_\parallel$ for a $96$-rung ladder. (a) The gray and purple plots correspond to the central charge and the discontinuity of the momentum distribution $Z_{k_F}$, respectively. The orange plot corresponds to the inverse correlation length. (b) Momentum distribution for the up-spin in the bonding band, $n(k)$, for different values of the density-assisted hopping ratio ($c_n$). (c) Power-law exponents for the pair ($\eta_d$), density ($\eta_n$) and spin ($\eta_{sz}$) correlation functions. The plots are $1/\eta$, hence a larger value means than the correlations decay more slowly. The error bars were estimated by changing the fitting range and obtaining the maximum deviation. With translucid lines we plot the results for $L=32, 48, 64, 80$ in panel (a), and $L=64, 80$ in (c) (more opaque lines correspond to larger systems).
• Figure 3: Entanglement entropy for a ladder with $L=96$ rungs, as (a) as a function of the bond at which the system is bisected, and (b) as a function of the logarithmic conformal distance, $\ln d(l|L)$. Orange lines on panel (b) correspond to our fitting function in Eq. \ref{['EQ_ccfit']}. The data corresponds to the case with $U=4t_\parallel$, $\tilde{t}_\perp=3t_\parallel$, $n=0.9375$ and $c_n=0.6$.
• Figure 4: Correlation functions with distance for a $L=96$ rungs ladder at $U=4t_{\parallel}$, $n=0.9375$, and $\tilde{t}_\perp=3t_{\parallel}$. Spin, density and pair correlation functions are shown. The data corresponds to a density-assisted hopping ratio with a value (a) $c_n=0.10$, (b) $c_n = 0.24$, and (c) $c_n = 0.40$. Dashed lines correspond to the fitted functions. (d) Momentum distribution for the up-spin [$n(k)$] in the bonding and antibonding band for different values of the density-assisted hopping ratio ($c_n$).
• Figure 5: Power-law exponents for the correlation functions for different system sizes as a function of $c_n$. The data corresponds to $U=4t$, $n=0.9375$, and effective perpendicular hopping is $\tilde{t}_\perp=3t$. We can see that with increasing system size, the product of the charge and pair exponents tends to one ($\eta_n \eta_d \to 1$) as expected in the thermodynamic limit.