Density-Matrix Renormalization Group Study of Pairing when Electron-Electron and Electron-Phonon Interactions Coexist: Effect of the Electronic Band Structure

The density-matrix renormalization group is used to study the pairing when both electron-electron and electron-phonon interactions are strong in the Holstein-Hubbard model at half filling in a region intermediate between the adiabatic (Migdal’s) and antiadiabatic limits. We have found (i) the pairing correlation obtained for a one-dimensional system is nearly degenerate with the charge density-wave correlation in a region where the phonon-induced attraction is comparable with the electron-electron repulsion, but (ii) pairing becomes dominant when we destroy the electron-hole symmetry in a trestle lattice. This provides an instance in which pairing can arise, in a lattice-structure dependent manner, from coexisting electron-electron and electron-phonon interactions.

Figure 1

Various correlation functions versus real-space distance in the Holstein-Hubbard model at half filling calculated for $(\lambda ,\omega )=(3.6,5.0)$ (in units where $t=1$, $\hslash =1$) with three values of $U$, indicated as A, B, and C in a schematic parameter space (top inset). Error bars are smaller than the size of each symbol. Only the data for $|i-j|=\mathrm{\text{odd}}$ are plotted because of an even-odd effect in correlation functions in open-boundary condition. Vertical arrows with (a),(b) correspond to the scans in Figs. 2a, 2b.

Figure 2

Calculated powers ($\eta$) for the correlation functions ($\propto r^{-\eta }$) against $U-\lambda$ for (a) $(\lambda ,\omega )=(0.4,5.0)$, (b) $(\lambda ,\omega )=(3.6,5.0)$, and (c) $(\lambda ,\omega )=(3.6,2.5)$ in an $L=40$, half-filled Holstein-Hubbard chain, where $U$ is varied along the dashed arrows in inset of Fig. 1. The gray shadow indicates $U<0$. (d) The result for the Hubbard model (with $\lambda =0$), where the curves for charge and on-site SC are degenerate.

Figure 3

(a) Correlation functions when the electron-hole symmetry is destroyed with $t^{\prime }=-0.5$ for $(U,\lambda ,\omega )=(2.0,3.6,5.0)$. (b) Exponents against $-t^{\prime }$ for various correlation functions for $(U,\lambda ,\omega )=(2.0,3.6,5.0)$, and (c) a similar plot for the attractive Hubbard model with $U=-1.6$. The inset in (b) depicts the trestle lattice.