Superconductivity, charge-density waves, antiferromagnetism, and phase separation in the Hubbard-Holstein model

By using variational wave functions and quantum Monte Carlo techniques, we investigate the interplay between electron-electron and electron-phonon interactions in the two-dimensional Hubbard-Holstein model. Here, the ground-state phase diagram is triggered by several energy scales, i.e., the electron hopping $t$, the on-site electron-electron interaction $U$, the phonon energy ${\omega }_0$, and the electron-phonon coupling $g$. At half filling, the ground state is an antiferromagnetic insulator for $U\gtrsim 2g^2/{\omega }_0$, while it is a charge-density-wave (or bipolaronic) insulator for $U\lesssim 2g^2/{\omega }_0$. In addition to these phases, we find a superconducting phase that intrudes between them. For ${\omega }_0/t=1$, superconductivity emerges when both $U/t$ and $2g^2/t{\omega }_0$ are small; then, by increasing the value of the phonon energy ${\omega }_0$, it extends along the transition line between antiferromagnetic and charge-density-wave insulators. Away from half filling, phase separation occurs when doping the charge-density-wave insulator, while a uniform (superconducting) ground state is found when doping the superconducting phase. In the analysis of finite-size effects, it is extremely important to average over twisted boundary conditions, especially in the weak-coupling limit and in the doped case.

Figure 1

Size scaling of the energy per site at half filling (with ${\omega }_0/t=1$, $\lambda /t=0.98$, and $U=0$) for periodic-periodic (red squares), periodic-antiperiodic (black circles), and twisted average (blue diamonds) boundary conditions. Error bars are smaller than the size of the symbols.

Figure 2

Size scaling of the energy per site at quarter filling (with ${\omega }_0/t=1$, $\lambda /t=2$, and $U=0$) for periodic-periodic (red squares) and twisted average (blue diamonds) boundary conditions. Error bars are smaller than the size of the symbols.

Figure 3

Ground-state phase diagram in the $(\lambda /t,U/t)$ plane for the Hubbard-Holstein model at ${\omega }_0/t=1$ (upper panel), ${\omega }_0/t=5$ (middle panel), and ${\omega }_0/t=15$ (lower panel). Antiferromagnetic (AF), charge-density-wave (CDW), and superconducting (SC) phases are present, the latter one being marked in blue. The calculations have been performed on the $12\times 12$ cluster with TABC. The dotted line $U=\lambda$ is also marked for reference.

Figure 4

Antiferromagnetic (red circles) and charge-density-wave (blue squares) parameters for the case with ${\omega }_0/t=1$. The cases with $\lambda /t=4.5$ (upper panel), where a first-order phase transition between these two insulators is present, and with $\lambda /t=2$ (lower panel), where a continuous phase transition takes place, are reported. The calculations have been performed on the $12\times 12$ cluster with TABC and error bars are smaller than the size of the symbols.

Figure 5

Antiferromagnetic (red circles), charge-density-wave (blue squares), and superconducting (black diamonds) parameters for the case with ${\omega }_0/t=5$. The cases with $\lambda /t=4.5$ (upper panel) and $\lambda /t=2$ (lower panel) are reported. In both cases, there is a small region where the ground state is superconducting with no charge-density-wave order. The calculations have been performed on the $12\times 12$ cluster with TABC and error bars are smaller than the size of the symbols.

Figure 6

Antiferromagnetic (red circles), charge-density-wave (blue squares), and superconducting (black diamonds) parameters for the case with ${\omega }_0/t=15$. The cases with $\lambda /t=4.5$ (upper panel) and $\lambda /t=2$ (lower panel) are reported. In this cases, there is a substantial region where the ground state is superconducting with no charge-density-wave order. The calculations have been performed on the $12\times 12$ cluster with TABC and error bars are smaller than the size of the symbols.

Figure 7

Energy per hole of Eq. (16) for $\lambda /t=2$ and various values of $U/t$ (upper panel) and for $U=0$ and various values of $\lambda /t$ (lower panel). In both cases ${\omega }_0/t=1$ and calculations are performed on the $12\times 12$ cluster with TABC. Error bars are smaller than the size of the symbols.

Figure 8

Variational parameter ${\mathrm{\Delta }}_{\mathrm{SC}}$ as a function of the electron density $n$, computed with periodic-antiperiodic (black circles), and twisted average (blue diamonds) boundary conditions on the $12\times 12$ cluster. Here, ${\omega }_0/t=15$ and $\lambda /t=0.98$. Error bars are smaller than the size of the symbols.

Figure 9

Energy per hole of Eq. (16) for $U=0$ and $\lambda /t=2$ (red squares), compared with $U/t=1.38$ and $\lambda /t=3.38$ (black circles), that corresponds to the same value of the effective interaction $U_{\mathrm{eff}}=U-\lambda$. In both cases ${\omega }_0/t=1$ and calculations are performed on the $12\times 12$ cluster with TABC. Error bars are smaller than the size of the symbols.

Figure 10

Energy per hole of Eq. (16) for $U=0$ and $\lambda /t=2$ for ${\omega }_0/t=1$ (red squares), 5 (black circles), and 15 (blue diamonds). Calculations are performed on the $12\times 12$ cluster with TABC. Error bars are smaller than the size of the symbols.