Competition among Superconducting, Antiferromagnetic, and Charge Orders with Intervention by Phase Separation in the 2D Holstein-Hubbard Model

Using a variational Monte Carlo method, we study the competition of strong electron-electron and electron-phonon interactions in the ground state of the Holstein-Hubbard model on a square lattice. At half filling, an extended intermediate metallic or weakly superconducting (SC) phase emerges, sandwiched between antiferromagnetic and charge order (CO) insulating phases. By carrier doping into the CO insulator, the SC order dramatically increases for strong electron-phonon couplings, but is largely hampered by wide phase separation (PS) regions. Superconductivity is optimized at the border to the PS.

Figure 1

Ground-state phase diagram of the half-filled Holstein-Hubbard model on a square lattice at (a) $\mathrm{\Omega }=8t$ and (b) $\mathrm{\Omega }=t$. The blue squares and red circles represent the boundaries of CO and AF, respectively. Error bars are drawn, but most of them are smaller than the symbol size. Solid lines are used to guide the eye. Dashed lines represent $\lambda =U/8t$. Based on the fact that if $\lambda =0$, the system is an AF insulator for any $U>0$ [37], we put the starting point of the AF boundary at the origin. The shaded region represents the intermediate PM or weak SC phase.

Figure 2

$S_s(\pi ,\pi )/N$ and $S_c(\pi ,\pi )/N$ as functions of $\lambda$ at (a) $(\mathrm{\Omega }/t,U/t)=(8,8)$ and (b) $(\mathrm{\Omega }/t,U/t)$ = (1,8), respectively. The vertical dashed line represents $U_{\text{eff}}=0$. The shaded region indicates the intermediate metallic (weakly SC) phase.

Figure 3

$E/N-\lambda$ curves of PM (weak SC) states crossing with (a) AF and (b) CO states at $(\rho ,\mathrm{\Omega }/t,U/t)=(1,8,8)$. The paths are along the vertical line at $U/t=8$ at the top right of Fig. 1. The curves are obtained by gradually changing $\lambda$ (in the direction indicated as colored arrows). Unimportant linear terms $f_1(\lambda )$ and $f_2(\lambda )\propto \lambda$ are subtracted from $E/N$ for clarity. The crossing points are indicated as black arrows.

Figure 4

Ground-state phase diagrams of the Holstein model in the $\delta \text{−}\lambda$ plane at (a) $\mathrm{\Omega }=8t$ and (b) $\mathrm{\Omega }=t$. In the vertical axis, $S_c(\pi ,\pi )/N$ (red squares) and $P_s^{\infty }$ (color plots) for $L=14$ are plotted in the CO and SC phases, respectively. Black squares in the bottom plane represent boundaries between the PS and $s$-wave SC regions. White areas denote the PS regions. Thick red lines at $\delta =0$ indicate the CO phase.

Figure 5

Physical quantities $S_c(\pi ,\pi )/N$, $P_s^{\infty }$, $\mu$, and $-{\chi }_c^{-1}$ as functions of doping $\delta$ at (a) $(\mathrm{\Omega }/t,U/t,\lambda )=(8,0,0.3)$ and (b) (8, 8, 1.3), respectively. The shaded area denotes the PS region, which was determined by the Maxwell construction. The dashed horizontal line in the middle panel is used for the Maxwell construction. The curves of $-{\chi }_c^{-1}$ were derived from the derivative of the $\mu -\delta$ curves (black curves) which were obtained by the seventh-order polynomial fit. The spinodal points ${\delta }_s$ are indicated as the arrows.