Doping-driven antiferromagnetic insulator-superconductor transition: A quantum Monte Carlo study

How superconductivity emerges in the vicinity of an antiferromagnetic insulating state is a long-standing issue of strong correlation physics. We study the transition from an antiferromagnetic insulator to a superconductor by hole doping based on a bilayer generalization of a Hubbard model. The projector quantum Monte Carlo simulations are employed, which are sign problem free both at and away from half filling. An anisotropic Ising antiferromagnetic Mott insulating phase occurs at half filling, which is weakened by hole doping. Below a critical doping value, antiferromagnetism coexists with the singlet superconductivity, which is a pairing across each rung with an extended $s$-wave symmetry. As doping further increases, the antiferromagnetic order vanishes, leaving only a superconducting phase. These results provide important information on how superconductivity appears upon doping the parent Mott-insulating state.

Figure 1

Generalized SZH model is defined on a bilayer square lattice. Its parameters include the intra- and interlayer hoppings of $t_{\parallel }$ and $t_{\perp }$, respectively, the on-site Hubbard interaction $U$, the interactions between two sites along each rung with $V$ in the charge channel, and $J_{\perp }$ and $J_z$ of the AF superexchanges along the $x\left(y\right)$ and $z$ directions, respectively.

Figure 2

QMC simulation results for the structure factors of (a) $\mathrm{A}{\mathrm{F}}_z$, (b) $\mathrm{A}{\mathrm{F}}_{x\left(y\right)}$, (c) SC, and (d) tPDW versus the doping $x$ as varying $L$. In the disordered phase for each order parameter, each structure factor shows very small size dependence when $L\gg \xi$ and converges to a value proportional to the square of correlation length ${\xi }^2$, while, in the ordered phase, it grows as $L\to \infty$.

Figure 3

Structure factors $F/L^2$ vs $1/L$ for both the $\mathrm{A}{\mathrm{F}}_z$ and SC orders. They are plotted at (a) $x=0$, (b) $x=\frac{1}{16}$, (c) $x=\frac{1}{8}$, and (d) $x=\frac{1}{4}$, respectively. The dashed lines are polynomial fittings to the QMC data from $L=6$ to $L=12$.

Figure 4

(a) Extrapolation of $F/L^2=\langle O^{†}O\rangle$ in the limit of $L\to \infty$ versus $x$, where $O$ represents operators for the $\mathrm{A}{\mathrm{F}}_z$, and SC order parameters. The $\mathrm{A}\mathrm{F}z$ ordering is suppressed beyond a critical doping $x_c\approx 0.11$ and the SC order coexists with the $\mathrm{A}{\mathrm{F}}_z$ one at small dopings $0<x<x_c$. (b) The single particle gap ${\mathrm{\Delta }}_{1p}$ at all doping levels and the spin gap ${\mathrm{\Delta }}_{A{F}_z}$ at $x>x_c$.

Figure 5

QMC simulations for the SU(2) symmetric model. The finite size scalings of the structure factors $F/L^2$ of the AF and SC orders, and the single-particle gap ${\mathrm{\Delta }}_{1p}$ at half filling (a) and the $1/16$ doping (b). The scales for ${\mathrm{\Delta }}_{1p}$ are along the axes on the right side of (a) and (b).

Figure 6

$\mathrm{\Delta }\tau$ dependence of the structure factors for various order parameters for $x=0$ (a) and $x=\frac{1}{4}$ (b) with $L=4$ (solid lines) and $L=6$ (dashed lines). $\mathrm{A}{\mathrm{F}}_z$ and $\mathrm{A}{\mathrm{F}}_{x\left(y\right)}$ represent the antiferromagnetic order along the $z$ direction and that along the $x$ or $y$ direction, respectively, and SC represents the superconducting order. The interacting parameter values are $U=2.5,V=t_{\perp }=0,J_{\perp }=1$, and $J_z=8$.

Figure 7

$\beta$ dependence of structure factors of various order parameters for $x=0$ (a) and $x=\frac{1}{4}$ (b) with $L=4$ (solid lines) and $L=6$ (dashed lines). The symbols and the interacting parameter values are the same as those presented in Fig. 6.

Figure 8

Spatial correlation functions at different doping levels with system size $L=4,6$, and 8. The correlation functions for the $\mathrm{A}{\mathrm{F}}_z$ and SC orders at $x=0$ are plotted in (a) and (b), respectively; those at $x=1/16$ are plotted in (c) and (d) ($x=1/18$ for the case of $L=6$), respectively; those at $x=1/4$ are plotted in (e) and (f), respectively. The squares of the order parameters obtained by the finite size scaling on the structure factors in the main text are plotted with dashed lines for comparison.

Figure 9

Imaginary time Green's functions. $G\left(\tau \right)$ is plotted at different dopings. Due to the particle-hole symmetry at half filling, the relation of $G\left(\tau \right)=-G(-\tau )$ is satisfied, while this symmetry is not held away from half filling. We employ $[-G(\tau \left)G\right(-\tau \left)\right]$ to extract the mean single particle gap, plotted in (b), which shows very weak size dependence.