Antiferromagnetic to superconducting phase transition in the hole- and electron-doped Hubbard model at zero temperature

The competition between $d$-wave superconductivity (SC) and antiferromagnetism (AF) in the high-$T_c$ cuprates is investigated by studying the hole- and electron-doped two-dimensional Hubbard model with a recently proposed variational quantum-cluster theory. The approach is shown to provide a thermodynamically consistent determination of the particle number, provided that an overall shift of the on-site energies is treated as a variational parameter. The consequences for the single-particle excitation spectra and for the phase diagram are explored. By comparing the single-particle spectra with quantum Monte Carlo and experimental data, we verify that the low-energy excitations in a strongly correlated electronic system are described appropriately. The cluster calculations also reproduce the overall ground-state phase diagram of the high-temperature superconductors. In particular, they include salient features such as the enhanced robustness of the antiferromagnetic state as a function of electron doping and the tendency towards phase separation into a mixed antiferromagnetic-superconducting phase at low doping and a pure superconducting phase at high (both hole and electron) doping.

Figure 1

Filling $n=⟨n_{i\sigma }⟩$ as a function of chemical potential $\mu$ obtained by integration of the spectral density [Eq. (7), solid lines] and via the derivative of $\Omega$ [Eq. (6), dashed lines]. Results are obtained, respectively, by considering ${\epsilon }^{\prime }$ as a variational parameter (a) and by setting ${\epsilon }^{\prime }=0$ (b). Calculations for the electron-doped case with $U=8$, nearest-neighbor hopping $t_{nn}=-1$ and next-nearest-neighbor hopping $t_{nnn}=0.3$.

Figure 2

Taken from Ref. 13. The dispersion of the peaks in the single-particle spectral weight from (a) QMC simulations of the Hubbard model at the temperatures indicated and at dopings $x=0.01$, $x=0.05$ [peaks in $A(\mathit{k},\omega )$ represented by error bars], and $x=0.13$ (solid circle). (b) ARPES experiments from underdoped and optimally doped materials [peak centroids in $A(\mathit{k},\omega )$], after Ref. 30.

Figure 3

Single-particle spectrum for the hole-doped ($x=0.01$ and $x=0.05$) case at $U=8$ $(t_{nn}=-1)$. For comparison with the QMC data, the next-nearest-neighbor hopping is set to $t_{nnn}=0$.

Figure 4

Single-particle spectrum for $U=8$ and $t_{nnn}=0.3$ $(t_{nn}=-1)$ in the hole-doped (a) and electron-doped (b) cases. Results are shown for dopings in the mixed $\mathrm{AF}+\mathrm{SC}$ state (see Sec. 4)—i.e., for $x=0.015$ (a) and $x=0.09$ (b), respectively.

Figure 5

(Color online) Evolution of the low-energy spectrum upon hole doping. Parameters are as in Fig. 4. The weight is obtained by integrating the low-energy $A(\mathit{k},\omega )$ spectrum down to 0.2 below the Fermi energy.

Figure 6

The same as in Fig. 5 but for the electron-doped case.

Figure 7

(Color online) Antiferromagnetic and superconducting order parameters $m$ and $\Delta$ and chemical potential $\mu$ as functions of hole doping $x$. $\Delta$ and $\mu$ are plotted for the $\mathrm{AF}+\mathrm{SC}$ (green, ${\Delta }_{\mathrm{AF}+\mathrm{SC}}$, ${\mu }_{\mathrm{AF}+\mathrm{SC}}$) as well as for the pure SC homogeneous solutions (blue, ${\Delta }_{\mathrm{SC}}$, ${\mu }_{\mathrm{SC}}$). Note that $\Delta$ is scaled by a factor of 5 for convenience. For $x<x_1$ the system exhibits a coexistence of AF and $d$-wave SC order. Phase separation occurs between the doping levels $x_1$ and $x_2$. For $x>x_2$ pure $d$-wave SC is realized. In the phase separation region $x_1<x<x_2$, the homogeneous solution becomes unstable and the system prefers to separate into a mixture of two densities corresponding to $x_1$ and $x_2$. The chemical potential ${\mu }_c$ is determined by the Maxwell construction shown in the upper figure. At ${\mu }^{*}$ the slope of the $\mathrm{AF}+\mathrm{SC}$ solution changes sign.

Figure 8

(Color online) Same as Fig. 7 but for electron doping. Note the enhanced robustness of the AF state and the strongly reduced scale $\Delta \mu \equiv ({\mu }^{*}-{\mu }_c)$ as compared to hole doping.

Figure 9

$\Omega$ vs $h_{\mathrm{AF}}^{\prime }$ (with the two other parameters $h_{\mathrm{SC}}^{\prime }$ and ${\epsilon }^{\prime }$ fixed at their stationary-point values) for different $\mu$ in the hole-doped case. Parameters: $U=8$ and $t_{nnn}=0.3$ $(t_{nn}=-1)$.