Effects of interaction strength, doping, and frustration on the antiferromagnetic phase of the two-dimensional Hubbard model

Recent quantum-gas microscopy of ultracold atoms and scanning tunneling microscopy of the cuprates reveal new detailed information about doped Mott antiferromagnets, which can be compared with calculations. Using cellular dynamical mean-field theory, we map out the antiferromagnetic (AF) phase of the two-dimensional Hubbard model as a function of interaction strength $U$, hole doping $\delta$, and temperature $T$. The Néel phase boundary is nonmonotonic as a function of $U$ and $\delta$. Frustration induced by second-neighbor hopping reduces Néel order more effectively at small $U$. The doped AF is stabilized at large $U$ by kinetic energy and at small $U$ by potential energy. The transition between the AF insulator and the doped metallic AF is continuous. At large $U$, we find in-gap states similar to those observed in scanning tunneling microscopy. We predict that, contrary to the Hubbard bands, these states are only slightly spin polarized.

Figure 1

(a) AF phase of the 2D Hubbard model in the $U\text{−}T\text{−}\delta$ space, for $t^{\prime }=-0.1$. (b) $T\text{−}U$ cut at $\delta =0$. (c) $T\text{−}\delta$ cut for $U=5$ and (d) $U=12$. (e) $\delta \text{−}U$ cut for $T=1/10$. The magnitude of $m_z$ is color coded [see SM [35] for $m_z\left(U\right)$ and $m_z\left(\delta \right)$ curves at different temperatures]. The paramagnetic to AF phase boundary is drawn where $m_z<0.045$. The red cross in (b) and (e) indicates the position of the Mott critical endpoint $(U_{\mathrm{MIT}},T_{\mathrm{MIT}})$ in the underlying normal phase.

Figure 2

(a) $T_N^d$ for $t^{\prime }=0,-0.1,-0.3,-0.5$. As a reference, we show with solid symbols the $T=0$ Hartree-Fock results of Ref. [47] for the critical $U$ of the AF onset. Crosses indicate the Mott endpoints in the underlying normal phase: Data are for $t^{\prime }=0$ (black), $t^{\prime }=-0.1$ (red), and $t^{\prime }=-0.5$ (green). (b) $T_N^d$ vs $t^{\prime }$ at $\delta =0$ for $U=5,12$. (c) $T_N^d$ vs $\delta$ at $U=16$ for $t^{\prime }=-0.1,-0.5$.

Figure 3

Local DOS (a), (e) along with spin projections (b), (f) for different doping levels as obtained from analytically continued [55] data. Vertical bars indicate the position of the Bogoliubov peaks. Occupation vs chemical potential $n\left(\mu \right)=1-\delta \left(\mu \right)$ for different temperatures (c), (g). Horizontal ticks mark the values of dopings that correspond to the values shown in (a), (e). Doping dependence of the difference in potential, kinetic, and total energies between the AF and the underlying normal phase (d), (h). Data are for $t^{\prime }=-0.1$ and $U=5$ (top panels) and $U=12$ (bottom panels). For more data on $N\left(\omega \right)$, see Figs. 4 and 5 in SM [35].