Magnetization, $d$-wave superconductivity, and non-Fermi-liquid behavior in a crossover from dispersive to flat bands

We explore the effect of inhomogeneity on electronic properties of the two-dimensional Hubbard model on a square lattice using dynamical mean-field theory (DMFT). The inhomogeneity is introduced via modulated lattice hopping such that in the extreme inhomogeneous limit the resulting geometry is a Lieb lattice, which exhibits a flat-band dispersion. The crossover can be observed in the uniform sublattice magnetization which is zero in the homogeneous case and increases with the inhomogeneity. Studying the spatially resolved frequency-dependent local self-energy, we find a crossover from Fermi-liquid to non-Fermi-liquid behavior happening at a moderate value of the inhomogeneity. This emergence of a non-Fermi liquid is concomitant of a quasiflat band. For finite doping the system with small inhomogeneity displays $d$-wave superconductivity coexisting with incommensurate spin-density order, inferred from the presence of oscillatory DMFT solutions. The $d$-wave superconductivity gets suppressed for moderate to large inhomogeneity for any finite doping while the incommensurate spin-density order still exists.

Figure 1

Upper panel: schematic representation of the inhomogeneity introduced by modulated hopping. The solid lines represent the hopping amplitude $(1+\alpha )t$, while the dashed lines represent $(1-\alpha )t$. The square drawn by the solid line is the smallest possible unit cell that captures magnetic and superconducting order parameters emerging at finite Hubbard interaction. In the limit $\alpha =1$, the square lattice with modulated hopping turns into the Lieb lattice. Lower panel: the noninteracting density of states (DOS) of the inhomogeneous lattice as a function of the normalized energy parameter $\tilde{\omega }$ (see text) for different choices of $\alpha$. The density of states evolves from square-lattice behavior to Lieb-lattice one. In the inset: DOS for the pure Lieb lattice.

Figure 2

Magnetic order parameter $m$ for $A$ site (upper panel), $B$ and $C$ sites (middle panel), and $D$ site (lower panel) for varying $\tilde{U}$ and different inhomogeneity $\alpha$ at zero temperature. The Hubbard interaction $U$ has been scaled by $t_{+}=(1+\alpha )t$.

Figure 3

Upper panel: phase diagram of the inhomogeneous Hubbard model showing staggered magnetization $m_s$ for different inhomogeneity parameter $\alpha$ and interaction $U$. Lower panel: uniform magnetization for the set of $\alpha$ and $U$. Here, $m_{\text{F}}$ is maximal for $\alpha \to 1$ and $U\to 0$.

Figure 4

Upper panel: uniform magnetization $m_{\text{F}}$ for varying $\tilde{U}$ and different $\alpha$. Lower panel: $m_{\text{F}}$ for varying $\alpha$ for different $U$.

Figure 5

Upper panel: double occupancy of $B/C$ sites for varying interaction $\tilde{U}$ and different $\alpha$ at $\beta =1/T=20$. Lower panel: double occupancy of $A$ sites.

Figure 6

Upper panel: imaginary part of the local self-energy, i.e., $-\text{Im}\mathrm{\Sigma }\left(i{\omega }_n\right)$ vs Matsubara frequency ${\omega }_n$, for sites $B$ and $C$, for different $\alpha ,T=0.05$ and $U=0.75$. For these parameters the system is in the nonmagnetic metallic regime [68]. Lower panel: $-\text{Im}\mathrm{\Sigma }\left(i{\omega }_n\right)$ vs Matsubara frequency ${\omega }_n$, for site $A$, for the same parameter as upper panel.

Figure 7

In the main panel: $\text{Im}\mathrm{\Sigma }\left(i{\omega }_{n=0}\right)$ vs $T$ at different sites for $\alpha =0.95$ and $U=2$. A similar plot is shown for $\alpha =0.4$ in the inset. For these parameters the system is in the nonmagnetic metallic regime [68].

Figure 8

Upper panel: average dSC order parameter for set of inhomogeneity $\alpha$ and chemical potential $\mu$ for the two-dimensional Hubbard model on the inhomogeneous square lattice with modulated hopping at $U=6.0$. The circles are the data points where we have carried out the plaquette $\text{DMFT}+\text{ED}$ calculations. The dashed lines are guides to the eye separating different regions. The lines are only qualitative and do not actually correspond to a phase boundary. For solid circles, we obtain a converged DMFT solution while open white circles represent the data set for which DMFT solutions are finite and oscillatory. The color code assigned to the solid circles represents the magnitude of the dSC order parameter. Lower panel: averaged magnetic order as a function of inhomogeneity $\alpha$ and chemical potential $\mu$.

Figure 9

Upper panel: the order parameters $m_{\text{avg}},n_{\text{avg}}$, and ${\mathrm{\Delta }}_{\text{avg}}$ for varying DMFT iterations for $\alpha =0.05$. All quantities show oscillatory behavior. Lower panel: same order parameters for varying DMFT iterations for $\alpha =0.5$. Here, $m_{\text{avg}}$ and $n_{\text{avg}}$ oscillate with the DMFT iteration, while ${\mathrm{\Delta }}_{\text{avg}}$ converges to zero.

Figure 10

Main panel: uniform dSC order parameter for converged DMFT solutions ${\mathrm{\Delta }}_{\text{avg}}$ vs doping $x$ for different values of $\alpha$ and $U=6.0$ at $T=0$. Uniform dSC order monotonically decreases with increasing $x$. Magnitude of dSC is smaller for larger inhomogeneity at given $x$ and is zero for $\alpha \ge 0.25$. In the inset: uniform dSC order parameter for varying $\alpha$ for given $x=0.135$.