Collective modes in the paramagnetic phase of the Hubbard model

The charge dynamical response function of the Hubbard model is investigated on the square lattice in the thermodynamic limit. The obtained charge-excitation spectra consist of a continuum, a gapless collective mode with anisotropic zero-sound velocity, and a correlation-induced high-frequency mode at $\omega \approx U$. The correlation function is calculated from Gaussian fluctuations around the paramagnetic saddle point within the Kotliar and Ruckenstein slave-boson representation. Its dependence on the on-site Coulomb repulsion $U$ and density is studied in detail. An approximate analytical expression of the high-frequency mode, which holds for any lattice with one atom in the unit cell, is derived. Comparison with numerical simulations, perturbation theory, and the polarization potential theory is carried out. We also show that magnetic instabilities tend to vanish for $T\gtrsim t/6$, and finite-temperature phase diagrams are established.

Figure 1

Doping dependence of kinetic energy ${\varepsilon }_0$ and coupling scale $U_0$ in the metallic phase at $T=0$ and for $U=100t$ at $T=t/10$.

Figure 2

Saddle-point variable $x$ and renormalization factor $z_0^2$ for different values of coupling $U$ and doping $\delta$ at $T=0$ (solid line) and $T=t/10$ (dashed line).

Figure 3

Region of the phase diagram with a degeneracy of the saddle-point solution at different temperatures. The solid line with its critical end point indicates the metal-to-bad-metal first-order transition taking place when increasing the coupling.

Figure 4

Instability of the paramagnetic phase toward incommensurate magnetic ordering (top) or phase separation (bottom) at different temperatures. The shaded area corresponds to values of doping and coupling for which the static spin/charge susceptibility can take negative values.

Figure 5

Real part of the function $f^s\left(k\right)$ at temperature $T=t/100$ for different values of coupling and doping, plotted for momenta along the path linking $\mathrm{\Gamma }=(0,0)$, $X=(\pi ,0)$, and $M=(\pi ,\pi )$.

Figure 6

Frequency dependence of the charge susceptibility ${\chi }_c\left(k\right)$ and the Lindhard function ${\chi }_0\left(k\right)$ for $\mathbf{k}=(\frac{\pi }{2},\frac{\pi }{2})$ at doping $\delta =0.1$, coupling $U=4t$, and temperature $T=t/100$. With $z_0^2\approx 0.91$, the particle-hole continuum ends at ${\omega }_{\mathrm{cont}}\left(\mathbf{k}\right)\approx 5.14t$.

Figure 7

Imaginary part of the charge susceptibility for $U/t=0$, 1, 4, and 12 from top to bottom, plotted for momenta along the path linking $\mathrm{\Gamma }=(0,0)$, $X=(\pi ,0)$, and $M=(\pi ,\pi )$. Parameters: $T=t/100$ and $\delta =0.1$.

Figure 8

Imaginary part of the charge susceptibility for doping values $\delta =0.1$, 0.5, and 0.9 from top to bottom. Parameters: $T=t/100$ and $U/t=8$.

Figure 9

Imaginary part of ${\chi }_c\left(k\right)$ (solid line) and of ${\chi }_0\left(k\right)$ (dotted line) for small momentum $\mathbf{k}=0.001(cos\theta ,sin\theta )$ with angle $\theta =0$ (a), $\pi /10$ (b), and $\pi /4$ (c), at moderate coupling $U=2t$ and strong coupling $U=12t$. Parameters: $\delta =0.1$ and $T=0$.

Figure 10

Zero-sound velocity at zero temperature as a function of the coupling in the $X$ and $M$ directions, for doping $\delta =0.1$ (triangle), 0.5 (dot), and 0.9 (square).

Figure 11

Evolution of the zero-sound velocity around the metal-to-insulator transition, in the $X$ (symbols) and $M$ directions (solid lines).

Figure 12

Left: Coupling dependence of the UHB mode dispersion for momenta along the $\mathrm{\Gamma }-M$ path, at temperature $T=t/100$ and doping $\delta =0.1$; the dashed line shows the maximum energy of the response continuum, which is reached at $M$. Right: Imaginary part of ${\chi }_c\left(k\right)$ at coupling $U/t=1$, 3, and 5.

Figure 13

Dispersion of the UHB mode for momenta along the $\mathrm{\Gamma }-M$ path at temperature $T=t/100$, and doping $\delta =0.5$ and 0.9. The dashed line shows the upper edge of the response continuum at $M$.

Figure 14

Doping dependence of the coupling scale $U_0$ and the inverse-mass renormalization factor $z_0^2$ at temperature $T=0$ (dotted line), $t/10$ (solid line), $t/3$ (dashed line), and $t$ (dot-dashed line) for different values of on-site interaction $U$.

Figure 15

Comparison between the imaginary part of ${\chi }_c\left(k\right)$ (top) and ${\chi }_{\mathrm{RPA}}\left(k\right)$ (bottom) at moderate coupling $U=t$ and strong coupling $U=8t$. Parameter: $T=t/100$.