Generic susceptibilities of the half-filled Hubbard model in infinite dimensions

Around a metal-to-insulator transition driven by repulsive interaction (Mott transition) the single-particle excitations and the collective excitations are equally important. Here we present results for the generic susceptibilities at zero temperature in the half-filled Hubbard model in infinite dimensions. Profiting from the high resolution of dynamic density-matrix renormalization at all energies, results for the charge, spin, and Cooper-pair susceptibilities in the metallic and the insulating phase are computed. In the insulating phase, an almost saturated local magnetic moment appears. In the metallic phase a pronounced low-energy peak is found in the spin response. It is the precursor of the magnetic moment in the insulator.

Figure 1

(Color online) Phase diagram of the Mott transition of a half-filled Hubbard model in infinite dimensions with semielliptic density of states at zero temperature as function of the interaction $U$. The critical value below which the insulator ceases to exist is $U_{c1}=2.38\pm 0.02D$. The critical value above which the metal ceases to exist is $U_{c2}=3.07\pm 0.1D$ (Refs. 13, 14). But between $U_{c1}$ and $U_{c2}$ the metal has the lower energy so that the insulator is only metastable (Ref. 13).

Figure 2

(Color online) Positive imaginary part ${\chi }_{+}^{\text{charge}}$ of the local charge susceptibility in the insulating phase as function of frequency for various values of the interaction $U$. The arrow points in the direction of increasing interaction.

Figure 3

(Color online) Positive imaginary part ${\chi }_{+}^{\text{charge}}$ of the local charge susceptibility as function of frequency for various values of the interaction $U$ in the metallic phase. The arrow points in the direction of increasing interaction. Note the different scale of the response compared to the response in the insulator in Fig. 2.

Figure 4

(Color online) Positive imaginary part ${\chi }_{+}^{\text{charge}}$ of the local charge susceptibility as function of frequency for interaction $U=2.8D$ in the metallic and insulating phases.

Figure 5

(Color online) Positive imaginary part ${\chi }_{+}^{\text{charge}}$ of the local charge susceptibility as function of frequency for values of the interaction $U$ in the metallic phase, but close to the transition to the insulator. The main contribution is extrapolated as if there were no additional shoulders (thin solid lines). Then the shaded area is attributed to spectral weight of the shoulder.

Figure 6

(Color online) Spectral weight $W$ attributed to the shoulder in the metallic charge response as function of the quasiparticle weight $Z$. Solid line is a quadratic fit with $c=0.758$.

Figure 7

(Color online) Positive imaginary part ${\chi }_{+}^{\text{spin}}$ of the local spin susceptibility in the insulating phase as function of frequency for various values of the interaction $U$. The arrow points in the direction of increasing interaction. The Lorentzian at zero frequency represents the example at $U=4D$ for the broadened $\delta$ peak occurring in the insulator.

Figure 8

(Color online) The total weight (circles) and four times the square of the local $S_z$ component (squares) of the spin response ${\chi }_{+}^{\text{spin}}\left(\omega \right)$ in the insulating phase (filled symbols) and in the metallic phase (open symbols). In the insulator the local $S_z$ component is taken from the weight $A$ of the $\delta$ peak as in Eq. (18). In the metal it is taken to be the weight of the dominant low-energy peak (see Figs. 9, 10). Dark lines interpolate the data in the insulator while the light lines interpolate and extrapolate the data in the metal.

Figure 9

(Color online) Positive imaginary part ${\chi }_{+}^{\text{spin}}$ of the local spin susceptibility in the metallic phase as function of frequency for various small values of the interaction $U$. The arrow points in the direction of increasing interaction. Note the quickly increasing peak at low frequencies and the quickly vanishing spectral weight at higher frequencies.

Figure 10

(Color online) Positive imaginary part ${\chi }_{+}^{\text{spin}}$ of the local spin susceptibility in the metallic phase as function of frequency for various values of the interaction $U$ close to the transition to the insulator. The arrow points in the direction of increasing interaction. The peak at low frequencies prevails completely; note the scale on the $y$ axis.

Figure 11

(Color online) Static susceptibility ${\chi }^{\text{spin}}\left(0\right)$ from our dynamic data via Eq. (19) (black crosses) compared to data obtained by exact diagonalization at very small temperature (red circles), adapted from Fig. 44 in Ref. 8. The black cross at $U=0$ corresponds to the analytic result ${\chi }^{\text{spin}}\left(0\right)=16/\left(3\pi D\right)$. Lines are guides to the eyes only.

Figure 12

(Color online) Positive imaginary part ${\chi }_{+}^{\text{spin}}$ of the local spin susceptibility as function of frequency for interaction $U=2.8D$ in the metallic and insulating phases.