Local moment approach as a quantum impurity solver for the Hubbard model

The local moment approach (LMA) has presented itself as a powerful semianalytical quantum impurity solver (QIS) in the context of the dynamical mean-field theory (DMFT) for the periodic Anderson model and it correctly captures the low-energy Kondo scale for the single impurity model, having excellent agreement with the Bethe ansatz and numerical renormalization group (NRG) results. However, the most common correlated lattice model, the Hubbard model, has not been explored well within the LMA+DMFT framework beyond the insulating phase. Here in our work, within the framework we complete the filling-interaction phase diagram of the single band Hubbard model at zero temperature. Our formalism is generic to any particle filling and can be extended to finite temperature. We contrast our results with another QIS, namely the iterated perturbation theory (IPT) and show that the second spectral moment sum rule improves better as the Hubbard interaction strength grows stronger in LMA, whereas it severely breaks down after the Mott transition in IPT. For the metallic case, the Fermi liquid (FL) scaling agreement with the NRG spectral density supports the fact that the FL scale emerges from the inherent Kondo physics of the impurity model. We also show that, in the metallic phase, the FL scaling of the spectral density leads to universality which extends to infinite frequency range at infinite correlation strength (strong coupling). At large interaction strength, the off half-filling spectral density forms a pseudogap near the Fermi level and filling-controlled Mott transition occurs as one approaches the half-filling. As a response property, we finally study the zero temperature optical conductivity and find universal features such as absorption peak position governed by the FL scale and a doping independent crossing point, often dubbed the isosbestic point in experiments.

Figure 1

Spectral densities and their scaling collapse. (a) Abrikosov-Suhl resonance appears at the Fermi level $\left(\omega =0\right)$ and the resonance width decreases with increasing $x$ or $U$. (Inset) Full spectra for the same. (b) Scaling collapse of spectral densities when the frequency axis is scaled by the low-energy scale ${\omega }_L=Z{t}_{*}$. Note that the collapse deviates from the noninteracting curve $\left(U=0\right)$ almost immediately away from the Fermi level.

Figure 2

Reason for noncollapse with the noninteracting DoS. Dashed lines (blue in color) and full lines (red in color) are the imaginary and real part of the self-energy, respectively. The static part $\mathrm{\Sigma }\left(0\right)=U/2$ has been subtracted from the real part. $\text{Im}\mathrm{\Sigma }\left(\omega \right)$ grows far more rapidly from $\omega =0$ in LMA (main panel) than that in IPT (inset) suggesting that not only in the strong coupling regime, but also in the intermediate correlation regime, incoherent scattering effects become important at energies even slightly away from the Fermi level. The interaction strength for the LMA, $U=1.13t_{*}(x=0.2t_{*})$; for the IPT, $U=3.0t_{*}$.

Figure 3

(Bottom inset) The peak position of $\text{Im}{\mathrm{\Pi }}_{\sigma -\sigma }\left(\omega \right)$ gives rise to an energy scale ${\omega }_m$ for various $U$'s. (Main panel) The position becomes universal when the frequency is scaled by ${\omega }_L=Z{t}_{*}$ signifying a proportional relation between $Z$ and ${\omega }_m$. (Top inset) $Z$ vs ${\omega }_m$ plot is a nice straight line with slope 1.59.

Figure 4

Mott transition reflected from the spectral density evolution as $U$ changes at zero temperature (for the hypercubic lattice). (Main panel) Quasiparticle resonance shrinks as $U$ is increased. At $U=3.56t_{*}$, a clear gap opens up at the Fermi level signaling the Mott metal-to-insulator transition (MIT). The gap gets enhanced as $U$ is increased further. (Right panel) Starting from a Mott insulator if $U$ is decreased, the gap at the Fermi level decreases and eventually closes at $U<2.38t_{*}$. (Left panel) A finite gap still persists at $U=2.38t_{*}$. Therefore an extrapolation method is required to find the critical value $U_{c1}$ where insulator-to-metal transition (IMT) happens (see Fig. 5).

Figure 5

Decay of the low-energy scale ${\omega }_L=Z{t}_{*}$ with increasing $U/t_{*}$ (solid line). Inset shows the same in log scale. ${\omega }_L$ seems to vanish at $U_{c2}=2.64t_{*}$. Note that we have been able to reach a value of the low-energy scale $\sim {10}^{-4}{t}_{*}$, which requires very high precision calculations. The dashed line shows that the spectral gap ${\mathrm{\Delta }}_g$ of the Mott insulator decreases linearly with decreasing $U/t_{*}$ and closes at $U_{c1}=2.36t_{*}$.

Figure 6

Second spectral moment $M_2$ in (a) IPT and (b) LMA for various interaction strengths. The insets show the respective absolute values of relative error.

Figure 7

Scaling universality at strong correlation strength: As $U\to U_{c2}^{-}$, the collapse of the spectral density extends all the way after the frequency being scaled by ${\omega }_L$, despite being limited to the low-energy scale regime, which could be order of ${\omega }_L$. For example, the scaling agreement between spectral densities at $U=2.51t_{*}$ and $U=2.60t_{*}$ runs up to $\sim 30{\omega }_L$. The inset shows the same spectral densities without the scaling.

Figure 8

Evolution of spectral densities for $x=0.5t_{*}$ $\left(U\simeq 1.5t_{*}\right)$ at $n\ne 1$ (away from half-filling), for the hypercubic lattice. Parent (half-filled, i.e., $n=1)$ phase is metallic. (a) Evolution from $n\le 1$ to $n\to 0$. (b) Evolution from $n\ge 1$ to $n\to 2$.

Figure 9

(a) Phase diagram on the occupancy-interaction $\left(n\text{−}U\right)$ plane. The region is bounded along the filling axis by empty orbital and doubly occupied band insulator lines. The metal emerging by doping a Mott insulator side is known as the filling controlled (FC) metal $\left(U>U_{c2}\right)$ and remaining region is the bandwidth controlled (BC) metal since interaction is low. In case of LMA, $U_{c2}=2.64t_{*}$. (b) Disappearance of quasiparticle residue $Z$ as occupancy approaches the half-filling value $n=1$ in IPT (inset, $U=5.0t_{*})$ and LMA (main, $U=2.9t_{*})$.

Figure 10

(Main) Spectral densities at various $U/t_{*}$'s at asymmetry parameter $\eta =0.3$. A pseudogap (a gap close to the Fermi level) forms in the spectral density at $U=2.9t_{*}$. (Inset) Scaling universality at strong interaction values at $\eta =0.25$: $U/t_{*}=2.89$, 2.92, 2.96, and 3.00. The universal region extends to very large values of $\omega /{\omega }_L$ and the universal scaling form is seen to be very different from the renormalized noninteracting Gaussian form.

Figure 11

Scaling behavior in the optical conductivity $\sigma \left(\omega \right)$ for the asymmetry parameter $\eta =0.3$ and different interaction strengths. The first absorption peaks arise at $\omega ={\omega }_L$ (inset), as revealed from the scaling of the frequency axis by ${\omega }_L$, leaving all peaks appearing at $\omega /{\omega }_L=1$ (main).

Figure 12

(Main) LMA results for optical conductivity with various hole concentrations: $\delta =1-n$. (Inset) Optical conductivity for ${\mathrm{Sm}}_{1-x}{\mathrm{Ca}}_x{\mathrm{TiO}}_3$ where $\tilde{W}$ is the bandwidth obtained from a tight-binding calculation, mentioned in Ref. [56], normalized to that of ${\mathrm{LaTiO}}_3$.

Figure 13

(a) Scaling agreement with the NRG spectral densities. Main panel shows that at half-filling, spectral density from LMA (pink line) agrees with that from NRG (red dash) up to $\omega \sim 2{\omega }_L$, while IPT spectral density (green dotted dash) only agrees in the vicinity of the Fermi level. Inset shows that the agreement extends further in the p-h asymmetric case and ends near the onset of the Hubbard bands. (b) (Main) Mott gap $\left({\mathrm{\Delta }}_g\right)$ closes as the interaction strength $U$ decreases leading to an insulator-to-metal transition at $U_{c1}$. $U_{c1}$ is higher in DDMRG (Karski et al.), SCPT (Gebhard et al.), and IPT, while VLMA (Kauch and Byczuk) and LMA show $U_{c1}\simeq 1.45t_{*}$ and $U_{c1}\simeq 1.6t_{*}$, respectively. (Inset) Spectral densities as functions of frequency for various methods at $U=4t_{*}$ showing Mott gaps at the Fermi level. The gaps in VLMA and LMA appear larger compared to that in DDMRG, SCPT, and IPT although the position of the Hubbard bands remain more or less the same.

Figure 14

(Main) Spectral densities against frequency for NRG, VLMA, and SEA-LMA at $U=2t_{*}$. Although the weight of Hubbard bands differ, both VLMA and SEA-LMA show resemblance with NRG in low-energy behavior (width of quasiparticle resonance) at $U=2t_{*}$. (Inset) The same at $U=1t_{*}$ for Luttinger LMA (L-LMA), VLMA by golden section search (G), VLMA by Kauch and Byczuk (K-B), and SEA-LMA. All show close resemblances, however, L-LMA misses the Luttinger pinning at the Fermi level.