Local theory for Mott-Anderson localization

The paramagnetic metallic phase of the Anderson-Hubbard model (AHM) is investigated using a nonperturbative local moment approach within the framework of dynamical mean-field theory with a typical medium. Our focus is on the breakdown of the metallic phase near the metal-insulators transition as seen in the single-particle spectra, scattering rates, and the associated distribution of Kondo scales. We demonstrate the emergence of a universal, underlying low-energy scale, $T_K^{\mathrm{peak}}$. This lies close to the peak of the distribution of Kondo scales obtained within the metallic phase of the paramagnetic AHM. Spectral dynamics for energies $\omega \lesssim T_K^{\mathrm{peak}}$ display Fermi liquid universality crossing over to an incoherent universal dynamics for $\omega \gg T_K^{\mathrm{peak}}$ in the scaling regime. Such universal dynamics indicate that within a local theory the low to moderately low-energy physics is governed by an effective, disorder renormalized Kondo screening.

Figure 1

Distribution of Kondo scales. In the main panel, the evolution of the $T_K$ distribution as a function of $W$ for $U=1.2$ is shown on a linear-log scale. The distributions are peaked and (sharply) bounded from below by $T_K^{\mathrm{peak}}$, the scale associated with the respective particle-hole symmetric limit of the effective impurity problem embedded in the typical medium. The shaded region highlights the narrow range of $T_K$'s spanned by small $W$ in contrast to the higher $T_K$ long tails spanned by larger $W$'s. (inset) $T_K^{\mathrm{peak}}$ is plotted as a function of $W$ for $U=1.2,1.8,2.7$. While, $U=1.2,1.8$ correspond to $U<U_{c2}, U=2.7>U_{c2}$. Hence, in the $W=0$ limit, $U=1.2$ and 1.8 correspond to Fermi liquids with $T_K^0=T_K^{\mathrm{peak}}\approx 0.025$ and 0.007, respectively. On the other hand, $U=2.7$ corresponds to a Mott insulator with $T_K^0=T_K^{\mathrm{peak}}=0$.

Figure 2

Distribution of the renormalized site energy, ${\epsilon }_i^{*}={\epsilon }_i+\mathrm{Re}{\mathrm{\Sigma }}_i\left(0\right)$, with ${\epsilon }_i=V_i-U/2$, plotted for $U=1.2$ and $W=1.8,2.1,2.575,\text{and}2.625$. A pronounced weight is observed around ${\epsilon }_i^{*}=0$ that represents the particle-hole symmetric limit. The initially broad peak becomes narrower and grows in intensity as $W$ is increased. Such an evolution of ${\epsilon }_i^{*}$ as a function of increasing $W$ indicates that a majority of sites tend to attain a $T_K$ close to that of the particle-hole symmetric limit. This can be correlated with the skewed nature of $P\left(T_K\right)$ in Fig. 1 as $W$ is increased. Note that the entire range is not shown.

Figure 3

A schematic of the surface formed by the $T_K^{\mathrm{peak}}$ as a function of $U$ and $W$.

Figure 4

Universal scaling of $-\mathrm{Im}{\mathrm{\Sigma }}_{\mathrm{ave}}\left(\omega \right)$. (Left) The $-\mathrm{Im}{\mathrm{\Sigma }}_{\mathrm{ave}}\left(\omega \right)$ with the static part $[a_0=-\mathrm{Im}{\mathrm{\Sigma }}_{\mathrm{ave}}\left(0\right)]$ subtracted is plotted for $U=1.2$, on an energy scale, ${\omega }^{\prime }$, with the bare frequency, $\omega$ rescaled by $T_K^{\mathrm{peak}}$ that has been obtained from the respective $P\left(T_K\right)$ plots. (Left inset) The same sets of data are plotted on a bare energy scale, $\omega$. (Right) The $-\mathrm{Im}{\mathrm{\Sigma }}_{\mathrm{ave}}\left(\omega \right)$ with the static part $[a_0=-\mathrm{Im}{\mathrm{\Sigma }}_{\mathrm{ave}}\left(0\right)]$ subtracted is plotted for $U=2.7$ on the rescaled energy scale, ${\omega }^{\prime }$. (Right inset) The same sets of data are plotted on a bare energy scale.

Figure 5

A schematic demonstrating the self-energy scattering dynamics corresponding to different energy regimes as a function of disorder, $W$ within the metallic phase. It should be noted that the quantity $\xi$ separating the “incoherent universal” and the “incoherent nonuniversal” regimes is just symptomatic of the proximity to the metal-insulator transition. In strong coupling as $W\to W_c$, where $W_c$ is the critical disorder strength for the metal-insulator transition when approached from the metallic side, $\xi \to \infty$ in the scaling regime.

Figure 6

Evolution of the density of states. The arithmetic mean of the density of states (DoS) also known as the average DoS (ADoS) (solid red line, green shaded region) and the geometric mean of the DoS also known as the typical DoS (TDoS) (solid black line, turquoise shaded region) at various $W$ for (a) $U=1.2$ and (b) 2.7. Similar to the $U=0$ scenario, when the disorder $W$ is small, both the ADoS and the TDoS produce almost the same density of states and as $W$ increases the TDoS gets suppressed over all energy scales.

Figure 7

Comparison of the band center ($\omega =0$) value of the TDoS and the ADoS as a function of $W$.

Figure 8

A qualitative phase diagram of the Anderson-Hubbard model, within the TMT-DMFT framework. We approach from the metallic side and identify the phase boundary based on the behavior of the peak $T_K^{\mathrm{peak}}$ of the distribution of Kondo scales and/or the band center value of the typical density of states, ${\rho }_{\mathrm{typ}}\left(0\right)$. For example, both ${\rho }_{\mathrm{typ}}\left(0\right)\to 0$ and $T_K^{\mathrm{peak}}\to 0$ as the Mott-Anderson phase boundary is approached from the metallic side. On the other hand, only $T_K^{\mathrm{peak}}\to 0$ as the Mott phase boundary is approached from the metallic side.

Figure 9

Self-energy within the LMA for the single impurity Anderson model. Diagrammatic representation of the dynamic self-energy, $\mathrm{\Sigma }\left(\omega \right)$ retained within the LMA in practice. The diagrams are expressed in terms of the polarization bubble, ${\mathrm{\Pi }}^{\overline{\sigma }\sigma }\left(\omega \right)$. Wavy line: interaction $U$, double line: renormalized host/medium propagator, and hatched region: transverse spin polarization propagator bubble. See Eq. (A5) and the associated text.