Hybridizing localized and itinerant electrons: A recipe for pseudogaps

In a system where selective Mott localization is realized, some electrons show a gap to charge excitations while others do not. A hybridization between these two kind of electrons will lead to a smoothening of this sharp difference and can even bring the system back to a complete delocalization. We show here that there is a large region of parameters at finite hybridization where the selective localization persists and the system shows a partial filling of the selective gap with incoherent states, giving rise to a pseudogap. This result is illustrated here in a two-orbital Hubbard model with Hund's coupling, but is based on quite general assumptions and should hold for a larger class of systems, and possibly be a paradigm for the pseudogap mechanism in cuprates.

Figure 1

Electronic compressibility of electrons in the localized orbital ($U/D=1.4$, $D_N/D_W=0.5$, $J=U/4$), calculated by numerical differentiation of $n\left(\mu \right)$ of Fig. 7. The incompressible fluid found in absence of hybridization ($V=0$) due to a clean gap in the orbitally resolved spectrum is replaced at nonzero $V$ by a phase in which the compressibility is reduced but nonvanishing. Excitations filling the gap are incoherent, and the pseudogap remains clearly delineated.

Figure 2

$V\text{−}U$ phase diagram of the half-filled two-band Hubbard model with $D_N/D_W=0.5$ and $J=U/4$. The solid lines correspond to the metal-insulator boundaries. For weak coupling and hybridization, the system is metallic. For high $V$ and $U$, the system is band insulating with a low-spin $S=0$, and a Mott insulator with a high-spin $S=1$, respectively. For intermediate coupling and weak-to-intermediate hybridization, there is an orbital-selective phase, where only the narrow band localizes, even at finite hybridization. The color scale represents the interorbital spin-spin correlation function $C_{WN}$, which is negative for the singlet state, and positive for the triplet. The metallic phase has a crossover from the limit in which the screened local moment is very small (labeled LS metal) to the one in which it is very close to the $S=1$ value favored by Hund's coupling (HS metal). The dashed line indicates the corresponding low-to-high-spin transition that occurs in the atomic limit, which is a good estimator of the crossover region.

Figure 3

Imaginary part of the Green's functions for $V=0.15$, for different values of the interactions. Notice that for $U/D=1.5$ the system is in an orbital-selective Mott phase.

Figure 4

Quasiparticle residue as a function of $U$. Solid (dashed) lines correspond to the wide (narrow) orbital.

Figure 5

Spin-spin correlation functions.

Figure 6

Wide orbital self-energy in the metallic ($U/D=0.7$) and selective ($U/D=1.5$) phases (for $V/D=0.2$). Red dashed line shows the fit with the non-Fermi-liquid $\Sigma$ of Eq. (6), and the green dashed one corresponds to a polynomial fit. The inset shows that below the critical $U$ of the metallic-OSP transition, the polynomial fit, corresponding to the Fermi-liquid solution, is better than the logarithmic one, and the opposite behavior starts at the transition, suggesting that the metal is in a non-Fermi-liquid state.

Figure 7

Charge density as a function of the chemical potential for $U/D=1.4$. Circles (squares) correspond to $V=0$ ($V=0.1$). The solutions for $\mu <0$ were obtained by a particle-hole symmetry.

Figure 8

Imaginary part of the Green's functions for $U/D=1.5$ and $V/D=0.2$, for different values of the fictitious temperature ${\beta }^{-1}$.

Figure 9

Possible zero-temperature phases in cases where coherence is restored at $T=0$. Solid lines correspond to the hybridized bands, while dotted lines correspond to the case $V=0$. As interactions increase, the bands get narrower and the effective hybridization decreases. When interactions are high enough, depending on the value of ${\Sigma }_{NW}\left(0\right)$, either (i) a Mott-like gap opens in the narrow band, renormalizing $V_{\mathrm{eff}}=0$, and only one band crosses the Fermi level; or (ii) a lower energy scale arises, opening a Kondo gap for any $V_{\mathrm{eff}}\ne 0$.