Spectral properties of strongly correlated multi-impurity models in the Kondo insulator regime: Emergent coherence, metallic surface states, and quantum phase transitions

We investigate the real-space spectral properties of strongly correlated multi-impurity arrays in the Kondo insulator regime. Employing a recently developed mapping onto an effective correlated cluster problem makes the problem accessible to the numerical renormalization group. The evolution of the spectrum as a function of cluster size and cluster site is studied. We applied the extended Lieb-Mattis theorem to predict whether the spectral function must vanish at the Fermi energy developing a true pseudogap or whether the spectral function remains finite at $\omega =0$. Our numerical renormalization group spectra confirm the predictions of the theorem and shows a metallic behavior at the surface of a cluster prevailing in arbitrary spatial dimensions. We present a conventional minimal extension of a particle-hole symmetric Anderson lattice model at $U=0$ that leads to a gapped bulk band but a surface band with mainly $f$-orbital character for weak and moderate hybridization strength. The change in the site-dependent spectra upon introducing a Kondo hole in the center of the cluster is presented as a function of the hole-orbital energy. In particular, the spectral signatures across the Kosterlitz-Thouless-type quantum phase transition from a singlet to a local moment fixed point are discussed.

Figure 1

Sketch of the geometric setup: An array of $N_f=7$ correlated $f$ orbitals (green) are locally coupled to a $1d$ tight binding chain (blue) with the number of conduction band orbitals, $N_c$, larger than $N_f$ to define a band continuum.

Figure 2

$c$-spectral function for the central position in a $N_f=7$ MIAM where $U=-2{\epsilon }_f=10{\mathrm{\Gamma }}_0$ in comparison with the DMFT solution of Kondo lattice model with the identical Kondo coupling $g=\rho J=\rho J_{\text{SW}}=0.25$ obtained by the Schrieffer-Wolff transformation [34].

Figure 3

Spectral function ${\rho }_{l_c}^f\left(\omega \right)$ of the central $f$ orbital in a dense array comprising (a) $N^f=2$, (b) $N_f=3$, (c) $N_f=4$, (d) $N_f=5$, (e) $N_f=6$, and (f) $N_f=7$ impurities. In case of even $N_f$, both central $f$ orbitals are equivalent due to parity symmetry. Parameters: ${\epsilon }_f=-U/2=-5{\mathrm{\Gamma }}_0, {\epsilon }^c=0, D=10{\mathrm{\Gamma }}_0, \mathrm{\Lambda }=2$, and $N_s=8000$.

Figure 4

Spectral functions ${\rho }^f\left(\omega \right)$ for an impurity array comprising $N_f=7f$ orbitals. The panels depict the spectral function of the individual $f$ orbitals starting at the outermost in panel (a) and ending at the center in panel (d). Parameters as in Fig. 3.

Figure 5

Effective spectral functions ${\tilde{\rho }}^f\left(\omega \right)$ for an impurity array comprising $N_f=7$ uncorrelated $f$ orbitals that are located at ${\epsilon }_l^f=0$ to maintain half filling. The panels depict the spectral function of the individual $f$ orbitals starting at the outermost in (a) and ending at the center in panel (d).

Figure 6

The conduction electron spectral functions ${\rho }_l^c\left(\omega \right)$ for the same parameters as in Fig. 4 $l=1$, (b) $l=2$, (c) $l=3$, (d) $l=4$, The Green's functions were calculated via Eq. (13).

Figure 7

Same as in Fig. 4 but with ${\epsilon }_f=-U/2=-10{\mathrm{\Gamma }}_0$.

Figure 8

Sketch of a strip of a $2d$ MIAM model for a Kondo insulator with a infinite large $y$ axis but a finite-size $x$ axis to define a surface. The correlated orbitals are depicted in green, the uncorrelated in blue. Note that on the surface the correlated orbitals are absent.

Figure 9

(a) The $k_y$-dependent band structure for the periodic stripe in Hartree mean field theory, ${\tilde{\epsilon }}^f={\epsilon }^f+U/2=0$ of a Kondo insulator model depicted in Fig. 8 and $V_l/t=0.5$ for a stripe width of $N_x=50$. The inset on the left-hand side reveals the small hybridization gap and the insulating behavior. Clearly visible are the two metallic surface bands that are not crossing each other, one for positive and one for negative $k_y$. (b) Probability of the edge $f$ and $c$ orbital contribution at the Fermi energy to the metallic band as function of $V/t$. The inset depicts this probability contribution for $V/t=1$ as function of the site along the $x$ axis on a logarithmic scale.

Figure 10

Sketch of the geometric setup with a Kondo hole: An array of $N_f=7$ correlated $f$ orbitals (green) are locally coupled to a $1d$ tight-binding chain (blue) with $N_c\gg N_f$. We replace the center correlated orbital with one uncorrelated orbital (red) and introduce ${\epsilon }^h$ as its single orbital energy.

Figure 11

Spectral function of the outermost $f$ orbital in an $N_f=7$ impurity array with an uncorrelated hole orbital in its center at ${\epsilon }_c=0$. For the correlated $f$ orbitals, ${\epsilon }_f=-U/2=-5{\mathrm{\Gamma }}_0$ holds, and the different panels show different on-site energies of the hole orbital: (a) ${\epsilon }^h/{\mathrm{\Gamma }}_0=0.0$, (b) ${\epsilon }^h/{\mathrm{\Gamma }}_0=1.0$, (c) ${\epsilon }^h/{\mathrm{\Gamma }}_0=2.0$, (d) ${\epsilon }^h/{\mathrm{\Gamma }}_0=2.1$, (e) ${\epsilon }^h/{\mathrm{\Gamma }}_0=2.2$, (f) ${\epsilon }^h/{\mathrm{\Gamma }}_0=5.0$. The QCP occurs at around ${\epsilon }_c^h/{\mathrm{\Gamma }}_0\approx 2.25$.

Figure 12

Same as Fig. 11 but for the second $f$ orbital.

Figure 13

Evolution of ${\rho }_{3,\sigma }^f\left(\omega \right)$ neighboring the hole site for the same parameters as in Fig. 11.

Figure 14

Evolution of the Kondo hole spectrum ${\rho }_{4,\sigma }^f\left(\omega \right)$ in the cluster center for the same parameters as in Fig. 11.

Figure 15

Zoom of the spectral functions around the Fermi energy near the QCP located at ${\epsilon }_c^h/{\mathrm{\Gamma }}_0\approx 2.25$. (a) $l=1$, (b) $l=2$, (c) $l=3$, (d) $l=l_c=4$.