Strongly correlated multi-impurity models: The crossover from a single-impurity problem to lattice models

We present a mapping of various correlated multi-impurity Anderson models to a cluster model coupled to a number of effective conduction bands capturing its essential low-energy physics. The major ingredient is the complex single-particle self-energy matrix of the uncorrelated problem that encodes the influence to the host conduction band onto the dynamics of a set of correlated orbitals in a given geometry. While the real part of the self-energy matrix generates an effective hopping between the cluster orbitals, the imaginary part, or hybridization matrix, determines the coupling to the effective conduction electron bands in the mapped model. The rank of the hybridization matrix determines the number of independent screening channels of the problem, and allows the replacement of the phenomenological exhaustion criterion by a rigorous mathematical statement. This rank provides a distinction between multi-impurity models of the first kind and of the second kind. For the latter, there are insufficient screening channels available, so that a singlet ground state must be driven by the intercluster spin correlations. This classification provides a fundamental answer to the question of why ferromagnetic exchange interactions between local moments are irrelevant for the spin-compensated ground state in dilute multi-impurity models, whereas the formation of large spins competes with the Kondo scale in dense impurity arrays, without evoking a spin density wave. The low-temperature physics of three examples taken from the literature are deduced from the analytic structure of the mapped model, demonstrating the potential power of this approach. Numerical renormalization group calculations are presented for up to five-site clusters. We investigate the appearance of frustration-induced non-Fermi-liquid fixed points in the trimer, and demonstrate the existence of several critical points of Kosterlitz-Thouless type at which ferromagnetic correlations suppress the screening of an additional effective spin-$1/2$ degree of freedom.

Figure 1

Schematic partitioning of the models in four different categories: I denotes the limit of a single-impurity problem, section 2 the MIAMs of the first kind, section 3 the MIAMs of the second kind, and section 4 the MIAM with infinite number of correlated lattice sides recovering the periodic Anderson model.

Figure 2

MIAM with 2d simple cubic lattice, containing uncorrelated lattice orbitals (green) and correlated impurities (blue). The individual panels depict different geometries of the impurity configuration leading to a different number of decoupled orbitals $N_{\text{free}}=N_f-\mathrm{rank}\left({\mathit{\Gamma }}_{\sigma }\right)$. (a) $N_{\text{free}}=0$, (b) $N_{\text{free}}=1$, (c) $N_{\text{free}}=1$, and (d) $N_{\text{free}}=2$.

Figure 3

The diluted periodic Anderson model in 1d: the local impurities are only connected to the B sublattice.

Figure 4

Phase diagram for the low-energy model with $C_3$ symmetry as function of the tunneling element $t^{\text{eff}}$ between the correlated orbitals and ${\mathrm{\Gamma }}^{\text{off}}/{\mathrm{\Gamma }}_0=0.2$, $U/{\mathrm{\Gamma }}_0=20$, ${\epsilon }^f/{\mathrm{\Gamma }}_0=-10$, $D/{\mathrm{\Gamma }}_0=10$. Small values of $t^{\text{eff}}$ result in a FM or weak AF exchange interaction and a FL. Intermediate $t_{c,1}^{\text{eff}}<t^{\text{eff}}<t_{c,2}^{\text{eff}}$ lead to an AF exchange and the frustrated Kondo regime. For large $t^{\text{eff}}>t_{c,2}^{\text{eff}}$ the PH asymmetry destroys the NFL fixed point on a crossover scale (purple line points) which is exponentially small for $(t^{\text{eff}}-t_{c,2}^{\text{eff}})\to 0^{+}$.

Figure 5

Effective tunneling element $t^{\text{eff}}=\mathrm{Re}{\mathrm{\Delta }}^{\text{off}}(-\mathrm{i}{0}^{+})$ (black) and ${\mathrm{\Gamma }}^{\text{off}}=\mathrm{Im}{\mathrm{\Delta }}^{\text{off}}(-\mathrm{i}{0}^{+})$ (light blue) as function of the dimensionless distance $R{k}_F$ for an isotropic linear dispersion ${\epsilon }_{\vec{k}}$ in (a) 2d and (b) 3d.

Figure 6

(a) Impurity contribution to the entropy for $T\to 0$ as function of the band center for three particle-hole-symmetric impurities with $U/{\mathrm{\Gamma }}_0=10,{\epsilon }_i^f=-U/2$, and $D/{\mathrm{\Gamma }}_0=10.$ (b) The two spin-spin correlation functions for the same parameter. The center impurity is labeled $i=2$, the two outer correlated sites $i=1,3$.

Figure 7

$S_{\mathrm{imp}}$ as function of $|{\epsilon }_c|/D$ for (a) $N_f=4$ and (b) $N_f=5$ and a fixed temperature $T/{\mathrm{\Gamma }}_0={10}^{-11}$ (corresponding to a fixed number of NRG iterations $N=50$) for particle-hole-symmetric impurities with $U/{\mathrm{\Gamma }}_0=10$. For $N_f=4$ we find three regions and three KT quantum phase transition points; for $N_f=5$ four regions separated by a KT-type phase transition are identified.

Figure 8

(a) Low-temperature scale $T_0$ for particle-hole-asymmetric three-impurity models with $U/{\mathrm{\Gamma }}_0=50,D/{\mathrm{\Gamma }}_0=10$, and four different values of ${\epsilon }^f$. (b) The critical value ${\epsilon }_c^c$ as a function of ${\epsilon }^f$.

Figure 9

(a) The two low-temperature scales $T_0$ and $T_1$ vs ${\epsilon }_c$ in the three-impurity model. $T_1$ (blue) is defined as crossover temperature approaching the unstable LM fixed point with $I=1/2$ and $T_0$ (red) denotes the crossover temperature from the unstable $S=1/2$ LM fixed point to the singlet fixed point. The crossover temperature for the multi-impurity system without coupling to the conduction band channels is added as a black line. (b) Low-energy fixed-point spin-spin correlation functions $〈{\vec{S}}_1{\vec{S}}_2〉$ and $〈{\vec{S}}_1{\vec{S}}_3〉$ vs ${\epsilon }_c$. Parameters: particle-hole-symmetric impurities with $U/{\mathrm{\Gamma }}_0=30,{\epsilon }_l^f=-U/2$, $D/{\mathrm{\Gamma }}_0=10$.

Figure 10

Schematic diagram of the tight-binding hopping parameters between the tree impurities.

Figure 11

The two low-temperature scales $T_0$ and $T_1$ vs ${\epsilon }_c$ in the $N_f=5$ MIAM. $T_1$ (blue) denotes crossover temperature approaching the lowest unstable LM fixed point and $T_0$ (red) crossover temperature from the unstable LM fixed point to the singlet fixed point. The crossover temperature for the multi-impurity system without coupling to the conduction band channels is added as a black line. Parameters: as in Fig. 9.

Figure 12

$\text{exp}(S_{\text{imp}}/k_{\text{B}})$ phase diagram for (a) $N_f=3$ and (b) $N_f=4$ impurities, separated by $\mathrm{\Delta }R=2a$ and plotted against ${\epsilon }_c/D$ and $U/{\mathrm{\Gamma }}_0$ for a constant temperature $T/{\mathrm{\Gamma }}_0={10}^{-15}$, ${\epsilon }_l^f/{\mathrm{\Gamma }}_0=-3$, and $D/{\mathrm{\Gamma }}_0=10$.