Validity of the local self-energy approximation: Application to coupled quantum impurities

We examine the quality of the local self-energy approximation, applied here to models of multiple quantum impurities coupled to an electronic bath. The local self-energy is obtained by solving a single-impurity Anderson model in an effective medium that is determined self-consistently, similar to the dynamical mean-field theory (DMFT) for correlated lattice systems. By comparing to exact results obtained by using the numerical renormalization group, we determine situations where “impurity-DMFT” is able to capture the physics of highly inhomogeneous systems and those cases where it fails. For two magnetic impurities separated in real space, the onset of the dilute limit is captured, but RKKY-dominated interimpurity singlet formation cannot be described. For parallel quantum dot devices, impurity-DMFT succeeds in capturing the underscreened Kondo physics by self-consistent generation of a critical pseudogapped effective medium. However, the quantum phase transition between high- and low-spin states upon tuning interdot coupling cannot be described.

Figure 1

Schematic of the models and mappings used. (a) Two separated impurities on the surface of a 3D cubic lattice, mapped to a problem involving two Wilson chains in NRG. (b) Mapping to a single-impurity problem in a self-consistently determined effective medium, via the local self-energy approximation (impurity-DMFT). (c) Quantum dots coupled in parallel between source and drain leads, mapped to a multi-impurity but single-channel problem in NRG.

Figure 2

Comparison between exact NRG results (solid black lines) and impurity-DMFT results (dashed red lines) for the real-space 2IAM. The local impurity spectral function at $T=0$ is plotted as $-\text{Im}{\mathrm{\Gamma }}_{\text{loc}}{G}_{11,\sigma }^{\text{imp}}\left(\omega \right)$ vs $\omega /D$. The two impurities are taken to be on the (100) surface of a (semi-) infinite 3D cubic lattice, separated by the vector $\mathbf{R}\equiv (R_x,R_y)$. Clockwise from top-left panel: $\mathbf{R}=(0,1),(1,1),(0,4),(2,2)$. Impurity parameters are $U/D=0.5, \epsilon =-U/2$, and $V/D=0.075$.

Figure 3

Impurity-DMFT effective medium, $\mathrm{\Delta }\left(\omega \right)/\left(\pi V^2\right)$ vs $\omega /D$ for the systems plotted in Fig. 2.

Figure 4

Exact NRG results for the PQD model consisting of $N$ coupled quantum dots. Plotted is the (even orbital) impurity spectral function $-\text{Im}{\mathrm{\Gamma }}_{\text{loc}}{G}_{e,\sigma }^{\text{imp}}\left(\omega \right)$ vs $\omega /D$ at $T=0$ for $N=1,2,3,4,5$ (black, red, blue, green, and purple lines, following the arrow). The inset shows the rescaled spectra satisfying the pinning condition (19). Impurity parameters are $U/D=0.5, \epsilon =-U/2, V/D=0.06$ (and $v=h=0$).

Figure 5

Comparison of impurity-DMFT results (red lines) and exact NRG (black lines) for PQD models with $N=2,3,4,5$. The main panel in each case shows the rescaled even-orbital spectrum $-\text{Im}{\mathrm{\Gamma }}_{\text{loc}}{G}_{e,\sigma }^{\text{imp}}\left(\omega \right)$ vs $\omega /T_K$ at $T=0$, while the lower insets show the unscaled (raw) spectra. The upper insets show the detailed low-energy behavior. Here $T_K$ is defined via $\text{Im}{G}_{e,\sigma }^{\text{imp}}(\omega =T_K)=0.95\text{Im}{G}_{e,\sigma }^{\text{imp}}(\omega =0)$. The dot parameters are $U/D=0.5, \epsilon =-U/2, V/D=0.05$ (and $v=h=0$).

Figure 6

(top panel) DMFT effective medium $\mathrm{\Delta }\left(\omega \right)/\left(\pi V^2\right)$ vs $\omega /D$ for PQDs with $N=2,3,...,10$ increasing in the direction of the arrow. Low-energy behavior follows a pseudogap power law $\mathrm{\Delta }\left(\omega \right)\sim {\left|\omega \right|}^{\alpha }$, with exponent $\alpha \sim N^{-\gamma }$ that depends on the number of dots (see inset, points). We find numerically $\gamma \simeq 0.43$ (inset, dashed line). The bare hybridization is shown as the red dotted line for comparison. (bottom panel) Corresponding local DMFT self-energy $-\text{Im}{\mathrm{\Sigma }}_{\sigma }^{\text{loc}}\left(\omega \right)$, showing the same low-energy power-law decay as $\mathrm{\Delta }\left(\omega \right)$. Same parameters as for Fig. 5.

Figure 7

Comparison of impurity-DMFT results (red dashed lines) and exact NRG (solid black lines) for the PQD model with $N=2$ (upper panels) and 3 (lower panels) in an applied magnetic field. Dot parameters $U/D=0.5, \epsilon =-U/2$, and $V/D=0.06$ such that $T_K^{h=0}\sim {10}^{-9}D$. Shown for $v=0$ but $h/D={10}^{-10},{10}^{-9},{10}^{-8}$, and ${10}^{-7}$. Plotted for $\sigma =\uparrow$.

Figure 8

DMFT effective medium $\mathrm{\Delta }\left(\omega \right)/\left(\pi V^2\right)$ vs $\omega /D$ for PQDs with $N=2,3,4$, and 5 (increasing with the arrow) for $U/D=0.5, \epsilon =-U/2, V/D=0.06$, and $h/D={10}^{-8}$ ($v=0$).

Figure 9

Comparison of impurity-DMFT results (red dashed lines) and exact NRG (solid black lines) for the PQD model with $N=2$ (left panels) and 3 (right panels) with finite interdot coupling $v\ne 0$ across the quantum phase transition. Upper panels show high-spin phase with $v=0.005D<v_c$. Lower panels show low-spin phase with $v=0.01D>v_c$. Shown for fixed $U/D=0.5, \epsilon =-U/2$, and $V/D=0.06$ (but $h=0$).