Mesoscopic Anderson box: Connecting weak to strong coupling

We study the Anderson impurity problem in a mesoscopic setting, namely the “Anderson box,” in which the impurity is coupled to finite reservoir having a discrete spectrum and large sample-to-sample mesoscopic fluctuations. Note that both the weakly coupled and strong coupling Anderson impurity problems are characterized by a Fermi-liquid theory with weakly interacting quasiparticles. We study how the statistical fluctuations in these two problems are connected, using random matrix theory and the slave boson mean-field approximation (SBMFA). First, for a resonant level model such as results from the SBMFA, we find the joint distribution of energy levels with and without the resonant level present. Second, if only energy levels within the Kondo resonance are considered, the distributions of perturbed levels collapse to universal forms for both orthogonal and unitary ensembles for all values of the coupling. These universal curves are described well by a simple Wigner-surmise-type toy model. Third, we study the fluctuations of the mean-field parameters in the SBMFA, finding that they are small. Finally, the change in the intensity of an eigenfunction at an arbitrary point is studied, such as is relevant in conductance measurements. We find that the introduction of the strongly coupled impurity considerably changes the wave function but that a substantial correlation remains.

Figure 1

The evolution of the energy levels and wave functions as a function of coupling strength for a realization drawn from the GOE (calculated using infinite-$U$ SBMFT). (a) Energy levels near the Fermi energy $\mu =0$ for coupling $V_0$ (from 0.423 to 1.0). (b) Zoom of a few levels above the Fermi energy. (c) The wave function amplitudes ${\left|\psi \left(\mathbf{R}\right)\right|}^2$ corresponding to the energy levels in (b) for an arbitrary position $\mathbf{R}\ne 0$. Parameters: band width $D=3$, $E_d=-0.7$, $\Delta =0.0075$, and $T=0.005$.

Figure 2

The analog of Fig. 1 for the GUE; parameters are the same. Note that the variation is smoother for these GUE results than for the GOE in Fig. 1.

Figure 3

The distribution of the mean-field parameters, critical coupling, and effective Wilson number from the SBMFT calculation for both GOE (blue solid line) and GUE (red dash-dotted line). (a) $\eta$, (b) $\xi$, (c) critical coupling $J_K^c$, and (d) the effective Wilson number $W^{*}=T_K{\chi }_0\left(0\right)$. The vertical black dashed lines mark the values for the corresponding bulk flat-band system. The following parameters were used: band width $D=3$, $E_d=-0.7$, $V_0=0.6$, and $T=0.005$. The mean level spacing is $\Delta =0.01$, and the Kondo temperature in the bulk limit is $T_K^{\mathrm{bulk}}\simeq 0.092$.

Figure 4

Variance of the mean-field parameters $\eta$ (left panel) and $\xi$ (right) as the size of the system is changed. The change in system size is quantified through the mean level separation $\Delta$. As expected, the fluctuations are smaller in the GUE compared to the GOE, and the dependence of the variance of the fluctuations on $\Delta$ is nearly linear. Here we use $D=3$, $E_d=-0.7$, and $V_0=0.8$.

Figure 5

The distribution of $S$ (which includes both $|{\lambda }_{\kappa }-{\epsilon }_i|/|{\epsilon }_{i+1}-{\epsilon }_i|$ and $|{\lambda }_{\kappa }-{\epsilon }_{i+1}|/|{\epsilon }_{i+1}-{\epsilon }_i|$) for the resonant level model (a) GOE and (b) GUE and for the SBMFT treatment of the infinite-$U$ Anderson model, (c) GOE and (d) GUE. (a, Insert) The cumulative distribution of the $V_0=0.9$ GOE data compared to the toy model. The dashed lines in (a) and (b) are the result of the toy model; those in (c) and (d) show the RLM result for $V_0=0.9$ and $1.3$, respectively. Parameters: $D=3$, ${\epsilon }_0=0$ for the RLM and $E_d=-0.7$ in the SBMFT, 5000 realizations are used, and there are 500 energy levels within the band.

Figure 6

Wave-function correlator ${\mathcal{C}}_{i,\kappa \left(i\right)}$ for the noninteracting RLM ($\overline{\eta }=1$). (a) GOE and (c) GUE, as a function of the average distance between ${\epsilon }_i$ and ${\epsilon }_0$. (b) GOE and (d) GUE, as a function of rescaled average distance $\epsilon =(i\Delta -D/2)/{\Gamma }_0$. The dashed lines are the result of Eq. (90) in which the wave-function fluctuations are taken into account but the energy levels are assumed bulklike. Parameters: $D=3$, impurity energy level ${\epsilon }_0=0$, 5000 realizations, and 500 energy levels within the band.

Figure 7

Wave-function correlation, ${\mathcal{C}}_{i,\kappa \left(i\right)}$, for the SBFMT approach to the infinite-$U$ Anderson model. (a) GOE and (c) GUE, as a function of the average distance from the middle of the band. (b) GOE and (d) GUE, as a function of rescaled average distance $\delta \overline{\epsilon }=[(i\Delta -D/2)-({\mathcal{E}}_0\left(\xi \right)-\mu \left)\right]/\Gamma \left(\eta \right)$. The black lines labeled RLM are results for the noninteracting RLM at $V_0=0.5$. The dashed lines are the result of Eq. (90) in which the wave-function fluctuations are taken into account but the energy levels are assumed bulklike. Parameters: $D=3$, impurity energy level $E_d=-0.7$, 5000 realizations, and 500 energy levels within the band.