Anderson orthogonality and the numerical renormalization group

Anderson orthogonality (AO) refers to the fact that the ground states of two Fermi seas that experience different local scattering potentials, say $|G_{\mathrm{I}}\rangle$ and $|G_{\mathrm{F}}\rangle$, become orthogonal in the thermodynamic limit of large particle number $N$, in that $|\langle G_{\mathrm{I}}|G_{\mathrm{F}}\rangle |\sim N^{-\frac{1}{2}{\Delta }_{\mathrm{AO}}^2}$ for $N\to \infty$. We show that the numerical renormalization group offers a simple and precise way to calculate the exponent ${\Delta }_{\mathrm{AO}}$: the overlap, calculated as a function of Wilson chain length $k$, decays exponentially $\sim e^{-k\alpha }$, and ${\Delta }_{\mathrm{AO}}$ can be extracted directly from the exponent $\alpha$. The results for ${\Delta }_{\mathrm{AO}}$ so obtained are consistent (with relative errors typically smaller than 1%) with two other related quantities that compare how ground-state properties change upon switching from $|{\mathrm{G}}_{\mathrm{I}}\rangle$ to $|{\mathrm{G}}_{\mathrm{F}}\rangle$: the difference in scattering phase shifts at the Fermi energy, and the displaced charge flowing in from infinity. We illustrate this for several nontrivial interacting models, including systems that exhibit population switching.

Figure 1

Anderson orthogonality for the spin-degenerate standard SIAM for a single lead [Eq. (10), $\mu \in \{\uparrow ,\downarrow \}$], with $\mu$-independent parameters ${\varepsilon }_{\mathrm{d}}$ and $\Gamma$ for ${\widehat{H}}_{\mathrm{I}}$ and ${\widehat{H}}_{\mathrm{F}}$ as specified in the panel (the full ${\varepsilon }_{\mathrm{d}}^{\mathrm{F}}$ dependence of ${\Delta }_{\mathrm{AO}}$ for fixed ${\varepsilon }_{\mathrm{d}}^{\mathrm{I}}$ is analyzed in more detail in Fig. 5). Several alternative measures for calculating the AO overlap are shown, using $z_{P{P}^{\prime }}\left(k\right)$ in Eq. (22) with $P^{(\prime )}\in \{G,K,\infty \}$, as defined in the text. All overlaps are plotted for even and odd iterations separately to account for possible even-odd behavior within the Wilson chain (thin solid lines with dots, and dashed lines, respectively, while heavy symbols identify lines with corresponding legends). If even and odd data from the same $z_{P{P}^{\prime }}\left(k\right)$ do not lie on the same smooth line, the combined data are also plotted (light zigzag lines) as guides to the eye. For large $k$, all AO overlaps exhibit exponential decay of equal strength. Separate fits of $e^{\lambda -\alpha k}$ to even and odd sectors are shown as thick solid lines, the lengths of which indicate the fitting range used. The values for ${\Delta }_{\mathrm{AO}}^2$ extracted from these fits using Eq. (19) are in excellent agreement with the displaced charge ${\Delta }_{\mathit{ch}}^2$, as expected from Eq. (8). The relative error is less than 1% throughout, with the detailed values specified in the legend, and $\langle 4\alpha /\ln \Lambda \rangle$ representing the averaged value with regard to the four measures considered.

Figure 2

AO exponents for the standard spin-degenerate SIAM with spin-asymmetric hybridization [Eq. (13), with $\mu \in \{\uparrow ,\downarrow \}$] as functions of ${\varepsilon }_{\mathrm{d},\mathrm{F}}$ (all other parameters are fixed as specified in the panel). The vertical dashed line indicates ${\varepsilon }_{\mathrm{d},\mathrm{I}}/U=-0.5$; at this line, the initial and final Hamiltonians are identical, hence all exponents vanish. The squared AO exponents for the individual channels ${{\Delta }_{\mathrm{AO}}^2}_{,\uparrow }$ (squares) and ${{\Delta }_{\mathrm{AO}}^2}_{,\downarrow }$ (dots) were calculated from Eq. (24). Their sum agrees (with a relative error of less than 1%) with ${\Delta }_{\mathrm{AO}}^2$ calculated from Eq. (19) (downward- and upward-pointing triangles coincide), confirming the validity of the addition rule for squared exponents in Eq. (6).

Figure 3

Determination of ${\Delta }_{\mathit{ch}}$, for the interacting resonant-level model of Eq. (14), for a single specific set of parameters for ${\widehat{H}}_{\mathrm{I}}$ and ${\widehat{H}}_{\mathrm{F}}$, specified in the figure legend (the ${\varepsilon }_{\mathrm{d},\mathrm{F}}$ dependence of ${\Delta }_{\mathrm{AO}}$ for fixed ${\varepsilon }_{\mathrm{d},\mathrm{I}}$ is analyzed in more detail in Fig. 4). We obtain ${\Delta }_{\mathit{ch}}$ (dashed line) by calculating ${\Delta }_{\mathit{sea}}^{\left(k\right)}+{\Delta }_{\mathit{dot}}$ and averaging the results for even and odd $k$. To reduce the influence of chain’s boundary regions, we take the average over the region between the vertical dashed lines.

Figure 4

Verification that ${\Delta }_{\mathrm{AO}}={\Delta }_{\mathit{ph}}={\Delta }_{\mathit{ch}}$ [Eq. (5)] for the spinless fermionic interacting resonant-level model [Eq. (14)]. All quantities are plotted as functions of ${\varepsilon }_{\mathrm{d},\mathrm{F}}$, with all other parameters fixed (as specified in the panel). The vertical dashed line indicates ${\varepsilon }_{\mathrm{d},\mathrm{I}}/U^{\prime }=0$. Heavy dots indicate the final occupation of the dot $n_{\mathrm{d}}$. The exponent ${\Delta }_{\mathrm{AO}}$ (light solid line) agrees well with ${\Delta }_{\mathit{ph}}$ and ${\Delta }_{\mathit{ch}}$ (triangles), with relative errors of less than $1\%$. The local and Fermi-sea contributions to the displaced charge ${\Delta }_{\mathit{ch}}$ are plotted separately, namely, ${\Delta }_{\mathit{dot}}$ (dashed line) and ${\Delta }_{\mathit{sea}}$ (dashed-dotted line). The latter is determined according to the procedure illustrated, for ${\varepsilon }_{d,\mathrm{F}}/U^{\prime }=-1.75$, in Fig. 3.

Figure 5

Anderson orthogonality for the single-lead, spin-symmetric SIAM [Eq. (13), with parameters as specified in the legend]. The energy of the $d$-level of the final system ${\varepsilon }_{\mathrm{d},\mathrm{F}}$ is swept past the Fermi energy of the bath, while that of the initial reference system is kept fixed in the Kondo regime at ${\varepsilon }_{\mathrm{d},\mathrm{I}}=-U/2$, indicated by vertical dashed line in panel (a) and in the inset to panel (b). Panel (a) shows, as function of ${\varepsilon }_{\mathrm{d},\mathrm{F}}$, the dot occupation per spin $n_{\mathrm{d}\mu }$ (dotted solid line), the contribution to the displaced charge by the Fermi sea ${{\Delta }_{\mathit{sea}}}_{\mu }$ (thin black line), the displaced charge ${\Delta }_{\mathit{ch}}^2$ (light solid line), and the parameters of the large-$k$ exponential decay $e^{\lambda -\alpha k}$ of $z_{GK}\left(k\right)$ as extracted from panel (b), namely, $\lambda$ (thick dashed line) and ${\Delta }_{\mathrm{AO}}$ (dark dashed line), derived from $\alpha$ via Eq. (19). Panel (b) shows the AO measure $z_{GK}\left(k\right)$ in Eq. (2) (light lines) for the range of ${\varepsilon }_{\mathrm{d},\mathrm{F}}$ values used in panel (a). The heavy lines shown on top for $k⩾64$ are exponential fits, the results of which are summarized in panel (a). The inset shows the relative error in the AO exponents $\delta {\Delta }^2\equiv ({\Delta }_{\mathrm{AO}}^2-{\Delta }_{\mathit{ch}}^2)/{\Delta }_{\mathit{ch}}^2$, i.e., the deviation between the light solid and dark dashed curves in panel (a); this error is clearly less than $1\%$ over the full range of ${\varepsilon }_{\mathrm{d}}$ analyzed.

Figure 6

Anderson orthogonality for a spinless (a) and spinful (b) two-lead SIAM, with dot levels of unequal width and a split level structure as defined in Eq. (28a) (all relevant model parameters are specified in the legends). In both cases, the higher level 2 is broader than the lower level 1 $({\Gamma }_2>{\Gamma }_1)$, leading to population switching as function of the average final level energy ${\varepsilon }_{\mathrm{d},\mathrm{F}}$. The fixed value of ${\varepsilon }_{\mathrm{d},\mathrm{I}}$ is indicated by the vertical dashed line. The inset to panel (a) shows a zoom into the switching region, clearly demonstrating that population switching occurs smoothly. For panel (b), a finite magnetic field $B$ causes a splitting between spin-up and spin-down levels, resulting in a more complex switching pattern. In both panels, ${\Delta }_{\mathrm{AO}}^2$ and ${\Delta }_{\mathit{ch}}$ agree very well throughout the sweep, with a relative error $\delta {\Delta }^2$ well below $1\%$.