Critical charge fluctuations in a pseudogap Anderson model

The Anderson impurity model with a density of states $\rho \left(\varepsilon \right)\propto {\left|\varepsilon \right|}^r$ containing a power-law pseudogap centered on the Fermi energy ($\varepsilon =0$) features for $0<r<1$ a Kondo-destruction quantum critical point (QCP) separating Kondo-screened and local-moment phases. The observation of mixed valency in quantum critical $\beta \text{−}{\mathrm{YbAlB}}_4$ has prompted study of this model away from particle-hole symmetry. The critical spin response associated with all Kondo destruction QCPs has been shown to be accompanied, for $r=0.6$ and noninteger occupation of the impurity site, by a divergence of the local charge susceptibility on both sides of the QCP. In this work, we use the numerical renormalization-group method to characterize the Kondo-destruction charge response using five critical exponents, which are found to assume nontrivial values only for $0.55\lesssim r<1$. For $0<r\lesssim 0.55$, by contrast, the local charge susceptibility shows no divergence at the QCP, but rather exhibits nonanalytic corrections to a regular leading behavior. Both the charge critical exponents and the previously obtained spin critical exponents satisfy a set of scaling relations derived from an ansatz for the free energy near the QCP. These critical exponents can all be expressed in terms of just two underlying exponents: the correlation-length exponent $\nu \left(r\right)$ and the gap exponent $\Delta \left(r\right)$. The ansatz predicts a divergent local charge susceptibility for $\nu <2$, which coincides closely with the observed range $0.55\lesssim r<1$. Many of these results are argued to generalize to interacting QCPs that have been found in other quantum impurity models.

Figure 1

Schematic phase diagram of the pseudogap Anderson model on the $\Gamma \text{−}{\varepsilon }_d$ plane for fixed $U$ and for band exponents (a) $0<r<\frac{1}{2}$, and (b) $r\ge \frac{1}{2}$. In (a), the SSC phase spans just the solid horizontal line at ${\varepsilon }_d=-U/2$.

Figure 2

$|Q_{\mathrm{loc}}|$ vs distance from the phase boundary along the $\Gamma$ and ${\varepsilon }_d$ axes for $r=0.6$. Filled (hollow) symbols represent points in the LM (${\mathrm{ASC}}_{-}$) phase. The linear variations on this log-log plot indicate power-law behavior in accordance with Eqs. (13).

Figure 3

Charge critical exponents plotted vs pseudogap exponent $r$: (a) $\tilde{\beta }$ obtained independently from the variation of $Q_{\mathrm{loc}}$ with respect to ${\varepsilon }_d$ and with respect to $\Gamma$, (b) $\tilde{x}$, and (c) $\tilde{\gamma }$. In all cases, the estimated nonsystematic error is smaller than the data symbol. Shading indicates the range within which each exponent is predicted to lie when the values of the correlation-length exponent $\nu$ from Table 1 are inserted into Eqs. (17).

Figure 4

Temperature-dependent part ${\chi }_c-{\chi }_c^{\mathrm{reg}}$ of the local charge susceptibility for band exponent $r=0.5$. The line fitted through the NRG data points corresponds to ${\chi }_c-{\chi }_c^{\mathrm{reg}}\propto T^{0.16}$.