Critical charge fluctuations in a pseudogap Anderson model

Experiments on heavy-fermion $\beta$-YbAlB$_4$ raise the possibility of critical destruction of the Kondo effect in a mixed-valence system. We consider a toy model of this phenomenon: the particle-hole asymmetric Anderson model with a pseudogapped density of states $\rho(\epsilon) \propto |\epsilon-\epsilon_F|^r$ where $\epsilon_F$ is the Fermi energy. The model exhibits a critical spin response at a quantum phase transition separating a Kondo phase from a non-Kondo (local-moment) phase, where the Kondo energy scale is driven continuously to zero on approach from the Kondo side [1]. This Kondo-destruction transition has recently been shown, for certain values of $r$, to be accompanied by a divergence of the charge susceptibility coming from either phase [2]. Here we present a systematic numerical renormalization-group study of the charge response as a function of $r$. The charge fluctuations are described by critical exponents that show nontrivial $r$ dependence. Over a range of $r$ values, these exponents satisfy hyperscaling equations consistent with a scaling anzatz for the critical free energy at an interacting quantum phase transition. [1] K. Ingersent and Q. Si, Phys. Rev. Lett. 89, 076403 (2002). [2] J. H. Pixley et al., Phys. Rev. Lett. 109, 086403 (2012).