Critical Local-Moment Fluctuations, Anomalous Exponents, and $\omega /T$ Scaling in the Kondo Problem with a Pseudogap

Experiments in heavy-fermion metals and related theoretical work suggest that critical local-moment fluctuations can play an important role near a zero-temperature phase transition. We study such fluctuations at the quantum critical point of a Kondo impurity model in which the density of band states vanishes as $|ϵ{|}^r$ at the Fermi energy ($ϵ=0$). The local spin response is described by a set of critical exponents that vary continuously with $r$. For $0<r<1$, the dynamical susceptibility at the critical point exhibits $\omega /T$ scaling with a fractional exponent, implying that the critical point is interacting.

Figure 1

Schematic phase diagram showing the vicinity of the quantum critical point of the pseudogap Kondo model.

Figure 2

${\chi }_{\mathrm{l}\mathrm{o}\mathrm{c}}(\omega =0)$ vs $T$ and ${\chi }_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\prime \prime }(T=0)$ vs $\omega$ at the symmetric critical point: NRG results for $\Lambda =3$, $r=0.1$ and 0.4 [25]. For a given $r$, ${\chi }_{\mathrm{l}\mathrm{o}\mathrm{c}}(\omega =0,T\ll 1)$ and ${\chi }_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\prime \prime }(|\omega |\ll 1,T=0)$ are described by equal exponents, $x=y$.

Figure 3

Comparison between NRG and small-$r$ results for the symmetric critical point. (a) ${\omega }^y{\chi }_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\prime \prime }(J=J_c)$ vs $\omega /T$ [25]: NRG results for $r=0.4$ and $T/D={10}^{-7}$, ${10}^{-6}$, ${10}^{-5}$, and ${10}^{-4}$ (symbols) and the prediction of Eq. (8) (solid line). (b) Exponent $1-x$ vs $r$: NRG data from Table  (symbols) and the prediction of Eq. (9), in the form of a spline fit (solid line) through numerical values for $({\rho }_0{J}_c{)}^2$, extrapolated to the continuum limit $\Lambda =1$ (see 18).