Upper critical dimension in a quantum impurity model: Critical theory of the asymmetric pseudogap Kondo problem

Impurity moments coupled to fermions with a pseudogap density of states display a quantum phase transition between a screened and a free moment phase upon variation of the Kondo coupling. We describe the universal theory of this transition for the experimentally relevant case of particle-hole asymmetry. The theory takes the form of a crossing between effective singlet and doublet levels, interacting with low-energy fermions. Depending on the pseudogap exponent, this interaction is either relevant or irrelevant under renormalization group transformations, establishing the existence of an upper-critical “dimension” in this impurity problem. Using perturbative renormalization group techniques we compute various critical properties and compare with numerical results.

Figure 1

Schematic RG flow diagram for the effective theory (2). The horizontal axis denotes the energy difference between doublet and singlet impurity levels, $s$, the vertical axis is the fermionic coupling $g$. (In the language of the infinite-$U$ Anderson model the $f$-electron energy is ${ϵ}_f\equiv s$, and the hybridization $V\equiv g$.) The thick lines correspond to continuous boundary phase transitions; the open circles are the critical fixed points. (a) $r>1$: $g$ is irrelevant, and the transition is a level crossing with perturbative corrections. (b) $r<1$: $g$ is relevant, and the transition is controlled by an interacting fixed point at ${g^{*}}^2=(1-r)∕3+\mathcal{O}{(1-r)}^2$ (4).

Figure 2

Feynman diagrams occurring in the perturbation theory for (2). Full∕wiggly∕dashed lines denote $f_{\sigma }∕b_s∕c_{\sigma }$ propagators. (a) Bare interaction vertex $g_0$. (b) $f_{\sigma }$ self-energy. (c) $b_s$ self-energy. (d) Diagrams entering the local susceptibility, needed to obtain the anomalous exponent ${\eta }_{\chi }$ (7).