Pseudogap Fermi-Bose Kondo Model

We consider a magnetic impurity coupled to both fermionic quasiparticles with a pseudogap density of states and bosonic spin fluctuations. Using renormalization group and large-$N$ calculations we investigate the phase diagram of the resulting Fermi-Bose Kondo model. We show that the Kondo temperature is strongly reduced by low-energy spin fluctuations, and make connections to experiments in cuprate superconductors. Furthermore, we derive an exact exponent for the critical behavior of the conduction electron $T$ matrix, and propose our findings to be relevant for certain scenarios of local quantum criticality in heavy-fermion metals.

Figure 1

Schematic RG flow diagram for the pseudogap Fermi-Bose Kondo model for small $ϵ$ and $r$, and ${\Delta }_s=0$. The axes denote the couplings to the bosonic ($\gamma$) and fermionic ($j$) baths. The model has three (meta)stable fixed points (solid dots): the local moment (LM), bosonic fluctuating (B-FL), and fermionic strong-coupling (F-SC) fixed points; the open dots are the critical fixed points F-CR and FB-CR; for details see text. The thick line corresponds to a line of continuous boundary phase transitions. In the metallic case, $r=0$, the LM and F-CR fixed points merge [1, 2].

Figure 2

Feynman diagrams for the Green’s function $G_T$ entering the conduction electron $T$ matrix. Full/dashed/wiggly lines denote $f$/$c$/$\varphi$ propagators. The double line is the full $f$ propagator, the shaded area the full vertex, and the open dots are the external sources.

Figure 3

Characteristic energy scale $T^{*}$ as a function of the bosonic spin gap ${\Delta }_s$, calculated for a fermionic host corresponding to a cuprate superconductor [13], using large-$N$ techniques. Solid line: $J_K=0.27\mathrm{e}\mathrm{V}$, $U=-0.15\mathrm{e}\mathrm{V}$, ${\gamma }_0=0.15\mathrm{e}\mathrm{V}$; Dash-dotted line: $J_K=0.38\mathrm{e}\mathrm{V}$, $U=0$, where $U$ is the additional impurity scattering potential. In comparison with NMR experiments, $T^{*}$ is equivalent to the Kondo temperature below which the impurity moment is screened; the horizontal axis can roughly be identified with doping (up to ${\Delta }_s\approx 50\mathrm{m}\mathrm{e}\mathrm{V}$) [14]. Note that the change in the fermionic band upon doping has little influence on $T^{*}$; an increase in the superconducting gap, ${\Delta }_0$, with underdoping will further suppress $T^{*}$.