Overscreened Kondo effect, (color) superconductivity, and Shiba states in Dirac metals and quark matter

We study the interplay between the Kondo effect and (color) superconductivity in doped Dirac metals with magnetic impurities and in quark matter with colorful impurities. We first point out that the overscreened Kondo effect arises in the normal state of these systems. Next the (color) superconducting gap is incorporated as a mean field and the phase diagram for a varying gap and temperature is constructed nonperturbatively. A rich phase structure emerges from a competition of effects unique to a multichannel system. The Kondo-screened phase is shown to disappear for a sufficiently large gap. Peculiarity of quark matter due to the confining property of non-Abelian gauge fields is noted. We also investigate the spectrum of sub-gap excited states, called Shiba states. Based on a model calculation and physical reasoning we predict that, as the coupling of the impurity to the bulk is increased, there will be more than one quantum phase transition due to level crossing among overscreened states.

Figure 1

Illustration of distinct Kondo effects: (a) exact screening, (b) underscreening and (c) overscreening. See the main text and Table 1 for further details.

Figure 2

The scaling function of the dimensionless coupling $\overline{G}$. The arrows indicate the flow as $D\to 0$.

Figure 3

Schematic phase diagram of Kondo impurities in Dirac superconductors for (a) $N_f=1$, (b) $N_f=2$ at weak coupling, and (c) $N_f=2$ at strong coupling. The double line ($===$) represents a first-order transition.

Figure 4

Dispersion relations of quasiparticles $\{\widehat{\omega }=\pm \sqrt{{\widehat{B}}_1^{\pm }(\widehat{k})},\pm \sqrt{{\widehat{B}}_2^{\pm }(\widehat{k})}\}$ for $\widehat{\mu }=0.5$ and ${\widehat{\mu }}_{\xi }=-0.1$. (a) $(\widehat{\mathrm{\Delta }},\widehat{V})=(0.05,0)$, (b) $(\widehat{\mathrm{\Delta }},\widehat{V})=(0,0.05)$, (c) $(\widehat{\mathrm{\Delta }},\widehat{V})=(0.05,0.05)$.

Figure 5

(a) Condensate $⟨\widehat{V}⟩$ and (b) thermodynamic potential $\widehat{\mathrm{\Xi }}$ with $(\widehat{\mu },{\widehat{\mu }}_{\xi },\widehat{G})=(0.5,0,10)$ at $\widehat{\mathrm{\Delta }}=0$. Here $\widehat{\mathrm{\Xi }}$ is normalized to 0 at the origin.

Figure 6

(a) Thermodynamic potential $\widehat{\mathrm{\Xi }}$ with $(\widehat{\mu },{\widehat{\mu }}_{\xi },\widehat{G})=(0.5,0,10)$ at $\widehat{T}=0$. The condensate drops to zero for larger $\widehat{\mathrm{\Delta }}$. (b) Phase diagram with $(\widehat{\mu },{\widehat{\mu }}_{\xi })=(0.5,0)$ at $\widehat{T}=0$. The Kondo singlet is formed in the shaded region only. The transition across the boundary of the shaded region [$\widehat{\mathrm{\Delta }}=2{\widehat{V}}_{T=\mathrm{\Delta }=0}(\widehat{G})$] is second order. As soon as ${\widehat{\mu }}_{\xi }\ne 0$ or $\widehat{T}\ne 0$, it becomes first order (Fig. 7). For comparison, we also show the phase boundary in the limit $\widehat{\mu }\to 0$ (thick dashed line).

Figure 7

Thermodynamic potential $\widehat{\mathrm{\Xi }}$ with $(\widehat{\mu },\widehat{G})=(0.5,10)$. A first-order phase transition occurs as $\widehat{\mathrm{\Delta }}$ is increased.

Figure 8

Condensate $⟨\widehat{V}⟩$ with $(\widehat{\mu },{\widehat{\mu }}_{\xi },\widehat{G})=(0.5,0,10)$. The phase transition is first order except on the $T=0$ axis where it is second order [cf. Fig. 6].

Figure 9

Schematic plots for the temperature dependence of $\mathrm{\Delta }$ in real materials. The dashed line denotes the curve for $\mathrm{\Delta }(T)$ while the dark area indicates the domain with a strong Kondo effect. Which of these possibilities is realized depends on microscopic details of the material and the impurity. The backreaction of impurities on the superconducting gap is neglected, assuming a sufficiently low density of impurities.

Figure 10

Spectrum of mid-gap excited states numerically obtained from (40) with $v\mathrm{\Lambda }/\mu =2$, in comparison with Shiba’s formula (45). Negative energy levels (not shown) appear symmetrically about the horizontal axis.

Figure 11

Dispersion relations of quasiparticles $\{\widehat{\omega }=\pm {\widehat{E}}_{\ell }(\widehat{k}),\ell =1,\dots ,6\}$ for $\widehat{\mu }=0.5$ and ${\widehat{\mu }}_{\xi }=-0.1$. (a) $(\widehat{\mathrm{\Delta }},\widehat{V})=(0.05,0)$, (b) $(\widehat{\mathrm{\Delta }},\widehat{V})=(0,0.05)$, (c) $(\widehat{\mathrm{\Delta }},\widehat{V})=(0.05,0.05)$.

Figure 12

(a) Thermodynamic potential $\widehat{\mathrm{\Xi }}$ and (b) condensate $⟨\widehat{V}⟩$ in the two-color two-flavor model with $(\widehat{\mu },{\widehat{\mu }}_{\xi },\widehat{G})=(0.5,0,5)$. The phase transition is first order except when $\widehat{T}=0$.

Figure 13

Spectrum of mid-gap excited states for the potential $U=\delta (\mathbf{x})(u_1{t}^3+u_2{\sigma }^3)$ in comparison with Shiba’s formula (45). Each level is doubly degenerate. Negative energy levels (not shown) appear symmetrically about the horizontal axis.

Figure 14

Feynman rules for the model (a3). The last two vertices are counterterms.

Figure 15

Feynman diagrams contributing to the $\beta$ function of $G$ at $\mathcal{O}(G^3)$. The diagram (c) is the leading one at $N_f\gg 1$.