Non-Fermi-liquid behavior in quantum impurity models with superconducting channels

We study how the non-Fermi-liquid nature of the overscreened multichannel Kondo impurity model affects the response to a BCS pairing term that, in the absence of the impurity, opens a gap $\mathrm{\Delta }$. We find that the low-energy spectrum in the limit $\mathrm{\Delta }\to 0$ actually does not correspond to the spectrum strictly at $\mathrm{\Delta }=0$. In particular, in the two-channel Kondo model, the $\mathrm{\Delta }\to 0$ ground state is an orbitally degenerate spin singlet, while it is an orbital singlet with a residual spin degeneracy at $\mathrm{\Delta }=0$. In addition, there are fractionalized spin-1/2 subgap excitations whose energy in units of $\mathrm{\Delta }$ tends toward a finite and universal value when $\mathrm{\Delta }\to 0$, as if the universality of the anomalous power-law exponents that characterize the overscreened Kondo effect turned into universal energy ratios when the scale invariance is broken by $\mathrm{\Delta }\ne 0$. This intriguing phenomenon can be explained by the renormalization flow toward the overscreened fixed point and the gap cutting off the orthogonality catastrophe singularities. We also find other non-Fermi-liquid features at finite $\mathrm{\Delta }$: the local density of states lacks coherence peaks, the states in the continuum above the gap are unconventional, and the boundary entropy is a nonmonotonic function of temperature. The persistent subgap excitations are characteristic of the non-Fermi-liquid fixed point of the model, and thus depend on the impurity spin and the number of screening channels.

Figure 1

(a) Subgap many-particle states in the two-channel Kondo (2CK) model with a fixed gap as a function of $g={\rho }_{0}J$. The energies are given with respect to the ground-state energy. $S$ is the total spin quantum number of the many-particle state. The parity of states (even and odd) refers to the channel inversion symmetry (channel $k=1$ maps into channel $k=2$ and vice versa). There are two spin-singlet “Kondo-screened” states where the spin forms a Kondo cloud predominantly with one of the two channels and is essentially decoupled from the other. There are two different spin-doublet states: one where the impurity is decoupled from the channels, and another where it strongly couples to the neighboring orbitals from the two channels to form a three-site antiferromagnetic Heisenberg spin chain. The point where these two states have equal excitation energy defines the self-dual point at $g=g^{*}$. (b) Eigenstates in the zero-bandwidth limit. (c) Approach toward the asymptotic universal spectrum in the zero-gap limit: as $\mathrm{\Delta }\to 0$, the doublet states do not merge with the continuum as in Fermi-liquid systems, but they converge to a finite $E/\mathrm{\Delta }$ value that is characteristic of the non-Fermi-liquid fixed point.

Figure 2

Subgap $S=1/2$ many-particle states in the 2CK model for a range of $\mathrm{\Delta }$ forming a geometric sequence with ratio $\sqrt{10}$.

Figure 3

Renormalization flow diagram with different scalings of the energy axis for the 2CK model. $N$ is the iteration number and $\mathrm{\Lambda }=2$ is the NRG discretization parameter.

Figure 4

Subgap many-particle states in the three-channel Kondo (3CK) models with (a) $S=1/2$ and (b) $S=1$. Here $\mathrm{\Lambda }=5$.

Figure 5

Temperature dependence of (a) impurity magnetic susceptibility and (b) impurity entropy in the 2CK model.

Figure 6

Channel symmetry breaking, $J_1\ne J_2$, in the 2CK model. (a) Evolution of the subgap states vs $\mathrm{\Delta }J$; the crossover point $T^{*}\sim \mathrm{\Delta }$ occurs for $\mathrm{\Delta }J\approx {10}^{-3}$. (b) Impurity spectral function ($\mathrm{Im}$ part of the $T$ matrix) for a range of $\mathrm{\Delta }J$.