Multichannel Kondo Effect in Superconducting Leads

TL;DR

The work presents an exactly solvable model of a spin $1/2$ impurity coupled to $n$ spin-singlet superconducting leads, revealing four distinct boundary phases arising from the competition between Kondo screening and bulk superconductivity. Using an exact Bethe Ansatz solution complemented by perturbative RG and thermodynamic Bethe Ansatz, the authors show that even with a gapped bulk, the impurity can exhibit overscreened non-Fermi-liquid behavior controlled by the $SU(2)_n$ WZW fixed point, and that the impurity entropy approaches fractional values $S_{ ext{imp}}(0)= ext{ln}[2 ext{cos}(rac{ ext{π}}{n+2})]$ in the Kondo and zero-mode phases, while yielding $S_{ ext{imp}}(0)= ext{ln} 2$ in YSR and unscreened phases. The phase diagram is organized by an RG-invariant parameter $d$, with four boundary phases distinguished by the structure of excitation towers and the presence of mid-gap bound states; the results generalize conventional multichannel Kondo physics to gapped superconducting environments and offer precise predictions for experimental signatures such as impurity entropy, Kondo scales, and subgap excitations. This work advances the understanding of impurity criticality in correlated, gapped hosts and provides a framework for exploring boundary conformal data in nonconformal settings through exact solutions.

Abstract

The traditional multichannel Kondo effect takes place when several gapless metallic electronic channels interact with a localized spin-$S$ impurity, with the number of channels $n$ exceeding the size of the impurity spin, $n>2S$, leading to the emergence of non-Fermi liquid impurity behavior at low temperatures. Here, we show that the effect can be realized even when the electronic degrees of freedom are strongly correlated and gapped. The system under consideration consists of a single spin-$\frac{1}{2}$ impurity coupled isotropically to $n$ spin singlet superconducting channels realized by one-dimensional leads with quasi-long-range superconducting order. The competition between the Kondo and superconducting fluctuations induces multiple distinct ground states and boundary phases depending on the relative strengths of the bulk and boundary interactions. Using the Bethe Ansatz technique, we identify four regimes: an overscreened Kondo phase, a zero-mode phase, a Yu-Shiba-Rusinov (YSR) phase, and a local-moment phase with an unscreened impurity, each with its own experimental characteristic. We describe the renormalization-group flow, the excitation spectrum, and the full impurity thermodynamics in each phase. Remarkably, even in the presence of a bulk mass gap, the boundary critical behavior in the Kondo phase is governed by the same exponents as in the gapless theory with the low-energy impurity sector flowing to the $SU(2)_n$ Wess-Zumino-Witten (WZW) fixed point, and the impurity entropy monotonically decreasing as a function of temperature. In both the overscreened Kondo and zero-mode phases, the residual impurity entropy is $S_{\mathrm{imp}}(T \to 0) = \ln[2\cos(π/(n+2))]$. In the YSR and unscreened phases on the other hand the impurity entropy exhibits non-monotonic temperature dependence and is effectively free at low temperatures with $S_{\mathrm{imp}}(T \to 0) = \ln 2$.

Figures (8)

• Figure 1: Experimentally realizable schematic of $n$ superconducting leads coupled (shown in red) isotropically via the Kondo exchange $J$ to a single spin-$\frac{1}{2}$ impurity (shown in green). Dashed lines indicate impurity–lead coupling, while braces labeled $g$ denote attractive spin–density-spin-density interactions between electrons of opposite chiralities.
• Figure 2: Phase diagram of the $n$-channel superconducting model with a spin-$\frac{1}{2}$ boundary impurity and even bulk particle number $N$ per flavor. The horizontal axis represents the RG-invariant parameter $d(J,g)$, real for $d\ge 0$ and purely imaginary $d=i\delta$ with $\delta>0$ to the right. For real $d$ or $0<\delta<\frac{1}{2}$ (Kondo phase), the impurity is overscreened by a multiparticle cloud, and the spectrum consists of a single continuous tower of eigenstates built upon the overscreened ground state. For $\frac{1}{2}<\delta<1$ (Zero-mode Phase I), the emergence of a purely imaginary boundary root $\lambda_d$ splits the spectrum into two distinct towers corresponding to configurations where the boundary mode is unoccupied or occupied. For $1<\delta<\frac{n}{2}$ (Zero-mode Phase II), higher-order boundary strings $\lambda_{\delta,\ell}^{(p)}=\lambda_\delta+i\ell$ with $\ell=1,\dots,p$ and $p=\lfloor\delta+\frac{1}{2}\rfloor$ appear, producing three towers of excitations. For $\frac{n}{2}<\delta<\frac{n}{2}+1$ (YSR phase), a quantum phase transition occurs: the impurity becomes unscreened in the ground state, the bound state acquires finite energy, and screening occurs only in the excited sector. Three towers persist, but some are lifted in energy due to hybridization with mid-gap states; the nature of this lifting differs between YSR Phase I ($\frac{n}{2}<\delta<\frac{n}{2}+\frac{1}{2}$) and YSR Phase II ($\frac{n}{2}+\frac{1}{2}<\delta<\frac{n}{2}+1$). For $\delta>\frac{n}{2}+1$, screening is impossible at any scale, and the system resides in the local moment phase, where the impurity spin remains asymptotically free and the three towers persist as asymptotically decoupled excitation sectors.
• Figure 3: Representative $\eta_p(\lambda)$ functions for small string indices $p$ in the two-channel ($n=2$) and three-channel ($n=3$) Gross--Neveu models. In both cases, $\eta_p(\lambda)$ decreases monotonically with increasing $p$, and the sequence terminates at the channel number, such that $\eta_{p=n}(\lambda)\to 0$. This behavior reflects the exact truncation of the $\eta$-string hierarchy at $p=n$, characteristic of the $n$-channel impurity problem.
• Figure 4: Impurity entropy $S_{\mathrm{imp}}$ in the overscreened Kondo phase as a function of the universal scaling variable $T/m$. (a) Representative behavior of the impurity entropy for the six-channel superconducting bulk. Colors denote the RG-invariant parameter $d$, demonstrating the universal scaling relation $S_{\mathrm{imp}} = S_{\mathrm{imp}}(d, T/m)$, which takes the value $S_{\mathrm{imp}}(T\to 0)=\ln(\sqrt{2+\sqrt{2}})$ as $T$ approaches zero and asymptotically reaches to $S_{\mathrm{imp}}(T\to \infty)=\ln 2$ at infinite temperature. (b) For the representative impurity parameter $d=0.25$, $S_{\mathrm{imp}}(T)$ is shown for different numbers of superconducting channels $n$, exhibiting the crossover from $S_{\mathrm{imp}}(0)=\ln \bigl[2\cos \bigl(\frac{\pi}{n+2}\bigr)\bigr]$ to $S_{\mathrm{imp}}(\infty)=\ln 2$.
• Figure 5: Impurity entropy $S_{\mathrm{imp}}$ in the Zero-mode phase as a function of the universal scaling variable $T/m$. (a) Representative behavior of the impurity entropy for the four-channel superconducting bulk. Colors denote the RG-invariant parameter $d$, demonstrating the universal scaling relation $S_{\mathrm{imp}} = S_{\mathrm{imp}}(d, T/m)$, which takes the value $S_{\mathrm{imp}}(T\to 0)=\ln\sqrt{3}$ as $T$ approaches zero and asymptotically reaches to $S_{\mathrm{imp}}(T\to \infty)=\ln 2$ at infinite temperature. When $\delta=0.6$ and $\delta=0.9$, the model is in Zero-mode Phase I where there are two towers and hence the impurity is computed from the sum of two towers given by the free energies in Eq.\ref{['T1contZM']} and Eq.\ref{['T2contZM']}, whereas for $\delta=1.35$ and $\delta=1.75$, the model is in Zero-mode phase II where there are three towers and the free energy contributions from the three towers given in Eq.\ref{['eq:Fimp_all']} are used to compute the impurity entropy. (b) For the representative impurity parameter $\delta=0.75$, $S_{\mathrm{imp}}(T)$ is shown for different numbers of superconducting channels $n$, exhibiting the crossover from $S_{\mathrm{imp}}(0)=\ln \bigl[2\cos \bigl(\frac{\pi}{n+2}\bigr)\bigr]$ to $S_{\mathrm{imp}}(\infty)=\ln 2$. Here, since $\delta<1$, the impurity entropy comes from the combined contribution of two towers.