Kondo Signatures of a Quantum Magnetic Impurity in Topological Superconductors

We study the Kondo physics of a quantum magnetic impurity in two-dimensional topological superconductors (TSCs), either intrinsic or induced on the surface of a bulk topological insulator, using a numerical renormalization group technique. We show that, despite sharing the $p+ip$ pairing symmetry, intrinsic and extrinsic TSCs host different physical processes that produce distinct Kondo signatures. Extrinsic TSCs harbor an unusual screening mechanism involving both electron and orbital degrees of freedom that produces rich and prominent Kondo phenomena, especially an intriguing pseudospin Kondo singlet state in the superconducting gap and a spatially anisotropic spin correlation. In sharp contrast, intrinsic TSCs support a robust impurity spin doublet ground state and an isotropic spin correlation. These findings advance fundamental knowledge of novel Kondo phenomena in TSCs and suggest experimental avenues for their detection and distinction.

Figure 1

(a) Phase diagram of $\delta E$ in $\mu \text{−}U$ space for extrinsic TSCs. Energy unit is set by the cutoff in Eq. (10), $\mathrm{\Delta }=0.1$, and $U$ is reduced by $\pi {\xi }_0$, where ${\xi }_0=\pi V^2/2$. The transition points where $\delta E=0$ are highlighted by the dark-blue line. (b) $\delta E$ versus $U$ at select $\mu$ in extrinsic TSCs ($U_c/\pi {\xi }_0=0.11$ and 0.32 for $\mu =0.0$ and 0.2, respectively), compared with results for intrinsic TSC ($U_c=0$) and $s$-wave SC ($U_c/\pi {\xi }_0=1.44$). NRG parameters used are $N=400$, $N_z=10$, $\mathrm{\Lambda }=2.5$, $L_{\max }=25$.

Figure 2

(a) Phase diagram of $M_{\text{imp}}$ in $\mu \text{−}U$ space for extrinsic TSCs (same energy units as in Fig. 1 and $\mathrm{\Delta }=0.1$). $U_c(\mu )$ sets the phase boundary between $M_{\text{imp}}=0$ and $M_{\text{imp}}=1/2$, and $\mathrm{\Gamma }(\mu )$ is a crossover boundary between the RS and PKS regimes. Both quantities are reduced by $\pi {\xi }_0$, as indicated by a bar over each of them. Representative impurity LDOS at $\mu =0.2$ for (b) the RS, PKS, and DGS ($S_f^z=1/2$) regimes of extrinsic TSCs and (c) intrinsic TSCs. $\omega =0$ is at the chemical potential $\mu$ of the normal state. Spectral features are broadened in a log-Gaussian scheme with a width factor $b=0.01$ [63]. Contour plots of spin correlation function $C^x(\mathbf{r})$ in the PKS regime for extrinsic TSCs and in the corresponding DGS regime for intrinsic TSCs are shown as insets. Contour values are -0.10 to -0.07 with a step of 0.01 outward. NRG parameters used are $N=600$, $N_z=10$, $\mathrm{\Lambda }=2.5$, $L_{\max }=25$.

Figure 3

The scaling behavior of $T{\chi }_{\text{imp}}(T)$ for the impurity in (a)–(c) extrinsic and (d) intrinsic TSCs. (a) At $\mathrm{\Delta }=0$, the RG flow goes to a LM fixed point with $M_{\text{imp}}=1/2$ for $\mu =0$ and, when $\mu \ne 0$, to KS and RS fixed points with $M_{\text{imp}}=0$ for $U\ne 0$ and $U=0$, respectively. (b) At $\mathrm{\Delta }\ne 0$ and $\mu =0$, the RG flow goes to $M_{\text{imp}}=1/2$ (LM) and 0 (PKS) fixed points for $U>U_c$ and $U<U_c$, respectively. (c) At $\mathrm{\Delta }\ne 0$ and $\mu \ne 0$, an additional RS fixed point with $M_{\text{imp}}=0$ appears. (d) RG flows go to the LM fixed point for $U\ne 0$ at all $\mu$. NRG parameters used are $N=600$, $N_z=10$, $\mathrm{\Lambda }=1.8$, $L_{\max }=15$.