Impurity-induced quantum phase transitions and magnetic order in conventional superconductors: Competition between bound and quasiparticle states

We theoretically study bound states generated by magnetic impurities within conventional $s$-wave superconductors, both analytically and numerically. In determining the effect of the hybridization of two such bound states on the energy spectrum as a function of magnetic exchange coupling, relative angle of magnetization, and distance between impurities, we find that quantum phase transitions can be modulated by each of these parameters. Accompanying such transitions, there is a change in the preferred spin configuration of the impurities. Although the interaction between the impurity spins is overwhelmingly dominated by the quasiparticle contribution, the ground state of the system is determined by the bound-state energies. Self-consistently calculating the superconducting order parameter, we find a discontinuity when the system undergoes a quantum phase transition as indicated by the bound-state energies.

Figure 1

Our setup of two magnetic impurities at ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$ in an $s$-wave superconductor with classical spins ${\mathbf{S}}_1$ and ${\mathbf{S}}_2$, respectively, oriented at a relative angle $\theta$. As a result of the magnetic exchange couplings, $J_1$ and $J_2$, YSR bound states form within the bulk gap ${\mathrm{\Delta }}_0$. When the distance between the impurities, $r$, is larger than the coherence length of the superconductor the energies are $E_1$ and $E_2$ but get changed to ${\epsilon }_1$ and ${\epsilon }_2$ as $r$ decreases and the bound states hybridize with each other.

Figure 2

Energy of the YSR bound states for the identical magnetic impurities oriented ferromagnetically (solid and dashed) and antiferromagnetically (dotted) as well as the energy difference $\delta \mathcal{E}$ (thick solid lines) as a function of the distance $r$ between impurities. When $\alpha =0.5$ (top panel), the system remains antiferromagnetic $\left(\delta \mathcal{E}>0\right)$, while for $\alpha =0.9$ (lower panel), the magnetic configuration oscillates between being antiferromagnetic and ferromagnetic. For convenience, these two configurations are separated by the vertical dotted lines.

Figure 3

The energy of the bound states $({\epsilon }_1,{\epsilon }_2)$ and the total YSR state energy $\left[\mathcal{E}\right(\theta \left)\right]$ as a function of relative angle $\theta$ for $k_{F}r=1,{\alpha }_1=0.5$, and ${\alpha }_2=1$. The change of quantum ground state at $\theta \approx \pm \pi /2$ is indicated by vertical dotted lines.

Figure 4

The difference in the energy $\delta {\mathcal{E}}_{gr}$ between ferro- and antiferromagnetic configurations of the system consisting of two identical impurities of coupling strength $\overline{J}$ as a function of the distance between impurities, $r/a$, for (a) $\overline{J}/t=1$ and (b) $\overline{J}/t=2.5$ found self-consistently (red dots) and not self-consistently (blue dots), respectively. Inset: enlarged area of (a) for large distances. The difference in the energy $\delta \mathcal{E}$ between ferro- and antiferromagnetic configurations including only the YSR bound state (green dots) is found self-consistently. The parameters used are $N_x\times N_y=33\times 25,\mu /t=1$, and ${\mathrm{\Delta }}_0/t=0.1$.

Figure 5

The total energy of the ground state for two identical impurities $\left(\overline{J}={\overline{J}}_1={\overline{J}}_2\right)$ as a function of the angle $\theta$ between magnetic moments found numerically for (a) $\overline{J}/t=2.17$ (ferromagnetic ordering) and (b) $\overline{J}/t=2$ (antiferromagnetic ordering) at the distance $r/a=6$. We additionally plot (c) $\mathcal{E}\left(\theta \right)$ for $\overline{J}/t=2.17$ also at $r/a=6$. Other parameters are the same as in Fig. 4.