Quantum Yu-Shiba-Rusinov dimers

Magnetic adatoms on a superconducting substrate undergo a quantum phase transition as their exchange coupling to the conduction electrons increases. For quantum spins, this transition is accompanied by screening of the adatom spin. Here, we explore the consequences of this screening for the phase diagrams and subgap excitation spectra of dimers of magnetic adatoms coupled by hybridization of their Yu-Shiba-Rusinov states and spin-spin interactions. We specifically account for higher spins, single-ion anisotropy, Ruderman-Kittel-Kasuya-Yosida coupling, and Dzyaloshinsky-Moriya interactions relevant in transition-metal and rare-earth systems. Our flexible approach based on a zero-bandwidth approximation provides detailed physical insight and is in excellent qualitative agreement with available numerical-renormalization group calculations on monomers and dimers. Remarkably, we find that even in the limit of large impurity spins or strong single-ion anisotropy, the phase diagrams for dimers of quantum spins remain qualitatively distinct from phase diagrams based on classical spins, highlighting the need for a theory of quantum Yu-Shiba-Rusinov dimers.

Figure 1

Phase diagrams of a spin-$1/2$ YSR dimer as a function of RKKY coupling $J_z$ and YSR energy $E_{\text{YSR}}$ for different anisotropies of the exchange coupling $K_{\perp }/K_z$. The anisotropy of the RKKY interaction is chosen to match the exchange anistropy, $K_{\perp }/K_z=J_{\perp }/J_z$. Phase diagrams are obtained from the zero-bandwidth Hamiltonian in Eq. (5). The color scale (see scale bar) indicates the expectation value of the number of bound quasiparticles $F$ as defined in Eq. (9). Black dashed lines indicate phase boundaries, at which the spin and/or fermion parity quantum numbers of the ground state change discontinuously. (a) Ising exchange with nonzero $K_z$ and $K_{\perp }=0$, corresponding to a classical-spin model of the adatom. Phase boundaries and crossovers essentially depend only on the sign of the RKKY coupling, reflecting the absence of screening of the adatom spin in the classical model. (b) Heisenberg exchange $K=K_{\perp }=K_z$. Phase boundaries and crossovers depend on the magnitude of the RKKY coupling. This phase diagram qualitatively reproduces the results of NRG simulations in Ref. [48]. (c) Dominant longitudinal and (d) dominant transverse anisotropic couplings as indicated in the panel. The singly screened phase (white) reduces in extent as $K_{\perp }/K_z$ increases. Parameters: $(K_z,K_{\perp })=100t(cos\theta ,sin\theta )$, (a) $\theta =0$, (b) $\theta =\frac{\pi }{4}$, (c) $\theta =0.05$, and (d) $\theta =\frac{\pi }{2}-0.1$, $V=0.9(K_z+2K_{\perp })/4$.

Figure 2

Illustrative level scheme of spin-$1/2$ dimers with isotropic exchange and ferromagnetic RKKY coupling $(J<0)$. The low-energy spectrum of the uncoupled dimer (left) is labeled by the fermion parities $P_j$ and effective spins $S_{\mathrm{eff},j}$ of the monomers as $(P_1,S_{\mathrm{eff},1})(P_2,S_{\mathrm{eff},2})$. For $E_{\text{YSR}}<0$ the local singlet state ($F\simeq 2$, blue) is the ground state. Nonzero RKKY interaction (center) couples the monomer states into states of total parity and spin $(P_{\text{tot}},S_{\text{tot}})$. This affects only the unscreened state ($F\simeq 0$, red), which splits into molecular singlet and triplet. For sufficiently large $\left|J\right|$, the molecular triplet becomes the ground state. Finally, hybridization of the YSR states splits the odd-fermion-parity states ($F=1$, black) into symmetric and antisymmetric states. For sufficiently large hybridization $\tilde{t}$ this leads to the singly screened ground state.

Figure 3

Excitation spectra of a spin-$1/2$ dimer as a function of YSR energy $E_{\text{YSR}}$ for isotropic (a) ferromagnetic (FM) and (b) antiferromagnetic (AFM) RKKY coupling $J$. Tunneling excitations (purple arrows) flip fermion parity and change total spin by $\pm 1/2$. In the even-fermion-parity phases, tunneling is possible into the symmetric and antisymmetric doublets. In the doublet phase, all (low-energy) even parity states are in principle accessible by tunneling, resulting in three peaks (or more for anisotropic RKKY interaction, see Fig. 4). Parameters as in Fig. 1 with fixed RKKY coupling $J=-2t$ (a) and $J=2t$ (b), as indicated by gray dashed lines in Fig. 1.

Figure 4

Tunneling spectroscopy of spin-$1/2$ dimers in the doublet phase with isotropic exchange and ferromagnetic RKKY coupling. (a) Spectral functions for different symmetries of the RKKY coupling (see legend; offset for clarity). (b) Level schemes (not to scale) for Heisenberg, XXZ and XYZ RKKY coupling, emphasizing the increase in the number of resonances as the spin rotation symmetry is reduced. Parameters: $E_{\mathrm{YSR}}=-0.2t$, $\mathrm{\Delta }={10}^{6}t$, $V=3\mathrm{\Delta }$, $J=-0.3t$, Heisenberg: $\widehat{J}=J\mathrm{diag}(1,1,1)$, XXZ: $\widehat{J}=J\sqrt{3/2}\mathrm{diag}(1,1,0)$, and XYZ: $\widehat{J}=J\mathrm{diag}(\sqrt{2},1,0)$.

Figure 5

Effect of Heisenberg RKKY coupling on YSR dimers with higher spins $S$. (a),(b) Phase diagrams as a function of isotropic RKKY coupling $J$ and YSR energy $E_{\mathrm{YSR}}$ for $S$ as indicated in the panel. As for spin-$1/2$ dimers, RKKY coupling stabilizes unscreened dimers with $F\simeq 0$ (red). Unlike spin-$1/2$ dimers, the phase boundary between the approximately doubly screened $S_{\mathrm{tot}}=2S-1$ (blue, $F\simeq 2$) and singly screened $S_{\mathrm{tot}}=2S-1/2$ phases (white, $F=1$) is no longer approximately constant as partially screened dimers gain RKKY energy, see (c). As $S$ increases, the phase boundaries on the ferromagnetic side become more and more parallel. Gray dashed lines indicate value of the RKKY coupling in Fig. 6. (c) Level scheme for higher spins. RKKY coupling shifts the energies not only of the unscreened dimer, but also of the singly and doubly screened dimer. Parameters: $V=2\mathrm{\Delta }$, $\mathrm{\Delta }={10}^{6}t$.

Figure 6

Excitation spectra of YSR dimers for adatom spins (a) $S=1$ and (b) $S=3/2$ with isotropic ferromagnetic RKKY coupling. Allowed tunneling excitations (purple arrows) are determined by the selection rules $P_{\mathrm{tot}}{P}_{\mathrm{tot}}^{\prime }=-1$ and $S_{\mathrm{tot}}^z-{\left(S_{\mathrm{tot}}^z\right)}^{\prime }=\pm 1/2$. (a) For $S=1$, the selection rules permit four possible tunneling excitations from the (approximately) doubly screened ground state (blue, $F\simeq 2$), three from the singly screened (black), while only two survive when the ground state is unscreened (red, $F=0$). (b) $S=3/2$ shows corresponding behavior. Parameters: $K=100t$, $V=70t$, $J=-2t$, as indicated by gray dashed lines in Fig. 5 and 5.

Figure 7

Tunneling spectroscopy of spin-1 dimers in the doubly screened region with isotropic exchange and ferromagnetic RKKY coupling. (a) Spectral functions for different symmetries of the RKKY coupling (offset for clarity). For Heisenberg RKKY coupling (purple), the resonances originate from tunneling into the symmetric and antisymmetric quartet states $(+,\frac{3}{2})$. Tunneling into the symmetric and antisymmetric doublet states $(-,\frac{1}{2})$ is allowed, but beyond the energy range shown here (see Fig. 6). Upon breaking spin rotation symmetry, the symmetric and antisymmetric quartets split into two peaks each. The odd-parity states remain degenerate even for XYZ coupling as they are protected by time reversal symmetry. (b) Corresponding level schemes (not to scale). Parameters: $E_{\mathrm{YSR}}=-t$, $\mathrm{\Delta }={10}^{6}t$, $V=2\mathrm{\Delta }$, Heisenberg: $\widehat{J}=-\mathrm{diag}(t,t,t)/\sqrt{3}$, XXZ: $\widehat{J}=-\mathrm{diag}(t/2,t/2,t/\sqrt{2})$, and XYZ: $\widehat{J}=-\mathrm{diag}(0.4t,0.58t,t/\sqrt{2})$.

Figure 8

Phase diagrams for YSR dimers with higher spins $S$ (see panels) including single-ion anisotropy. The phases are labeled by $(P_{\mathrm{tot}},S_{\mathrm{tot}}^z,\mathrm{\Sigma })$, with $\mathrm{\Sigma }$ indicating the spatial parity of the dimer. Note that $E_{\mathrm{YSR}}=E_o-E_e$ contains single-ion anisotropy, as defined in Appendix pp2-s1. [(a)–(c)] Easy-axis anisotropy $D<0$. For ferromagnetic RKKY coupling, the phase diagrams are unchanged compared to Fig. 5 for isotropic couplings. For antiferromagnetic RKKY coupling, the phase boundaries are affected for $J\lesssim \left|D\right|$. [(d) and (e)] Easy-plane anisotropy $D>0$. The phase diagrams are strongly modified for $\left|J\right|\lesssim D$ with significant difference between integer and half-integer spins. For integer spins, weak RKKY coupling favors the screened phase, while for half-integer spins it strongly favors the unscreened phase. Integer spins have different spatial parity for the doubly screened phase, but equal parity in the unscreened phase. For half-integer spins, the roles are reversed. Parameters: $V=2\mathrm{\Delta }$, $\mathrm{\Delta }={10}^{6}t$, [(a)–(c)] $D=-10t$, and [(d)–(f)] $D=5t$.

Figure 9

Classical spin alignment for different screening scenarios in the presence of anisotropic coupling, easy-axis single-ion anisotropy, and DM interaction, as outlined in Sec. 3c. (a) Unscreened spins ${\mathbf{S}}_1$ and ${\mathbf{S}}_2$ align along the $z$ direction due to easy-axis single-ion anisotropy $D$. (b) A screened spin (here ${\mathbf{S}}_2$) aligns perpendicular to the $z$ direction due to transverse exchange coupling $K_{\perp }$, which dominates over the single-ion anisotropy. The singly screened case is favored by DM interactions with $\mathbf{M}$ lying in the $xy$ plane. (c) Two screened spins align in the $xy$ plane (here antiferromagnetically), associated with a gain in energy due to the transverse RKKY coupling. Quantum effects modify the classical picture, see Fig. 10.

Figure 10

Phase diagrams for YSR dimers with transverse ($xy$) couplings $K_{\perp }$ and $J_{\perp }$, easy-axis anisotropy $D$, and DM interaction $M$. [(a)–(c)] For vanishing DM interaction, the fermion parity $P_{\mathrm{tot}}$, the spin projection $S_{\mathrm{tot}}^z$, and the spatial parity $\mathrm{\Sigma }$ are good quantum numbers. For integer spin, small RKKY coupling $J_{\perp }\ll \left|D\right|$ favors the screened spin phases as expected from classical considerations, see Fig. 9. In contrast, for half-integer spins the screened phase is suppressed. Note also the alternation in phase boundaries associated with changes in spatial parity $\mathrm{\Sigma }$. The singly screened phase (white) is increasingly suppressed for higher spins due to the different spin alignments of unscreened and screened monomers, see Fig. 9. [(d) and (e)] For nonzero DM interaction with $\mathbf{M}=M\widehat{\mathbf{y}}$ orthogonal to the single-ion anisotropy, only $P_{\mathrm{tot}}$ and the discrete spin rotation $\mathrm{\Pi }$ remain good symmetries. DM interaction drastically enhances the singly screened phase (white) for integer spins, while it merely shifts the phase boundary/crossover for half-integer spins. Parameters: $V=2\mathrm{\Delta }$, $\mathrm{\Delta }={10}^{6}t$, $D=-15t$, [(a)–(c)] $M=0$, and [(d)–(f)] $M=5t$.

Figure 11

Phases of a $S=2$ YSR dimer with isotropic exchange coupling and weak isotropic RKKY interaction $J$. (a) and (c) show number of bound quasiparticles $F$ and (b) and (d) show total spin $S_{\mathrm{tot}}$ as a function of $J$ and $E_{\text{YSR}}$. Parameters: $\mathrm{\Delta }={10}^{6}t$, [(a) and (b)] $V=20\mathrm{\Delta }$, and [(c) and (d)] $V=0.5\mathrm{\Delta }$.

Figure 12

Phase diagrams for a spin-$3/2$ dimer with transverse ($xy$) couplings $K_{\perp }$ and $J_{\perp }$, easy-axis anisotropy $D$, and DM interaction $M$, see Sec. 3c, in the intermediate regime $S\left|D\right|\ll K_{\perp }\lesssim 4S^3\left|D\right|$. (a) $M=0$. Note that in contrast to Fig. 10 transverse RKKY coupling stabilizes the doubly screened dimer as the energy balance inverts for $K_{\perp }\lesssim 4S^4\left|D\right|$. (b) $M=3t$. In contrast to Fig. 10 a singly screened phase (white) arises. This requires $K_{\perp }\lesssim 4S^3\left|D\right|$, see Appendix pp2-s5. Parameters: $\mathrm{\Delta }=200t$, $V=2\mathrm{\Delta }$, and $D=-15t$.