Shiba States in Superconductors

• Shiba states are discrete electronic levels emerging within a superconductor's gap due to magnetic impurities, exhibiting particle–hole symmetry and marking local quantum phase transitions.
• They are analyzed using T-matrix formalism, Green’s function techniques, and numerical renormalization to reveal signatures like gap suppression and crossover to Andreev states.
• Their behavior depends critically on host density of states, impurity coupling strength, anisotropy, and spatial hybridization, offering pathways to topological superconductivity.

Searching arXiv for recent and foundational work on Shiba states to ground the article. Shiba states, or Yu–Shiba–Rusinov (YSR) states, are discrete electronic levels that appear inside the superconducting gap of a superconductor when a magnetic—or, more generally, a localized interacting—impurity is coupled to it. In the standard isotropic continuum model with constant normal-state density of states $\rho\equiv \rho(E_F)$, a point-like impurity produces a pair of particle–hole-symmetric in-gap bound states whose energies are controlled by the exchange coupling and the host density of states; the zero-energy crossing of that pair marks a local quantum phase transition (Basak et al., 2022). From that canonical setting, the subject has expanded to include self-consistent gap renormalization, multiorbital and lattice-specific Shiba multiplets, hybridized dimer and chain states, driven and cavity-coupled regimes, and generalized forms in topological, fractionalized, and strongly disordered superconductors (Meng et al., 2015).

1. Canonical single-impurity problem

The minimal formulation starts from an $s$-wave BCS host and a local magnetic impurity. In Nambu space, a single classical spin at $r=0$ acts as a local scattering potential, and the exact Green’s function may be written in $T$-matrix form as $$G(\omega_n)=G_0(\omega_n)+G_0(\omega_n)T(\omega_n)G_0(\omega_n)$$, with the bound-state energies given by the poles of the $T$-matrix (Meng et al., 2015). In the standard continuum limit, neglecting nonmagnetic potential scattering, the in-gap energies are

$$E_{\rm YSR}=\pm \Delta\,\frac{1-\alpha^2}{1+\alpha^2}, \qquad \alpha=\pi\,\rho(E_F)\,J,$$

or equivalently $\alpha=\pi \rho_0 J S$ when the impurity spin magnitude is written explicitly (Basak et al., 2022).

This formula encodes the basic phenomenology. For $|\alpha|<1$, the Shiba state lies inside the gap, $-\Delta<E_0<\Delta$, and approaches zero energy as $s$0. The zero crossing at $s$1 is equivalently expressed as a critical coupling $s$2 in the constant-DOS model (Basak et al., 2022). The real-space Shiba wavefunction decays exponentially on the scale $s$3, with a spatial envelope determined by dimensionality and oscillatory structure set by the Fermi momentum (Westström et al., 2016).

The single-impurity solution also provides the building block for later generalizations. In dense or dilute arrays, each impurity contributes a localized Shiba orbital; in multiorbital settings, each scattering channel may generate its own pair of in-gap poles; and in quantum-impurity problems the same subgap excitation is reinterpreted as a many-body singlet–doublet excitation rather than merely a classical bound state (Arrachea, 2021).

2. Quantum phase transition, anisotropy, and thermal spectroscopy

In quantum-impurity formulations, the zero-energy crossing is a many-body level crossing. For MnPc on Pb(111), the exchange coupling strength determines whether the ground state is a Kondo singlet or a spin multiplet, and magnetocrystalline anisotropy splits the Shiba resonances into triplets described at lowest order by

$s$4

with splittings $s$5 for an effective spin-1 manifold (Hatter et al., 2015). In that system, the spectral weights of the three Shiba peaks distinguish the two sides of the quantum phase transition: for the Kondo-screened regime one finds $s$6, while in the free-spin regime the peak areas follow Boltzmann factors $s$7 (Hatter et al., 2015).

Beyond the classical-spin approximation, single-ion anisotropy and spin fluctuations substantially modify the Shiba energy. In the weak-coupling regime, easy-axis anisotropy $s$8 suppresses transverse spin dynamics and drives the result toward the classical formula when $s$9, whereas hard-axis anisotropy $r=0$0 yields qualitatively different low-energy behavior: for half-integer $r=0$1 it produces an effective $r=0$2 impurity with renormalized couplings, while for integer $r=0$3 the nondegenerate $r=0$4 ground state pushes the Shiba resonance toward the continuum edge $r=0$5 (Andrade et al., 2017). This establishes that anisotropy is not a secondary correction but a direct control parameter for subgap spectroscopy.

Finite temperature does not simply broaden away the Shiba state. Numerical renormalization group calculations for the single-impurity Anderson model coupled to superconducting contacts show that the impurity spectral function decomposes as

$r=0$6

where the $r=0$7-peak and a continuous subgap background coexist at any finite temperature (Zitko, 2016). The continuum arises from inelastic exchange scattering of thermally excited Bogoliubov quasiparticles off the impurity, with total weight activated as $r=0$8 at low $r=0$9, while the $T$0-peak itself persists to temperatures of order $T$1 (Zitko, 2016). In the strong-hybridization regime, additional narrow peaks just below the gap edge appear with activated weight $T$2, tentatively interpreted as high-order Shiba states (Zitko, 2016).

3. Spatial structure, self-consistent order-parameter renormalization, and the Shiba–Andreev crossover

The canonical formula treats the superconducting gap as externally fixed, but self-consistent treatments show that impurities renormalize the order parameter locally. For weak exchange, the gap correction at the impurity position satisfies $T$3 for $T$4, while for two impurities the orientation-dependent interference term scales as

$T$5

with the sign set by whether the impurity spins are parallel or antiparallel (Meng et al., 2015). Relative spin orientation therefore controls the sign and amplitude of the inter-impurity gap correction.

At strong exchange, the gap must be solved self-consistently. In a two-dimensional square-lattice Bogoliubov–de Gennes calculation, the local order parameter obeys $T$6, and the numerics show that $T$7 under each impurity is gradually suppressed, initially $T$8, then jumps through zero at a critical $T$9, producing a local $$G(\omega_n)=G_0(\omega_n)+G_0(\omega_n)T(\omega_n)G_0(\omega_n)$$0-junction (Meng et al., 2015). Between two impurities, the gap is further renormalized and can even be enhanced relative to $$G(\omega_n)=G_0(\omega_n)+G_0(\omega_n)T(\omega_n)G_0(\omega_n)$$1 for certain $$G(\omega_n)=G_0(\omega_n)+G_0(\omega_n)T(\omega_n)G_0(\omega_n)$$2’s (Meng et al., 2015).

This self-consistent renormalization changes the very character of the bound state. When the local renormalized gap under the impurity becomes smaller than the Shiba energy $$G(\omega_n)=G_0(\omega_n)+G_0(\omega_n)T(\omega_n)G_0(\omega_n)$$3, the bound level lies above the local gap and becomes an Andreev state delocalized in the gap-suppressed region (Meng et al., 2015). Numerically, Shiba states are tightly localized directly at the impurity site, whereas Andreev states extend over the entire region where $$G(\omega_n)=G_0(\omega_n)+G_0(\omega_n)T(\omega_n)G_0(\omega_n)$$4 is suppressed, with a maximum sometimes between impurities (Meng et al., 2015). In proximitized molecular junctions, a related distinction appears in a different form: the excited state is mediated by correlated electron-hole (Andreev) pairs instead of Cooper pairs, yet the spectroscopy still exhibits the characteristic singlet–doublet Shiba structure (Island et al., 2016).

4. Band-structure control: singular density of states, sublattices, and unconventional hosts

The dependence of Shiba states on $$G(\omega_n)=G_0(\omega_n)+G_0(\omega_n)T(\omega_n)G_0(\omega_n)$$5 makes them acutely sensitive to host band structure. Across honeycomb, kagome, and Lieb lattices, the generic behavior remains $$G(\omega_n)=G_0(\omega_n)+G_0(\omega_n)T(\omega_n)G_0(\omega_n)$$6 with $$G(\omega_n)=G_0(\omega_n)+G_0(\omega_n)T(\omega_n)G_0(\omega_n)$$7 and $$G(\omega_n)=G_0(\omega_n)+G_0(\omega_n)T(\omega_n)G_0(\omega_n)$$8, but singular or vanishing densities of states produce drastic shifts in scale (Basak et al., 2022). At a Dirac point, where $$G(\omega_n)=G_0(\omega_n)+G_0(\omega_n)T(\omega_n)G_0(\omega_n)$$9, $T$0 diverges; near a Van Hove singularity, where $T$1 is large but finite, $T$2 is reduced moderately; and at a flat band, where $T$3, $T$4 (Basak et al., 2022). In the Lieb lattice, this tuning becomes strongly sublattice dependent: for an impurity on a corner site at $T$5, $T$6 is large because the flat-band projected DOS on the corner vanishes, whereas for an impurity on an edge site, $T$7 is very small because the flat-band projected DOS on edges diverges (Basak et al., 2022).

A complementary analytic treatment of two-dimensional $T$8-wave superconductors with Van Hove singularities and high-order Van Hove singularities shows that the functional form of the Shiba energy remains the same as in the constant-DOS case, but the effective couplings are enhanced by the singular DOS (Uldemolins et al., 2021). In that framework, the critical magnetic coupling is lowered by a factor $T$9, and the slope $$E_{\rm YSR}=\pm \Delta\,\frac{1-\alpha^2}{1+\alpha^2}, \qquad \alpha=\pi\,\rho(E_F)\,J,$$0 at the Shiba transition depends only on the bulk DOS factor $$E_{\rm YSR}=\pm \Delta\,\frac{1-\alpha^2}{1+\alpha^2}, \qquad \alpha=\pi\,\rho(E_F)\,J,$$1, not on $$E_{\rm YSR}=\pm \Delta\,\frac{1-\alpha^2}{1+\alpha^2}, \qquad \alpha=\pi\,\rho(E_F)\,J,$$2 or $$E_{\rm YSR}=\pm \Delta\,\frac{1-\alpha^2}{1+\alpha^2}, \qquad \alpha=\pi\,\rho(E_F)\,J,$$3 themselves (Uldemolins et al., 2021). The same analysis identifies a large-$$E_{\rm YSR}=\pm \Delta\,\frac{1-\alpha^2}{1+\alpha^2}, \qquad \alpha=\pi\,\rho(E_F)\,J,$$4 regime in which the transition becomes controlled by the nonmagnetic scattering amplitude $$E_{\rm YSR}=\pm \Delta\,\frac{1-\alpha^2}{1+\alpha^2}, \qquad \alpha=\pi\,\rho(E_F)\,J,$$5 rather than by the exchange coupling $$E_{\rm YSR}=\pm \Delta\,\frac{1-\alpha^2}{1+\alpha^2}, \qquad \alpha=\pi\,\rho(E_F)\,J,$$6 (Uldemolins et al., 2021).

Hydrogenated graphene realizes a more radical departure from the conventional paradigm. There, Yu–Shiba–Rusinov states can occur without a finite density of states at the Fermi energy and without magnetic atoms, because atomic hydrogen chemisorbed in graphene creates a weakly localized spin-$$E_{\rm YSR}=\pm \Delta\,\frac{1-\alpha^2}{1+\alpha^2}, \qquad \alpha=\pi\,\rho(E_F)\,J,$$7 moment and a defect-induced resonance near zero energy (Lado et al., 2015). In proximitized graphene, the resulting subgap states move linearly with $$E_{\rm YSR}=\pm \Delta\,\frac{1-\alpha^2}{1+\alpha^2}, \qquad \alpha=\pi\,\rho(E_F)\,J,$$8 for small $$E_{\rm YSR}=\pm \Delta\,\frac{1-\alpha^2}{1+\alpha^2}, \qquad \alpha=\pi\,\rho(E_F)\,J,$$9, with a parity-switching condition $\alpha=\pi \rho_0 J S$0, in sharp contrast to the $\alpha=\pi \rho_0 J S$1-independent $\alpha=\pi \rho_0 J S$2 of the textbook metallic formula (Lado et al., 2015). This suggests that Shiba phenomenology is controlled not only by exchange scattering, but also by how the impurity restructures the local low-energy Hilbert space of the host.

5. Hybridization in dimers, clusters, and chains

When two impurities are brought within the spatial extent of each other’s Shiba wavefunctions, their bound states hybridize. In the ferromagnetically aligned case, the two degenerate orbitals split into symmetric and antisymmetric combinations; scanning-tunneling spectroscopy on Mn dimers on Nb(110) resolves this splitting for each of the four single-adatom Shiba orbitals $\alpha=\pi \rho_0 J S$3 (Beck et al., 2020). In an antiferromagnetically aligned dimer, simple theory without spin–orbit coupling predicts a common shift but no splitting, yet on a surface with broken inversion symmetry the addition of Rashba-type spin–orbit coupling lifts that degeneracy and produces a finite splitting $\alpha=\pi \rho_0 J S$4 (Beck et al., 2020). The result is that the presence or absence of dimer splittings cannot be interpreted independently of spin–orbit coupling and surface symmetry.

A subgap Green’s-function formulation generalizes this to multiorbital clusters. In that approach, each crystal-field-split orbital channel $\alpha=\pi \rho_0 J S$5 carries its own $\alpha=\pi \rho_0 J S$6, $\alpha=\pi \rho_0 J S$7, form factor, and Shiba pair $\alpha=\pi \rho_0 J S$8; sufficiently separated impurities are then described by an effective Hamiltonian whose normal-hopping and anomalous-pairing matrix elements decay as $\alpha=\pi \rho_0 J S$9 and depend on geometry, orbital structure, and the relative orientation of classical moments (Arrachea, 2021). For parallel dimers, bonding and antibonding branches alternate crossing zero as the distance varies; for antiparallel dimers, the spectrum remains gapped; and for canted configurations both hopping and anomalous pairing are present, allowing parity switches when the effective hopping exceeds $|\alpha|<1$0 (Arrachea, 2021). The same formalism extends to trimers with frustration in the orientation of the magnetic moments (Arrachea, 2021).

In periodic chains, hybridized Shiba orbitals form subgap bands. A linear chain of impurities spaced by $|\alpha|<1$1 supports a non-linear eigenvalue problem whose matrix elements decay as $|\alpha|<1$2, and in translationally invariant cases the Fourier-transformed problem yields Shiba band dispersions $|\alpha|<1$3 (Westström et al., 2016). Ferromagnetic chains on substrates with Rashba spin–orbit coupling and helical chains with spiral spin texture can realize topological superconductivity, while vacancies act as missing impurities and bind low-lying “anti-Shiba” states below the band edge of the regular magnetic chain (Westström et al., 2016). A finite density of vacancies forms an anti-Shiba band whose proliferation can destroy the topological phase, with particular fragility in some dilute-chain parameter regions (Westström et al., 2016).

An antiferromagnetic chain supplies a distinct route to topology. In that setting, a weak Zeeman field together with a supercurrent in the substrate generates a staggered spin-current that converts preexisting topologically unprotected Shiba states into Majorana fermions localized at the chain ends (Heimes et al., 2014). In the resulting topological regime, the spin-polarization of the edge Majorana wavefunctions depends solely on the parity of the number of magnetic moments, providing a parity-controlled spin texture proposed as a signature in spin-polarized STM (Heimes et al., 2014).

6. Driven, cavity-coupled, and boson-assisted Shiba physics

Shiba states are not restricted to static impurities. For a classical magnetic moment precessing at frequency $|\alpha|<1$4 and angle $|\alpha|<1$5, a rotating-wave treatment yields a quasi-energy

$|\alpha|<1$6

to leading order in the deep-Shiba and small-drive limit (Kaladzhyan et al., 2016). In that regime, a nearby normal STM tip sees dc charge and spin currents that depend strongly on the precession frequency, precession angle, and the Shiba level position in the gap, and the charge current is proportional to the difference between the electron and hole wavefunctions of the Shiba state (Kaladzhyan et al., 2016). This makes the driven current a direct measure of particle–hole asymmetry without requiring a spin-polarized tip.

The same driven impurity produces a back-action on the magnetization. Time-dependent calculations show that Shiba states contribute both reactive and dissipative torques acting on the classical spin, that the dissipative part is controlled by the finite linewidth of the Shiba pole, and that the resulting torques can be detected through ferromagnetic resonance measurements (Mishra et al., 2020). In that formulation, the torque spectrum directly probes the even- and odd-frequency triplet pairings generated by the dynamics of the impurity (Mishra et al., 2020).

Cavity and bosonic couplings provide additional control channels. For a strongly correlated quantum dot coupled to superconducting leads and capacitively coupled to a microwave cavity, multiple Shiba states appear inside the gap, and the zero-temperature quantum phase transition is pinpointed by a sudden change in the transmission signal $|\alpha|<1$7 (Chirla et al., 2015). If, more generally, a bosonic mode couples to the tunnelling between an Anderson impurity and a superconducting host, an additional $|\alpha|<1$8-wave conduction channel opens and generates a second pair of odd-parity YSR states (Müller et al., 2022). The exchange couplings in the $|\alpha|<1$9- and $-\Delta<E_0<\Delta$0-wave channels depend sensitively on the bosonic state—ground state, few quanta, or classically driven Floquet state—so the YSR spectrum becomes controllable by phononics or photonics (Müller et al., 2022).

7. Material platforms and generalized Shiba regimes

Quantum dots and molecular junctions provide gate- and coupling-tunable realizations of Shiba physics. In an InAs nanowire quantum dot coupled to a normal lead and a superconducting aluminium lead, transport spectroscopy resolves subgap states whose gate and magnetic-field evolution are quantitatively fitted by numerical renormalization group calculations; the zero-bias crossing marks the singlet–doublet transition, and the extracted parameters include $-\Delta<E_0<\Delta$1, $-\Delta<E_0<\Delta$2, $-\Delta<E_0<\Delta$3, and $-\Delta<E_0<\Delta$4 (Jellinggaard et al., 2016). In proximity-induced superconducting molecular break junctions, an organic radical spin-$-\Delta<E_0<\Delta$5 impurity exhibits Shiba excitations on both sides of the singlet–doublet transition, with the quantum phase transition occurring at $-\Delta<E_0<\Delta$6 and the excited state mediated by Andreev pairs in the proximitized leads (Island et al., 2016).

Iron-based superconductors motivate still more anomalous forms. In FeSe$-\Delta<E_0<\Delta$7Te$-\Delta<E_0<\Delta$8, the Zeeman field of a magnetic impurity locally favors a spin-triplet inter-orbital pairing over the bulk spin-singlet intra-orbital pairing, producing topologically protected zero modes at the boundary between the two pairing regions (Ghazaryan et al., 2022). These zero modes form Kramers doublets that are insensitive to the direction of spin polarization or to the separation between impurities, which differs sharply from the fine-tuned zero crossing of the single-band classical formula (Ghazaryan et al., 2022). In FeTe$-\Delta<E_0<\Delta$9Se$s$00, scanning tunneling spectroscopy instead reveals several peaks above a smallest full gap $s$01; comparison with simulations indicates that the peaks above the first one are generalized Shiba states that spatially overlap, forming an amorphous impurity band embedded between $s$02 and the larger-gap scale (Lee et al., 2024).

A further extension arises in one-dimensional superconductors with spin–charge separation. There, a magnetic impurity interacts exclusively with the gapped spin sector and still drives a local quantum phase transition, but the tunneling spectrum is shaped by gapless charge modes (Moca et al., 2024). At zero temperature, the local spectral function acquires a universal power-law threshold,

$s$03

which at half filling becomes $s$04 at criticality (Moca et al., 2024). In this regime the Shiba excitation is no longer a discrete $s$05-peak but a fractionalized threshold singularity. A plausible implication is that “Shiba state” has become a broader organizing concept for impurity-induced superconducting subgap structure, extending from localized BCS bound states to topological Kramers doublets, amorphous impurity bands, and fractionalized threshold spectra when the host itself departs from the conventional metallic $s$06-wave paradigm.