Protection of Unconventional Superconductivity from Disorder

Unconventional superconductivity is a desirable state of matter due to its potential for high transition temperatures $T_{\mathrm{c}}$ and associated favorable superconducting properties. However, the sign-changing nature of the order parameter of unconventional superconductors renders their condensates fragile to disorder, an inevitability in real materials. We uncover the generic properties of electronic band structures and associated Bloch weights able to support robust unconventional superconductivity. We demonstrate this property in several case studies of the kagome and Lieb lattices, showing how unconventional superconductors exhibit unusually weak $T_{\mathrm{c}}$ suppression by disorder, despite featuring fully compensated sign-changing order parameters. We contrast these results with those for unconventional superconductivity on the square and honeycomb lattices, which are unable to protect the condensates from disorder. Finally, we discuss material candidates for which this effect may be realized.

💡 Research Summary

In this work the authors address a long‑standing puzzle in unconventional superconductivity: why sign‑changing order parameters, which are central to high‑temperature and many exotic superconductors, are typically extremely fragile to disorder such as point defects. By combining symmetry analysis of multi‑sublattice Bloch states with a standard Abrikosov‑Gor’kov (AG) treatment of impurity scattering, they uncover a generic mechanism that can protect unconventional condensates from pair‑breaking.

The starting point is a generic tight‑binding Hamiltonian on a lattice with several sublattices, (H=\sum_{\mathbf k,\alpha\beta} h_{\alpha\beta}(\mathbf k) c^{\dagger}{\alpha\mathbf k}c{\beta\mathbf k}). Diagonalising this Hamiltonian yields band eigenstates (u_{n\alpha}(\mathbf k)) that encode the weight of each sublattice (\alpha) in band (n). The authors show that, because the Hamiltonian respects the space‑group symmetry (G), the Bloch eigenvectors must transform according to irreducible representations (irreps) of the little group at each high‑symmetry (\mathbf k) point. For one‑dimensional irreps the transformation law forces certain components of (u_{n\alpha}(\mathbf k)) to vanish exactly at specific (\mathbf k) points – a “symmetry‑enforced zero”.

These zeros have direct consequences for impurity scattering. Point‑like, non‑magnetic impurities couple locally to a particular sublattice via a potential (V h^{\rm imp}\alpha). In the Born approximation the self‑energy on sublattice (\alpha) is proportional to the Fermi‑surface average of (|u{n\alpha}(\mathbf k)|^2). If (|u_{n\alpha}(\mathbf k)|^2) is zero over a sizable region of the Fermi surface, the corresponding scattering rate (1/\tau_\alpha) is strongly suppressed. Consequently, even for a sign‑changing order parameter (\Delta(\mathbf k)) that is fully compensated ((\sum_{\mathbf k}\Delta(\mathbf k)=0)), the impurity‑induced pair‑breaking term can become negligible when scattering predominantly connects states with the same sign of the gap. This deviates from the standard AG prediction, which assumes uniform scattering and yields a rapid suppression of the critical temperature (T_c).

To illustrate the principle, the authors study three lattice families: the honeycomb lattice (two sublattices), the kagome lattice (three sublattices forming a corner‑sharing network), and the Lieb lattice (a square‑derived lattice with a flat band). For the honeycomb case the sublattice weights are constant ((|u|^2=1/2)) and all unconventional pairings follow the AG curve. In contrast, on the kagome lattice the (E_2) (d‑wave) pairing exhibits a markedly weaker (T_c) suppression because the Bloch weights on the three sublattices are highly anisotropic and vanish along symmetry‑protected lines. On the Lieb lattice the authors focus on a (B_1) ((d_{x^2-y^2})) order parameter. When impurities reside on the 2c Wyckoff positions (sublattices A and C) the Bloch weight on sublattice A is zero along certain directions, leading to a finite anomalous Green’s function component that mimics an s‑wave response and thus protects (T_c). Impurities on the 1b site (sublattice B) do not enjoy this protection and the AG behavior is recovered.

The key guiding rule that emerges is: unconventional order parameters that transform trivially under the site‑symmetry group of the impurity‑hosting sublattice are robust against point‑like disorder. This rule is independent of the detailed hopping parameters; it follows solely from orbital content and Wyckoff positions. The authors verify its generality by extending the analysis to non‑trivial orbitals and longer‑range hoppings in the Supplementary Material.

Finally, the paper discusses realistic material platforms where this mechanism could be operative. Kagome superconductors such as AV₃Sb₅ (A = K, Rb, Cs) are highlighted as promising candidates: if their pairing belongs to the (E_2) (d‑wave) channel, the superconducting state should be unusually resilient to vacancy or substitutional disorder. For the Lieb lattice, the authors point to cuprate superconductors (which are effectively inverse‑Lieb systems) and to engineered inverse‑Lieb structures where the dominant orbitals reside on the “A” sublattice. In such systems, oxygen or other point defects would have a reduced pair‑breaking effect, potentially explaining the observed robustness of (d_{x^2-y^2}) superconductivity in the presence of disorder.

In summary, the work provides a symmetry‑based, band‑structure‑centric explanation for disorder‑protected unconventional superconductivity, identifies concrete lattice geometries where the effect is strongest, and proposes several real‑world material families where experimental verification should be feasible. This insight opens a new route for designing high‑(T_c) superconductors that combine unconventional pairing symmetries with intrinsic immunity to the inevitable disorder present in real crystals.