Kagome Superconductors: Topology & Correlations

• Kagome superconductors are quantum materials defined by a 2D lattice of corner-sharing triangles that produce flat bands, Dirac points, and van Hove singularities.
• They exhibit intertwined orders such as charge density waves, nematicity, and magnetism, which can be tuned by pressure, doping, and magnetic fields.
• Their multiband superconductivity and topologically nontrivial states offer a promising platform for probing Majorana modes and novel quantum phenomena.

Kagome superconductors constitute a class of quantum materials in which the kagome lattice geometry—characterized by two-dimensional nets of corner-sharing triangles—enforces unique electronic structures with flat bands, Dirac crossings, van Hove singularities (vHs), and nontrivial topology. The archetypes, AV₃Sb₅ (A = K, Rb, Cs), and emerging analogues such as MPd₅, LaIr₃Ga₂, and CeRu₂, display unconventional superconductivity often intertwined with charge density waves (CDWs), nematic order, magnetism, and time-reversal symmetry breaking (TRSB). These platforms have become central for investigating the interplay between electronic correlations, frustration, topology, and superconductivity in quantum materials.

1. Crystal, Electronic, and Topological Structure

Kagome superconductors typically crystallize in highly symmetric structures dominated by perfect or near-perfect planar kagome nets:

• AV₃Sb₅ (A=K, Rb, Cs): Space group P6/mmm. V atoms form an ideal 2D kagome net in the ab-plane, sandwiched between Sb layers, with intercalated alkali-metal ions (Zhao et al., 2021, Hu et al., 2023, Wilson et al., 2023).
• MPd₅ (M=Ca, Sr, Ba): Also P6/mmm, with Pd forming the kagome layers (Li et al., 21 Feb 2025).
• LaIr₃Ga₂: Kagome network of heavy Ir atoms, P6/mmm; pronounced spin–orbit coupling from 5d orbitals (Gui et al., 2021).
• CeRu₂: Cubic Laves phase (Fd–3m), Ru atoms in stacked (111) kagome planes (Deng et al., 2022).

The unique lattice geometry strictly dictates the low-energy band structure:

• Flat bands (nondispersive, high DOS), typically from destructive quantum interference in the hopping around the kagome network.
• Dirac points at Brillouin-zone corners (K points), protected by lattice symmetry; in AV₃Sb₅ the Dirac point is found at E_D ≈ –0.30 eV relative to E_F (Hu et al., 2023).
• vHs at M points; these singularities very often occur within tens of meV of E_F due to the band filling, enhancing susceptibility to instabilities (Wilson et al., 2023, Luo et al., 2024).
• Topological band inversion yielding $\mathbb{Z}_2$ indices and protected surface states traversing the E_F window (Gui et al., 2021, Hu et al., 2023, Li et al., 21 Feb 2025).

2. Charge Density Wave, Nematic, and Magnetic Orders

Charge Density Wave (CDW):

• AV₃Sb₅ compounds universally host a CDW transition at T_CDW ≃ 80–104 K (Zhao et al., 2021, Wilson et al., 2023). The order is 2×2×2 or 2×2×4, combining in-plane “star-of-David” and “trihexagonal” modulations; the CDW reconstructs the Fermi surface, gaps out part of the vHs-derived states, and reduces nesting (Wilson et al., 2023, Hu et al., 2023).
• Landau free energy for the CDW order parameters Φ_i (three M-points): $$F[Φ] = α\sum_i|\Phi_i|^2 + β\sum_i|\Phi_i|^4 + γ\sum_{i<j}|\Phi_i|^2|\Phi_j|^2 + η(\Phi_1\Phi_2\Phi_3 + c.c.)$$ (Wilson et al., 2023).
• Time-Reversal Symmetry Breaking: The CDW state is chiral, as shown by μSR relaxation, Kerr effect, and STM reciprocity violations (Gu et al., 2021, Yang et al., 2024). A chiral-current or "flux" order parameter emerges:

$$\chi_{ij} = i \langle c_i^{\dagger} c_j - c_j^{\dagger} c_i \rangle \neq 0$$

assigning opposite fluxes to the two sublattice loops.

• Electronic Nematicity: SI-STM demonstrates that rotational (C₆) symmetry breaking persists even after CDW long range order is fully suppressed via Ti or Sn doping, manifesting as nanoscale C₂ nematic puddles that constitute the “parent” symmetry-broken state throughout the AV₃Sb₅ phase diagram (Xu et al., 27 Nov 2025).

Magnetism: In CeRu₂, ferromagnetic order coexists with superconductivity, both arising from a Ru-derived kagome flat band near E_F. The flat-band induced Stoner instability and the coexistence regime are tunable by pressure and Ru occupancy (Deng et al., 2022).

3. Superconductivity: Gap Structure, Pairing Symmetry, and Collective Modes

• Critical Temperatures:
  • AV₃Sb₅: T_c = 0.9–2.5 K, reaching T_c,max ≈ 5.6 K under pressure (Zhao et al., 2021, Wilson et al., 2023, Xu et al., 19 Feb 2025).
  • MPd₅: T_c = 1.5–2.6 K (phonon-mediated) (Li et al., 21 Feb 2025).
  • LaIr₃Ga₂: T_c ≈ 5.2 K (Gui et al., 2021).
  • CeRu₂: T_c ≈ 5–6 K, with magnetic order at T_N ≈ 40 K (Deng et al., 2022).
• Gap Symmetry and Structure:
  • Bulk Probes & ARPES: Demonstrate nearly isotropic, nodeless gaps on all Fermi surfaces—Δ ≈ 0.5–0.8 meV corresponding to 2Δ/k_B T_c ≈ 2.8–3.5, consistent with BCS values, robust to the suppression of long-range CDW (Zhong et al., 2023, Hu et al., 2023).
  • Thermal Conductivity: In pristine CsV₃Sb₅, ultralow-T κ/T reveals a finite residual linear term κ₀/T at H=0, characteristic of nodal superconductivity (d-wave or s_± with accidental nodes); a rapid, nearly √H field-dependence is observed (Volovik effect) (Zhao et al., 2021). Subsequent ARPES and STM suggest both nodeless (s- or s+is) and line-node (d-wave) channels may coexist or compete depending on CDW and disorder (Zhong et al., 2023, Jiang et al., 25 Apr 2025).
  • Multiband Character: STM/STS shows two coexisting gaps—a dominant, nearly isotropic gap on the Sb-p pocket, and a smaller, highly anisotropic gap on the V-d derived pockets. Under Ta doping, gaps merge and become fully isotropic as CDW is suppressed (Hu et al., 2024).
  • Collective Modes: High-resolution tunneling spectroscopy identifies a sharp subgap bosonic collective mode, interpreted either as a Leggett mode (relative-phase oscillation between multiband condensates) or a Bardasis–Schrieffer mode (subleading channel pairing fluctuation), surviving the full suppression of CDW (Hu et al., 2024).
• Pairing Symmetry Landscape:
  • Theoretical Modeling: BdG and tight-binding analyses on the kagome lattice classify possible pairings by irreducible representations of $C_{6v}$ (A₁: s-wave; E₂: d+id; E₁: p+ip, etc.), finding unambiguous signatures in ARPES, LDOS, and impurity spectra to distinguish between isotropic s-wave, chiral d+id, p+ip, and frustrated TRS-breaking states (Liu et al., 2024, Jiang et al., 25 Apr 2025).
  • Frustrated Superconductivity: On-site and NN pair-hopping interactions allow a $2\pi/3$ sublattice phase difference (“frustrated” state), producing a sixfold amplitude modulation and global TRSB, yet preserving a full gap and Hebel–Slichter peak (Jiang et al., 25 Apr 2025).
  • Disorder Effect: Increasing disorder induces a continuous (non-nodal) transition from the TRS-breaking frustrated state to an isotropic s-wave, accounting for the experimentally observed sensitivity of gap structure to impurities (Jiang et al., 25 Apr 2025).
• Pressure and Magnetic Field Tuning:
  • The superconducting critical temperature displays one or multiple domes as a function of pressure in CsV₃Sb₅ (peaking at P ≈ 0.8 GPa and again at >11.4 GPa as CDW is suppressed and re-entrant) (Zhao et al., 2021); in CeRu₂, Tc recovers at high pressure with the emergence of a secondary superconducting transition (Deng et al., 2022).
  • Magnetic field and field-history dependent effects, such as reorientation of vortex lattices and multiband vortex bound state signatures (Y-type and X-type zero-bias states), further reveal the role of multiband anisotropy (Huang et al., 2024).
• Supercurrent Interference and Diode Effect:
  • Little–Parks oscillations in KV₃Sb₅ confirm conventional 2e Cooper pairing; anomalous interference patterns, phase-coherent domain structures, and global critical current effects demonstrate the existence of spatially varying superconducting order (Wu et al., 12 Oct 2025).
  • In CsV₃Sb₅, a zero-field superconducting diode effect occurs—critical current is nonreciprocal (I_c⁺ ≠ |I_c⁻|)—signaling simultaneous inversion and TRSB, indicative of chiral or mixed-parity pairing (Yang et al., 2024). K₁₋ₓV₃Sb₅/Nb and RbV₃Sb₅ Josephson devices demonstrate magnetic hysteresis and phase-sensitive effects consistent with complex (d±id, p±ip, s+is) order parameters.

4. Van Hove Singularities, Flat Bands, and Correlated Instabilities

Kagome electronic structure features sharp vHs close to E_F, enhancing the density of states and favoring symmetry-breaking instabilities (CDW, SC, nematic):

• Emergent Flat Bands: High-resolution ARPES in AV₃Sb₅ reveals four branches of nearly perfect flat bands (FB1–FB4) at binding energies 0.07–0.7 eV below E_F, not accounted for by straightforward tight-binding or atomic localization, but evolving continuously with CDW ordering and directly tracking vHs energies (Luo et al., 2024).
• CDW–Driven Splitting: The 2×2 CDW potential lifts the vHs degeneracies, causing multiple nearly flat bands with enhanced DOS, boosting tendencies toward competing symmetry-broken phases (CDW, nematicity, SC) (Luo et al., 2024).
• Superconductivity Enhancement: The presence of multiple vHs and flat bands within a small energy window supports enhanced Cooper pairing and can stabilize competing pairing symmetries depending on doping, pressure, and disorder (Luo et al., 2024, Wilson et al., 2023).

5. Topological Superconductivity and Majorana Physics

Kagome systems offer distinct avenues for realizing topological superconductivity by leveraging their inherent band inversion, symmetry, and spin–orbit effects:

• Intrinsic Topological Surface States: ARPES observes Dirac-like, nontrivial surface bands at or near E_F in AV₃Sb₅, LaIr₃Ga₂, MPd₅, and computationally in AZr₃Pb₅ (Hu et al., 2023, Li et al., 21 Feb 2025, Yi et al., 2022, Gui et al., 2021).
• s-wave + Rashba SC for Helical/Chiral Majorana Modes: Tight-binding models demonstrate that even on-site/nearest-neighbor s-wave superconductivity, in the presence of Rashba spin–orbit coupling and/or 2×2 chiral-flux CDW order, stabilizes time-reversal-invariant helical or time-reversal-breaking chiral topological superconductivity—characterized by nontrivial $Z_2$ or Chern number phases supporting Majorana edge states (Mojarro et al., 12 Mar 2025).
• Experimental Probes: Tunneling and thermal transport experiments, along with field-tunable vortex spectroscopy, motivate future efforts to detect Majorana zero modes and confirm bulk topological invariants in kagome SCs (Huang et al., 2024, Mojarro et al., 12 Mar 2025).

6. Material Diversity and Design Principles

The kagome superconductor landscape has broadened considerably:

• Heavily Studied: AV₃Sb₅ family—unconventional SC coexisting with chiral CDW, vHs, and multiband effects (Wilson et al., 2023).
• SOC-dominated: LaIr₃Ga₂, with 5d states, enhanced SOC, and flat bands (Gui et al., 2021).
• Magnetic: CeRu₂ combines kagome superconductivity and itinerant Ru-derived flat-band magnetism (Deng et al., 2022).
• Phonon-driven & Topological: MPd₅ (M=Ca, Sr, Ba), AZr₃Pb₅—phonon-mediated s-wave SC, strong tuning of T_c by pressure/doping, and nontrivial topological indices (Li et al., 21 Feb 2025, Yi et al., 2022).

Materials design principles extracted from experiment and theory include:

• Tuning chemical potential and CDW wavevector to pin vHs/flat bands to E_F.
• Exploiting strong spin–orbit coupling (Ir, Pd, heavy Zr/Pb) to engineer topological surface and edge states.
• Strain, pressure, or carrier doping to traverse SC/metal/CDW/nematic phase boundaries and enhance T_c or access new correlated regimes (Li et al., 21 Feb 2025, Yi et al., 2022, Luo et al., 2024).

7. Outlook and Open Questions

Kagome superconductors define a rich landscape of unconventional superconductivity arising from frustrated geometry, flat bands, nontrivial topology, and intertwined quantum order:

• The dominant pairing symmetry is highly sensitive to CDW order, disorder, and band filling; transitions between frustrated TRSB states (six-fold modulations), nodeless s-wave, and chiral d+id or p+ip are all accessible depending on tuning (Yang et al., 2024, Jiang et al., 25 Apr 2025).
• The precise role of nematicity—now established as ubiquitous beyond the CDW regime—in stabilizing or competing with CDW and SC is an open direction (Xu et al., 27 Nov 2025).
• The nature and universality of Leggett or Bardasis–Schrieffer-like collective modes—now observed in multiband kagome systems—offer new spectroscopic handles on interband or subleading channel fluctuations (Hu et al., 2024).
• Corner Josephson, edge-state tunneling, and phase-sensitive thermal transport experiments are essential to directly confirm the predicted Majorana physics in the chiral and helical topological superconducting phases (Mojarro et al., 12 Mar 2025).
• Extending the kagome design to new chemical families (e.g., MPd₅, AZr₃Pb₅), with tunable topology and strong phononic or electronic coupling, promises further insights and possibly new platforms for topological quantum computing (Li et al., 21 Feb 2025, Yi et al., 2022).

Kagome superconductors thus represent a paradigmatic condensed matter system where geometric frustration, multiple competing orders, topology, and tunable correlated ground states intersect, with critical open questions poised for resolution through next-generation spectroscopy, quantum transport, and theoretical modeling.