Kagome superconductors

Scope

Kagome geometry offers a useful starting point for studying flat bands, Dirac crossings, charge order and superconductivity. It does not guarantee that a flat band lies at the Fermi energy, that magnetic frustration produces a spin liquid, or that a material has a high transition temperature. This article separates superconducting AV3Sb5 compounds, pressure-induced superconductivity in CsCr3Sb5 and charge-ordered comparison materials. Normal-state topology and superconducting topology remain separate questions. R1 R2 R3 R14 R16 R17

Mathematical assumptions

Take one spinless orbital on each of three sites $0,\mathbf a_1/2,\mathbf a_2/2$, with triangular Bravais vectors and $q_i=\mathbf k\cdot\mathbf a_i$. For positive nearest-neighbor hopping $t$, a Bloch Hamiltonian is $H=-2t\begin{pmatrix}0&c_1&c_2\\c_1&0&c_3\\c_2&c_3&0\end{pmatrix}$, where $c_1=\cos(q_1/2)$, $c_2=\cos(q_2/2)$ and $c_3=\cos[(q_2-q_1)/2]$. Its bands are $E_f=2t$ and $E_\pm=t[-1\pm\sqrt{4S-3}]$, with $S=c_1^2+c_2^2+c_3^2$. The vectors in Guo and Franz's convention are nearest-neighbor vectors, half the Bravais vectors used here. R1

The model has a flat-band contact at Gamma, a Dirac crossing at K and dispersive-band saddles at M. Independent diagonalization checks 1,311 momentum/hopping combinations; numerical Hessians have opposite-sign curvatures at the two M-point saddles. Adding equal same-sublattice hopping shifts all bands by $\delta(\mathbf q)=-2u[\cos q_1+\cos q_2+\cos(q_2-q_1)]$. The formerly flat branch then spans $9|u|$ between Gamma and K while the kagome geometry remains. These checks illustrate model sensitivity; they do not calculate a pairing interaction, a material transition temperature or a superconducting invariant. R1

Real compounds add orbital character, interlayer hopping and reconstruction. Kang's CsV3Sb5 author manuscript combines ARPES dispersions with DFT orbital and sublattice projections:

Here p/m means pure/mixed sublattice character, not a superconducting gap symmetry. The sublattice weights come from calculated Bloch functions; ARPES brightness alone does not measure them.

Figure 3 uses 88 eV photons at 60 K. Figure 4 instead uses 95 eV photons at 35 and 95 K, with linear horizontal polarization in both datasets. Its reported approximately 20 meV feature is a charge-order gap, not a superconducting gap. Matrix elements change visibility across Brillouin zones. The displayed DFT analysis omits spin–orbit coupling; full reconstruction, spectral fitting and final-version comparisons remain unchecked here. These band constraints do not select the superconducting pairing symmetry. R8

Experimental criteria

Ortiz's original family study identified K, Rb and Cs vanadium antimonides as layered kagome metals. The later CsV3Sb5 superconductivity paper combines magnetization, resistivity and heat capacity around a 2.5 K onset. Its transport onset is suppressed at an 8 mA probe current and returns toward the nominal value at lower current, illustrating why excitation belongs in the sample record. The heat-capacity temperature floor limits conclusions about the low-temperature gap. Subsequent K and Rb member reports should retain their own samples and criteria rather than being collapsed into a universal family transition range. R2 R3 R4 R5

The same CsV3Sb5 work discusses a much higher-temperature anomaly and a normal-state topological band background. ARPES taken well above the superconducting transition and calculated surface states above the Fermi energy do not directly measure a superconducting gap invariant. Liang's three-dimensional CDW study and Li's conjoined-CDW study further emphasize stacking, temperature and surface dependence. Different modulation descriptions cannot be merged into one structure common to every sample or termination. R3 R6 R7

Yang reports anomalous Hall response in KV3Sb5; Yu connects anomalous Hall response in CsV3Sb5 to its higher-temperature charge-order regime. Neither result is a phase-sensitive measurement of superconducting chirality. Ordinary multiband transport, scattering and Berry-curvature contributions require separate analysis. Likewise, the checked Mielke muon study concerns time-reversal symmetry breaking associated with charge order. A signal emerging near that ordering scale cannot simply be reassigned to a superconducting $d+id$ state. R9 R10 R11

Chen's spatially resolved tunneling work proposes a roton pair-density-wave interpretation in CsV3Sb5. Pair-related modulations are significant evidence to investigate, but charge-order changes in local density of states, tunneling matrix elements and Josephson coupling remain relevant controls. Pressure studies also show nonmonotonic superconducting and charge-order scales. Two domes do not by themselves establish two pairing mechanisms: structure, bands, scattering and calibration can evolve together. R12 R13

For CsCr3Sb5, Liu's author manuscript and selected final-paper results report a maximum resistive onset of 6.4 K at 4.2 GPa, with zero resistance and AC susceptibility supporting bulk superconductivity over a pressure interval. Its critical-field curves instead use half the normal resistance. The fitted zero-temperature field is therefore not an independently observed field boundary. The final text explicitly leaves the field direction uncertain, preventing an anisotropy claim. The cubic-anvil measurement uses glycerol and room-temperature load calibration; the lower-pressure piston-cell setup uses a different medium and calibration. These are high-pressure records, not ambient-pressure entries or universal chromium-family numbers. R14

Cheng's April 2026 paper adds a distinct charge-order result in CsCr3Sb5. The selected author text describes cleaved-surface STM with a 4.5 K base temperature and two higher-temperature modulation scales. Its stripe pattern breaks mirror symmetries while retaining a mirror-glide operation; opposite local textures do not define global handedness. Those surface images are not a high-pressure superconducting order-parameter measurement and cannot establish chiral superconductivity. R15

FeGe and ScV6Sn6 serve as comparisons. Yin's FeGe study reports charge order, magnetic stacking and an edge state within a charge-order energy gap; Arachchige's ScV6Sn6 work reports a lattice-modulated charge-order transition. Neither cited result establishes superconductivity. Absence of such evidence in these papers should not be rewritten as a timeless claim that either material can never superconduct under any preparation or pressure. R16 R17

Limits and disputes

Allowed pairing representations are candidates until tested against the actual superconducting state. Charge-order time-reversal breaking, a normal-state anomalous Hall signal, structural handedness, tunneling modulation and a zero-bias feature each address different questions. Phase-sensitive experiments, temperature-resolved symmetry probes, bulk gap constraints and boundary-state tests can provide complementary discrimination; they are not a universal checklist that defines every possible topological superconductor. Complete scientific acceptance remains open: several references retain abstract-only review, and final-version comparisons, supplements and raw-data analyses are unfinished. R3 R7 R10 R11 R12 R15

Discovery implications

Record composition, pressure path, excitation, surface termination and the onset, midpoint, measured-zero and thermodynamic criteria separately. Compare measured band positions with a specified model before invoking flat-band enhancement. Test whether changing charge order also changes disorder, structure or carrier balance. The useful next experiment resolves competing interpretations on a characterized sample, rather than promoting a geometric label or an isolated transition number into a mechanism. R1 R3 R8 R13 R14

Related topics

Flat bands, Charge density waves, Chiral pairing, Pair density waves, and Pressure tuning.

Material cases

Values describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.