Nodal pockets in a monolayer kagome lattice

The family A₂T₃X₄ combines an alkali site (K, Rb, Cs), a transition-metal site (Ni, Pd, Pt) and a chalcogen site (S, Se). The source grew five compounds by a flux method: Cs₂Ni₃S₄, Cs₂Ni₃Se₄, (Rb,Cs)₂Pd₃S₄, (Rb,Cs)₂Pd₃Se₄ and Cs₂Pt₃S₄. The transition-metal atoms form a kagome net, each coordinated by a low-symmetry planar arrangement of chalcogen atoms, and in the source’s reading that crystal field isolates a single d_xy orbital near the Fermi level, and the Fermi level lies between the filled d_xy and the empty d_zx orbitals: the crystals are semiconductors with a gap in the visible, coloured gold, olive gold, teal, dark magenta and silver across the five compounds. The layers are held by van der Waals bonds, so they exfoliate. Cs₂Pd₃S₄ flakes of one to five layers reach 10 to 30 µm across, and a single layer measures about 0.7 nm by atomic force microscopy. The source states that the flakes stay stable in an argon glovebox for at least 60 days and withstand 300 °C; its Extended Data Fig. 2 shows them after 10 days and up to 300 °C.

On exfoliated flakes 30 to 100 nm thick, photoemission (ARPES) with microfocused beams at two endstations, below 20 K and with hybrid-functional DFT beside it, shows three d_xy bands just below the Fermi level: the topmost is a quasiflat band, and the two below it meet at a Dirac point. In bulk Rb₂Pd₃S₄ the quasiflat band stays within 70 meV along Γ̄–M̄ and within 180 meV near K̄, and photon energies from 60 to 200 eV show it without dispersion along the out-of-plane momentum. Across the family the d_xy band width is larger for Pd than for Ni, the flat band is flatter for S than for Se, and the flattest width, 𝒲_flat ≈ 180 meV, belongs to A₂Pd₃S₄.

For nanofocused photoemission (nano-ARPES) of thin flakes the source assembles graphene/A₂Pd₃S₄/graphite heterotrilayers: the graphite gives a flat support and the graphene covers the flake. Only the covered flake shows a sharp S-2p doublet; an exposed flake on the same device shows a broadened, shifted spectrum. From the orientations of the Brillouin zones one heterotrilayer is twisted by 28.9° between graphene and flake and by 20.2° between flake and graphite. A thickness series compiled from Rb₂Pd₃S₄ (bulk and eight layers) and Cs₂Pd₃S₄ (four, two and one layer) resolves the non-dispersive flat band and the linearly dispersing Dirac bands down to one layer, with minimal dependence on thickness. In the monolayer the bands sharpen and lie within 420 meV of the Fermi level, a shift the source reads as charge transfer to the adjacent graphene and graphite.

A single-gated monolayer Cs₂Pd₃S₄ device, read in operando by nano-ARPES, moves the flat band over a 390-meV range as the gate voltage 𝒱_G runs from −7.5 to 11 V, and the band’s energy saturates beyond 𝒱_G < −2.5 V, which the source reads as its top reaching the Fermi level. The measurement runs at room temperature; the hBN dielectric conducts under the synchrotron light and the device has no top gate, which, the source states, together limit the gate to about 30% of what a standard transport device reaches. A dual-gated trilayer Rb₂Pd₃Se₄ device, read in two-terminal transport, is highly insulating for 0 < 𝒱_TG < 5 V. Between −2.5 and 0 V its resistance falls by more than three orders of magnitude and turns metallic; between −5.5 and −2.5 V the high-temperature resistance keeps falling while the temperature dependence turns insulating, a state the source places inside the partially filled flat band and names as a candidate sign of the onset of many-body effects. Its carrier densities come from a parallel-plate capacitor model.

The source’s Table 1 sets A₂Pd₃S₄ against twisted bilayer MoTe₂, its moiré column adopted from the source’s ref. 42: a flat-band width of 140 meV against 10 meV and a localisation length 𝔞_loc of 6.2 Å against 52 Å, both printed without their kind, a Coulomb estimate 𝔘_C = e²/4πε₀ε_r𝔞_loc of 460 meV against 19 to 35 meV, and fractional quantum anomalous Hall gaps of 14 to 29 meV, calculated, against 1 to 5 meV. The source’s calculations carry the monolayer further. The mean-field diagram of monolayer Rb₂Pd₃S₄ holds 120° triangular antiferromagnetism, noncoplanar and stripe phases and, over the largest portion of it, the flat-band ferromagnet, in which the flat band carries Chern number 1. Exact diagonalisation of the fractionally filled topmost Chern band of Cs₂Pd₃S₄, and of Rb₂Pd₃S₄ in its extended data, gives a threefold quasidegenerate ground state whose spectral flow returns only after a 6π flux insertion, and a many-body Chern number of 1/3. One flat-band electron per 1 nm cell gives 𝔫 ≈ 1 nm⁻², against 0.01 nm⁻² for one per 10 nm moiré cell.

There is one substance, the omnium, and the crystal is a driven volume of it. Its nuclei are the saturated density cores of that volume, its fixed pinning sites (P-⁠3). The transition-metal nuclei form the kagome net; the chalcogen and alkali nuclei and the nuclei of the covering graphene and the supporting graphite are clamps beside it. One layer holds one net, and the net holds the nodal architecture at every thickness the source measures.

Energy injected into the medium sets up standing density waves, and excitations lock into the nodal lines and surfaces where the transverse density gradient vanishes, as sand collects on a driven plate (P-⁠2). Eigenstates and orbitals are nodal patterns of the driven field, the geometry of a mode is set by the clamps and not by the medium, and localisation is trapping in disconnected nodal pockets. The isolated flat band is that trapping in the pockets of the clamp net. A band is a family of modes q of the electrons, three-strand closures, in one confinement, the crystal’s support, and each mode costs its mass (I-⁠5):

with Ω the confinement domain, ℬ its boundary data, χ the orientation and ζ the position on the inside–outside chart; the operator D is to be derived (O-⁠9). The flat band’s flatness is a cost the pockets fix: the crystal momentum moves it by less than the record’s bounds in bulk Rb₂Pd₃S₄, 70 meV along Γ̄–M̄, under twice the 40-meV resolution, and 180 meV near K̄, where the record measures a small dispersion. Out of plane it does not move within the resolution: the clamps lie in the plane, and the van der Waals gap adds no node to the pockets.

The source names the flat band’s cause destructive interference of electron hopping in a frustrated geometry. In FDT the wavefunction ψ = Ae^iφ is the projection of a helix onto the plane normal to the cone axis, and the universe is deterministic (I-⁠4). Round each hexagon the clamp net locks the projected helical phases, and the interference the source names is their cancellation, deterministic. In the source’s nearest-neighbour model that cancellation is exact: the flat band is the set of patterns whose amplitudes sum to zero on every triangle (X-⁠1, below).

Gating is one system-wide settlement (0.1a). The gate moves the cost of every mode of the stack together, the band’s closures keep their class (I-⁠5), and the gate field’s energy enters the stack’s total gravitating energy 𝓜 once (I-⁠7). The saturation beyond −2.5 V and the conducting illuminated dielectric are the same stack settling as one system; the clause names no objective for the settlement and assigns no share of the gate’s charge to the band, the dielectric or the graphitic layers. That the environment moved the monolayer’s bands to within 420 meV of the Fermi level is the source’s charge-transfer reading of a measured position.

Every energy of a flake enters its density ledger once (I-⁠7): its nuclei and its closures in the rest mass, the closures at the cost of each occupied mode, their band energies there and nowhere else; in E/c² what those costs leave out, the exchange and interaction compression the source writes as its Coulomb and Hubbard scales, and the gate field’s energy; the stack’s load in PV/c²; and the radiation at the bath temperature in aT⁴V/c².

The carrier density 𝔫 is a count per area and never this density. Table 1’s kagome Coulomb energy returns from the source’s own formula on its own inputs, 460 meV at 𝔞_loc = 6.2 Å with the exact diagonalisation’s ε_r = 5; the map from a charge to a force-law input is to be derived (O-⁠6), and no Coulomb scale comes from the force law here. The probes are one carrier at different regimes (I-⁠2): the 98 eV photons of the photoemission, whose stroke radius ħc/E is 2.0 nm, the 514 nm Raman laser at 2.41 eV, and the sulfur vibrations near 280 and 355 cm⁻¹ in the phonon regime. No force carrier is exchanged.

The calculated one-third state takes the analysis’s Laughlin reading through I-⁠5a’s fractional accounting, the mechanism of the fractional quantum Hall effect. The state is K = (3), rank one: its threefold quasidegeneracy is the |det K| = 3 of the K-matrix account, and its return after a 6π flux is the period 3 × 2π. The many-body 1/3 is the Hall readout, the charge covector paired with itself through K⁻¹, a reading and never a possession, and the strand ledger stays integral. In the analysis’s own pairing the source’s quantum metric and Berry curvature, band geometry over crystal momentum, sit on the scalar and the bivector grade of one lifted object; neither is a spacetime curvature.

X-⁠1, exploratory. In the source’s nearest-neighbour model, which its tight-binding fit extends, the adjacency of the kagome net is YY^T − 2I, with Y the incidence of sites and triangles: every site lies on two triangles and every triangle on three sites. The Hamiltonian H_nn = −𝔱_nn(YY^T − 2I), with 𝔱_nn > 0 the nearest-neighbour hopping, then has 2𝔱_nn at the top of its spectrum, the source’s topmost quasiflat band, exactly on the patterns whose amplitudes sum to zero on every triangle. The integer solutions of that constraint on one triangle form the lattice of Gram matrix ((2, −1), (−1, 2)), determinant 3, the matrix I-⁠5a writes for its seat lattice, against the honeycomb bond’s (2), determinant 2: the three is the triangle’s corner count, a structural parallel. The kagome band is never identified with I-⁠5a’s seat lattice, nor the parallel’s ℤ₃ with the source’s threefold degeneracy. An in-sample re-description of the source’s model, never counted with the tests.

Six tests stand where a core clause supplies what the consensus calculation lacks: a changed observable dictionary (P-⁠5) or a constitutive input (I-⁠7). Each names its observable, solves its defining condition, and fixes its comparator, design, decision and executed rates before any data. The core assigns none of these observables a magnitude, so every test predicts a sign or an existence: P-⁠5’s twist-to-moment forms and the isotope sensitivities are to be derived, and each amplitude below is a declared test point. The rates are those of three standard deviations, 0.00135 one-sided and 0.0027 two-sided. The tests carry the analysis’s labels, T-⁠1 to T-⁠5 and T-⁠7.

P-⁠5: imposing a physical twist on a material, by torsion or helical geometry, is establishing a magnetic field. The analysis reads an interlayer twist as such a twist, and in its reading a twist fixes a hand and the field’s direction comes from what the configuration carries beyond the hand: the device’s polar axis for a twisted bilayer, the face the sensor reads for a ribbon, the layer order for a heterotrilayer. Which hand gives which sign the core does not assign, so a first session fixes the sign by the direction of its rejection and a second, independent session decides one-sided in each direction.

1. T-⁠1
   The hand locks the anomalous Hall sign of twisted bilayers. Sixteen ionic-liquid-gated Hall bars of bilayer Cs₂Pd₃S₄, eight at Θ = +10° and eight at −10°, the hand of each fabrication slot drawn at random, gated to half filling of the flat band, where the source’s mean-field calculation for monolayer Rb₂Pd₃S₄ holds the flat-band ferromagnet; the analysis carries that calculation to bilayer Cs₂Pd₃S₄ as the block’s premise, compound and stack both changed. Each device is cooled twice through the ordering range, the cryostat’s residual field nulled and then set to +10 µT in one cooldown and −10 µT in the other, the vertical heat flow reversed between the two and the cooling rates a factor of ten apart; it is locked when both cooldowns give one sign of the zero-field ℛ_xy, referred to its polar axis. FDT: wherever the twist’s field exceeds the 10 µT set field by more than the spread of the spontaneous choice, every device locks, its sign fixed by its hand. Comparator: a twisted stack is even under time reversal, so the signs after the slower cooldowns are independent of the hands. The decision rejects the comparator when the lock count ℒ reaches 15, a binomial tail of 0.00026, and the hand–sign table of the locked devices and that of all sixteen devices’ slower cooldowns each pass the two-sided Fisher test at 0.0027; perfect separation gives 0.00016.
2. T-⁠2
   Static torsion makes a torsion-odd stray field over a pristine ribbon. A suspended monolayer Cs₂Pd₃S₄ ribbon, 10 µm by 2 µm, assembled and read without air or solvent, held at zero applied field in magnetic shielding, one clamp turned to Θ_c = ±5°, read at its centre by scanning SQUID magnetometry in a flux-locked loop, which returns the signed field and puts no light on the ribbon. FDT: static torsion establishes a field, B_odd = [B(+Θ_c) − B(−Θ_c)]/2 ≠ 0, its sign set by the torsion sense. Comparator: zero, since static torsion is even under time reversal and the pristine monolayer, its pristine state read from the source’s trilayer Rb₂Pd₃Se₄ device, is nonmagnetic; the same sequence on the ribbon cut at the rotated clamp cancels the clamp’s own background where the cut leaves it unchanged. One hundred alternating cycles on each; the Welch statistic 𝒮_W decides two-sided against the critical value 3.04 at 198 degrees of freedom.
3. T-⁠7
   A twisted heterotrilayer and its mirror image differ by a time-odd Kerr rotation. Graphene/Cs₂Pd₃S₄/graphite stacks with a pristine monolayer, both interfaces turned one way at the source’s twists, (+28.9°, +20.2°), and as the mirror image, (−28.9°, −20.2°). Modulo 60° the mirror graphene interface lies at 31.1°, 1.1° from the achiral 30°, and the graphite one at 39.8°. Zero-area Sagnac interferometry, which in linear response reads only the time-odd rotation, reads each stack after zero-field cooling at four incident powers, and each stack’s Φ_Kerr is the zero-power intercept of a weighted fit quadratic in the power, which removes any light-borne moment linear or quadratic in the power. FDT: Φ_odd = [Φ_Kerr(stack) − Φ_Kerr(mirror)]/2 ≠ 0, its sign set by the hand and referred to the layer order. Comparator: zero. Thirty-two stacks per hand; the Welch statistic decides two-sided at 0.0027.

I-⁠7: for purely density-coupled equilibrium observables a heavier isotope gives a larger R_s, a higher α and a larger response, opposite in sign to reduced-mass scaling. An isotope substitution is a fixed-geometry perturbation, so the density input is the A-⁠1 step at fixed field radius, Δln α_ext = Δln 𝓜 per formula unit, and the lattice constants of both isotopologues are measured, any difference entering as its own geometric channel. The observable’s sign, with E_FD the observable of T-⁠3 below, m_iso the substituted species’ mass and R_room the substituted nuclei’s skin radius, is that of the sensitivity-weighted sum,

and each block stands at the test point of a positive, dominating density sensitivity S_dens. A species-resolved consensus calculation of the zero-point renormalisation, done before the session, fixes the comparator’s sign s_ZP and its value. The block discriminates where that sign is negative; where it is positive the density share and the comparator carry one sign and the block retires, and where the calculated consensus shift lies within three of its standard uncertainties of zero, no session runs until a calculation fixes a sign.

1. T-⁠3
   The Pd isotope moves the flat band against the Dirac point in the sign the comparator excludes. The observable is E_FD = E_FB − E_DP, the flat band’s top above the Dirac point, in Cs₂Pd₃S₄ grown from enriched ¹⁰⁴Pd and ¹¹⁰Pd, Δln 𝓜 = 0.0252 per formula unit for fully enriched lots, read by micro-ARPES at the source’s 98 eV and linear horizontal polarisation below 20 K, the two isotopes alternated in one session. Twelve flakes per isotope, each from its own growth, the growths interleaved in a furnace order drawn at random; the Welch statistic decides one-sided against the comparator’s sign, critical value 3.38 at 22 degrees of freedom. The lattice constant, the nuclear size, the finite nuclear mass and the enriched lots’ chemistry each enter the comparator once.
2. T-⁠4
   The Rb isotope moves the same observable in the sign the comparator excludes. E_FD in Rb₂Pd₃S₄ grown from enriched ⁸⁵Rb and ⁸⁷Rb, with T-⁠3’s photon condition and channels. The source reads the alkali site as marginal to the bands; the density channel reads its mass, Δln 𝓜 = 0.00645, so the sensitivity that gives T-⁠3 a share of two flake standard deviations gives 0.51 here. Twelve growths per isotope reject that at 0.022; 150 reject it at 0.83 at a zero lot bound and 0.71 at a bound of 0.05 flake standard deviations, against the critical value 3.03 at 298 degrees of freedom.
3. T-⁠5
   The S isotope moves the optical gap in the sign the comparator excludes. The absorption onset E_opt of bulk-like Cs₂Pd₃S₄ flakes grown from enriched ³²S and ³⁴S, Δln 𝓜 = 0.0111, with T-⁠3’s design; the exciton binding enters the comparator once. The measured anion coefficients of sp-bonded gaps are positive, +0.79 ± 0.05 meV/amu for the free exciton of β-ZnS and 0.71 to 0.95 meV/amu for S in CdS, and at a positive comparator sign the block retires. No measured class fixes the sign for the d–d gap of Cs₂Pd₃S₄, so the species-resolved calculation does.

Executed under the declared noise models, T-⁠1’s comparator points reject at a rate of 0 and its FDT point at 1.00, 0.87 with 2% of the signs misread; a fair device-fixed bias that locks every device leaves the decision at 0.00052. T-⁠2’s comparator rejects 0.0027 with or without a clamp background of 0.3 reading standard deviations, which cancels in the difference, against 0.97 at a ribbon field of 0.5 reading standard deviations; the ribbon’s mean alone, which is not used, rejects 0.85 under that background. T-⁠3 rejects 0.00125 at the comparator’s boundary with the lots’ share at its bound, and 0.81 at a density share of two flake standard deviations and a zero bound. T-⁠7’s intercept leaves 0.27 of a light-borne share growing as the square root of the power, 0.18 of a cubic one and 0.37 of a quartic one, so a light-borne hand difference of two offset standard deviations raises the comparator’s rejection to 0.035, 0.015 and 0.086 in the three cases, and the decision rejects 0.93 at a twist-borne hand difference of that size. Every block’s ablation is live: with the FDT input in place the encoding is +1, with that input at its consensus value it is −1, and their difference, 2, exceeds the sign threshold 1.

FDT agrees with the measured record of general relativity and quantum mechanics without reservation. For the same source and data it returns the record’s numbers unchanged: the flat-band bounds, the monolayer’s position below the Fermi level, the gating span, the insulator-to-metal transition, the twist angles and the one-layer step are accepted in full and read, never altered. The source’s kagome Coulomb estimate, its two per-cell counts and its capacitor density return from its own formulas on its own inputs, the identity map: 460 meV at 𝔞_loc = 6.2 Å and ε_r = 5, 𝔫 = 1/𝔞_cell² for both per-cell counts, and −1.5 × 10¹³ cm⁻² at the end of the gate axis of the source’s Fig. 4e. Table 1’s moiré Coulomb column is the source’s ref. 42 and stands as printed, and the abstract’s 220-fold stands as the abstract’s figure. Every calculation is carried as calculated and every fit as fitted, in sample where the source set it against the same spectra. In the gravitational window the nuclei and the flake read laboratory-small compactness on A-⁠1, and for the same stress tensor and data FDT and general relativity solve the same field equation, G_μν = κT_μν; the source’s quantum geometry is band geometry over crystal momentum and enters no curvature slot. In the consensus limit the tests set the added twist field and the added density sensitivity to zero, their consensus values, and the differences come from those two inputs alone.