Nodal order and magnetic twist in hexalayer graphene

In rhombohedral stacking each graphene layer sits displaced by one carbon–carbon bond relative to the one beneath. The stacking hybridises the inner layers strongly and leaves one sublattice of the top layer and one of the bottom layer electronically active, and with six layers these carry nearly flat bands. The hexalayer was exfoliated, its rhombohedral domains identified by Raman spectroscopy with a 532 nm laser and isolated by anodic oxidation lithography with an atomic force microscope, then picked up by dry transfer in the sequence graphite, hBN, hexalayer, hBN, graphite, with graphite top and bottom gates. The maps run against the carrier density 𝔫 and the perpendicular electric field ℰ. Three devices gave similar results. The dilution refrigerator’s base temperature is 9 mK.

The source’s 12-band tight-binding model, with its printed hoppings, gives the single-particle picture. With no potential between the layers the conduction and valence bands overlap substantially, and the band-edge states lie equally on the top and bottom layers; a potential polarises the band edges onto one layer or the other and, made larger, separates the bands into a gap. The conduction band is close to a parabola and the valence band an inverted Mexican hat. In the measured longitudinal resistance a transport gap opens at charge neutrality once ℰ exceeds about 100 mV nm⁻¹. Below that threshold lies the semimetal, with a correlated insulator at charge neutrality near ℰ = 0.

Between 𝔫 = 0 and 𝔫 = −1.8 × 10¹² cm⁻² a triangular region symmetric about ℰ = 0 hosts orbital magnetism: the Hall resistance is hysteretic in both the magnetic and the electric field, with a butterfly-shaped electric-field loop like that of pentalayer graphene. At the triangle’s tail the resistance drops sharply near 170 mK, and the differential resistance shows a gap that a magnetic field suppresses. The source names this superconducting-like state SC1. It persists to an in-plane field of about 6 T, more than ten times the Pauli limit of about 300 mT that the source estimates from the observed T_c. A second such state, SC2, sits near the phase boundary that extends from the comet region described below. The resistance reaches zero in neither, and the source gives two candidate causes for the remainder: inadequate filtering, or a fragile superconducting state. Both states lie where electron and hole Fermi surfaces coexist.

Quantum oscillations map those Fermi surfaces. The source normalises each oscillation frequency as 𝖿_v = 𝖿_FFTe/(𝔫h), the number of Bloch states inside the Fermi surface relative to the carrier density, so that a doubly degenerate parabolic band peaks at 0.5. The fermiology shows twofold-degenerate Fermi surfaces, which the source reads as a two-by-two splitting into two flavoured bands, A and B, each twofold degenerate. In that picture, as ℰ grows, flavour A’s emergent gap shrinks while flavour B’s widens, until flavour A’s closes and inverts. In the field cut at 𝔫 = −1.26 × 10¹² cm⁻² the top-layer and bottom-layer Fermi surfaces cross at about 42 mV nm⁻¹. In the leaf-shaped region near SC1 two pockets, FS1 and FS2, appear and are gone by 500 mK, and a third, FS3, exceeds their sum by 0.5.

In a comet-shaped region near 𝔫 = 0 and ℰ ≈ −35 mV nm⁻¹, under a 50 mT stabilising field, the Hall resistance switches in steps as ℰ is swept, with both switches offset to the same side: near −34 mV nm⁻¹ going forward and −37 mV nm⁻¹ coming back. The loop’s polarity reverses with the sign of the perpendicular field, and the loop shrinks as that field grows. At 𝔫 = −0.93 × 10¹¹ cm⁻² the magnetic hysteresis reverses as ℰ is tuned: forward field sweeps lie above backward ones for ℰ ≥ −35.5 mV nm⁻¹ and below them for ℰ ≤ −36.2 mV nm⁻¹. The source reads the sharp switching as a spontaneous electric polarisation 𝒫 and names the state ferroelectric orbital magnetism, distinct from the ferrovalley magnetism of the triangle, whose moment 𝔐 = 𝖼_fv𝖵ℰ follows the valley polarisation 𝖵 and stays aligned with the external field. A phenomenological free energy with the coupling −𝗀_me𝒫𝔐𝖡, minimised at each field point from the minimum at the point before, reproduces the comet’s key features in the source’s simulations.

There is one substance, the omnium, and the hexalayer is a driven volume of it. Its carbon nuclei are the saturated density cores of that volume, its fixed pinning sites (P-⁠3). Each sits at the schematic skin value α_core → 1⁻ of its own interior chart, a reading kept apart from the same nucleus’s exterior compactness, far below 1, and both charts are declared. The rhombohedral stacking, each layer displaced by one carbon–carbon bond from the one beneath, sets where the clamps sit.

Energy injected into the medium excites standing density waves, and excitations lock into the nodal lines and surfaces where the transverse density gradient vanishes, ∇α = 0, as sand collects on a driven plate (P-⁠2). The flat-band states on the top and bottom sublattices, the only ones the stacking leaves electronically active, are such nodal patterns, and the clamps set their geometry, not the medium. The superconductivity whose features the source reports is a shared scattering-free nodal channel, and its onset is a global eigenmode snap of the standing-wave network. The source’s states keep their superconducting-like designation, and the finite resistance stands in full.

A Cooper pair is the tip-to-tip overlap of two opposite-helicity electron cones (P-⁠4). Opposite flow directions attract: the opposite-sense cones the settlement joins mesh like gears and pull tip to tip into a configuration of low curvature energy. The sign belongs to each cone’s circulation sense, not to the local velocity of adjacent surfaces, and the size is I-⁠1 work through a declared response kernel. The bare force law at condensed-matter compactness returns about 10⁻⁴¹ N, a correct small number and not the binding scale, which only a declared kernel reaches (I-⁠1); the pairing energy and the source’s Coulomb penalty for a uniform emergent gap each take their own kernel and domain. Which broken symmetry, spin, valley or a mix, groups the isospins into the two flavours the record leaves open, so the flavours keep their generic names, and orbital phase winding, spin and cone helicity stay distinct.

A magnetic field is a macroscopic helical density gradient of the omnium, the bivector grade of the density field coherently aligned along an axis (P-⁠5). The triangle’s and the comet’s moments are its projections, aligned along the stacking axis, the axis the Hall geometry reads. Mechanical vorticity and magnetic moment are two grade projections of one density-gradient topography, and no carrier is exchanged (I-⁠2).

Each energy of the source’s functional enters the device’s mass-equivalent once (I-⁠7). The held-moment term carries the source’s −𝖡𝔐 and the ferrovalley energy; the polarisation energy −ℰ𝒫 enters once; the coupling −𝗀_me𝒫𝔐𝖡 enters inside the same moment channel, not as a second copy; and the double wells enter together, once. The Coulomb penalty is an energy difference between candidate states and adds no term. A pattern whose rest energy depends on where it sits in a twist takes that energy as its clock factor (0.1c):

with 𝗆 the held moments, each at its settled value between switching events, and E_ind the energy of the moments the twist induces, the work of inducing them included. The gradient of E along the twist pushes; a uniform twist, a uniform level of 𝔪, turns a held moment and pushes nothing. The 50 mT stabilising field is such a level across the device: it turns the moments and pushes nothing.

Events do not commute, and a process run backward is never the mirror of the forward event (0.1a). The two branches of each hysteresis loop are two forward settlements from different starting states. The source’s local free-energy minimisation is its phenomenological prescription for that evolution; the omnium settles system-wide.

Each observable carries its own weight. Transport weights the current path and its contacts, an oscillation frequency weights its extremal orbit and the count of carrier states, and the density of states is a histogram of eigenstates; none of them is omnium density, and an average supplies no point value. The layer polarisation is a ratio of state weights, never a density coordinate. The total Berry phase of each valley, which scales as π𝖭 with the layer count 𝖭, is a phase winding at fixed field identity, never a strand count: the electrons are patterns of strand count 3 (I-⁠5), and the source’s electron and hole surfaces remain band language. On the A-⁠1 chart the whole device reads a small gravitational compactness, a correct reading, with the curvature of general relativity (I-⁠1, I-⁠3); no electronic gap is calibrated from it.

Each test compares neighbouring devices, so that common instrument response cancels in the pair. Three isolate the density channel that an isotope substitution adds at fixed geometry (I-⁠7); two isolate the field that a physical twist establishes (P-⁠5). Each fixes its referent devices, an observable solved by a stated criterion, the comparator it is read against, a threshold chosen in advance and rules for detection, exclusion and attribution. Each amplitude is a declared test point: the isotope tests set a constant density sensitivity, and the mirror tests a contrast for a physical twist field.

The isotope devices are made by the source’s method from carbon enriched to 99.0% in ¹²C or in ¹³C, of masses 12.000 and 13.003 u, so that the two mixtures differ by Δln M_at = 0.0787. Two source crystals of each composition serve four chips each; each of eight chips carries one light and one heavy device, positions alternated chip by chip, through three cooldowns: 16 devices and 24 paired cooldowns. The density input is the device’s own A-⁠1 exterior α_ext, moved at fixed pressure, reference radius and packing; at the test point where half of the device’s inventoried mass-equivalent is substituted carbon, Δln α_ext = 0.0401. I-⁠7 fixes the input’s sign: for a purely density-coupled equilibrium observable a heavier isotope gives a larger R_s, a higher α and a larger response, opposite in sign to reduced-mass scaling. The observable’s sign follows the sensitivities S_i = ∂ln O/∂ln x_i, integrated along the substitution path.

1. T-⁠1
   The isotope shift of the SC1 transition. At each device’s own SC1 maximum, T_c solves ℛ_xx(T_c) = ½ℛ_xx(300 mK) at zero applied field. The comparator adds every co-varying channel in Δln T_c: reduced mass, −0.040 to 0; heavy-carbon hyperfine pair breaking, −0.010 to 0; the interlayer and in-plane hoppings, +0.0002 to +0.0022; electron–lattice renormalisation, the source crystals and flavour A’s emergent gap, ±0.003, ±0.003 and ±0.005; −0.070 to +0.020 in all, rounded outward. At the test point the density channel adds a residual of 0.28 in ln T_c, against a design threshold of 0.2 from the eight-chip Student rule and the comparator’s half-width.
2. T-⁠2
   The isotope shift of the charge-neutral gap opening. Activation fits from 0.30 to 1.00 K on each field polarity give the two fields where the activation energy reaches zero, and their mean magnitude ℰ_g; each device uses its own measured hBN thicknesses and gate calibration. The comparator, from the hoppings, the conversion’s spacing term, electron–lattice renormalisation, the source crystals, the hyperfine channel and the emergent gap with screening, spans ±0.020 in Δln ℰ_g. The test point is a residual of 0.18, against a design threshold of 0.1.
3. T-⁠3
   The isotope shift of the Fermi-surface crossing. Each device is set to 𝔫 = −1.26 × 10¹² cm⁻², and its two layer-assigned frequency branches, tracked through oscillation spectra from 0.5 to 1.6 T, cross where their difference changes sign, at ℰ_x. The same ±0.020 interval bounds the comparator, the test point is again a residual of 0.18, and the threshold is 0.1.

Detection is a mean residual above the threshold; the residual of FDT’s sign is excluded when the mean residual plus the threshold lies below it. A detection is read as the density channel when X-ray diffraction at 4 K on all four source crystals places the interlayer log difference between −4.7 × 10⁻⁴ and −1.6 × 10⁻⁴ and the in-plane one between −1.5 × 10⁻⁴ and 0, and the two compositions’ Dingle temperatures agree within 10% of their mean.

The mirror devices add a twist. Twelve chips each hold a mirror pair made by the source’s method: top hBN at +5° and −5° from a shared mirror plane and bottom hBN at +10° and −10°, every angle measured after assembly to 0.10°, with gates, Hall bars and contacts mirrored and a fair coin assigning hand to position before fabrication. This analysis reads mirror-rotated encapsulation as a physical twist, and a physical twist establishes a field where the held moments sit (P-⁠5, 0.1c). The vertical mirror reverses hand and perpendicular field together, so the added field is odd in the hand; its sign and size are to be derived (O-⁠6), and both tests are two-sided.

1. T-⁠4
   Mirror-odd triangle remanence. At 𝔫 = −0.90 × 10¹² cm⁻² and ℰ = +5 mV nm⁻¹, cool each pair from 1.0 K to a measured carrier temperature of 20 mK at zero applied field and zero current, twenty times per chip, and read the remanent Hall resistance at 1 nA against the centre of the device’s own saturated loop. The hand contrast 𝖽 is the difference between the two hands’ fractions of positive readings. Its comparator is zero: joint reversal of magnetisation and field is a symmetry of the ferrovalley form and of forward relaxation into local minima, and composed with the mirror it maps one forward cooldown onto its partner.
2. T-⁠5
   Mirror-odd comet-step polarity. At 𝔫 = −0.46 × 10¹¹ cm⁻² and zero applied field, prepare each device at −45 mV nm⁻¹ and sweep forward to −30 mV nm⁻¹, across the switch near −34 mV nm⁻¹, twenty times, and count the sign of the Hall step. The residual applied field still couples through −𝗀_me𝒫𝔐𝖡 at a zero setting, but joint reversal of moment and field is a symmetry of the source’s whole functional, so the comparator contrast is again zero.

Both tests apply the exact randomisation test to the twelve chip contrasts, all 2¹² = 4096 sign flips, two-sided at p ≤ 0.0027. At the test point every chip reads 𝖽 = +0.6, and the attainable threshold is exactly 0.6, the lattice’s own critical value. A contrast of 0.6 is excluded when the tests shifted by +0.6 and by −0.6 both reject. A detection is read as the twist’s field when the remanence and the comet step give one sign of an equivalent perpendicular field, each against its own +50 mT reference, with the mirror, angle and in-plane conditions met.

Monte Carlo calibration under the declared noise models fixes how the rules behave. At the comparator interval’s upper end the isotope rules reject falsely at 0.00134 for T_c and 0.00132 for the fields, and with no density channel present they exclude a residual of the test-point size at 0.97 and 0.98. The mirror rule rejects falsely at 0.0018 with the hands randomised and 0.017 with them held fixed, and it excludes a contrast of 0.6 at 0.93 when there is no hand effect. Every probability is per observable.

X-⁠1, exploratory. The source’s own functional, ℱ = −𝖺_𝒫2𝒫²/2 + 𝖺_𝒫4𝒫⁴/4 − 𝖺_𝔐2𝔐²/2 + 𝖺_𝔐4𝔐⁴/4 − ℰ𝒫 − 𝖡𝔐 − 𝗀_me𝒫𝔐𝖡, with its dimensionless parameters (1, 1, 1, 1, 2.5) and each field point started from its predecessor’s minimum, reconstructs in sample the source’s local-minimum model of the comet’s coupled switching. At zero fields its branches sit at 𝒫 = ±1 and 𝔐 = ±1, a lost minimum marks each switch and the preceding branch supplies the history. Joint reversal of 𝔐 and 𝖡 at fixed 𝒫 and ℰ leaves it unchanged, the algebra the mirror comparator uses. An in-sample reconstruction, never counted with the tests.

FDT agrees with the measured record of general relativity and quantum mechanics without reservation. The limiting map holds the source’s stress tensor, geometry, band parameters, preparation history and observable dictionary fixed: it returns the same curvature (I-⁠3) and, in I-⁠4’s weak-field slow-motion layer, the same retained order. It carries the source’s tight-binding bands, density of states and layer polarisation as the source computed them; at the printed parameters and signs the 12-band Hamiltonian gives a 2.5 meV overlap at 0.5 meV per layer, and a larger potential separates the bands into a gap. The normalised frequency and its sum rule, the Lifshitz–Kosevich fit with its fitted background, the Pauli estimate and the local-minimum Landau calculation are the identity map on the source’s inputs, and none is counted among the predictions. The isotope tests set the added density sensitivity to zero in their consensus limit, the transition test keeping its reduced-mass and bounded channels; the mirror tests set the added encapsulation field to its conventional value and keep the residual applied field, whose symmetry-even part cancels. The prospective differences come from the added density response and the added twist field alone; a branch label or a small compactness generates none of them.