The chain and the cell

Pyrochlore spin ice states are the two-in-two-out configurations of Ising spins on the pyrochlore lattice (Bramwell and Gingras 2001), equivalently the Eulerian orientations of the diamond lattice whose bonds carry the spins. The smallest ice-preserving move reverses the six arrows around a hexagonal plaquette, a single spin flip creating a monopole pair instead (Castelnovo, Moessner, and Sondhi 2008). The chain studied here picks a hexagon uniformly at random and, if its arrows run head-to-tail, reverses all six with probability \(1/2\). It is reversible with the uniform stationary distribution on each flux sector, and under Henley’s classical-to-quantum correspondence its generator is the ring-exchange Hamiltonian of pyrochlore ice at its Rokhsar–Kivelson point (Henley 2004; Rokhsar and Kivelson 1988; Hermele, Fisher, and Balents 2004).

Cells are blocks of \((a,b,c)\) conventional cubic cells with periodic boundaries, holding \(16abc\) spins and as many hexagons, the hexagon set derived from the coordinates rather than read off the quotient graph. The winding \(\mathbf W\) is conserved and everything below sits at \(\mathbf W=0\), a sector never connected because frozen states exist in it at every size, so the object of study is its dynamically accessible component. At \((1,2,2)\), 64 spins and 64 hexagons give 221,628 zero-winding states in 205 components: a giant component of 221,424 states, whose transition matrix has 2,651,888 nonzeros, and 204 frozen singletons. That component is small enough to diagonalise rather than sample.

What an observable can see

Global arrow reversal \(R\) complements every arrow bit. It commutes with the transition matrix \(P\), and on the giant component it has no fixed point, so it acts as a free involution with 110,712 two-element orbits. Taking one representative \(r_j\) per orbit and its partner \(p_j\), the vectors \((e_{r_j}\pm e_{p_j})/\sqrt2\) span the even and odd subspaces, and \(P^{\pm}[j,k]=P[r_j,r_k]\pm P[r_j,p_k]\) are two symmetric blocks of size 110,712 whose spectra partition the spectrum of \(P\).

The autocorrelation of an observable \(O\) is \(\rho(t)=\langle y,P^ty\rangle/\langle y,y\rangle\) with \(y=O-\langle O\rangle\), a sum of decaying exponentials weighted by the squared overlaps of \(y\) on the eigenvectors, so an observable of definite parity has overlap exactly zero on every eigenvector of the opposite parity and its \(\tau_{\exp}\) is only a lower bound on \(1/\Delta\). That an observable’s autocorrelation time is not the chain’s is standard (Sokal 1997). What is unusual here is that a symmetry makes the shortfall exact, so no series length and no system size closes it.

The natural observable is the coarse-grained polarisation: with \(\sigma_b=2x_b-1\), \(\mathbf d_b\) the bond’s A-to-B displacement and \(\mathbf q^{(k)}=(2\pi/4c_k)\,\mathbf e_k\), \(P_j^{(k)}=N^{-1}\sum_b \sigma_b d_{bj}e^{i\mathbf q^{(k)}\cdot\mathbf r_b}\), whose transverse components \(j\neq k\) carry the Coulomb-phase slow mode (Henley 2010; Isakov et al. 2004). Every \(\sigma_b\) changes sign under \(R\), so \(\mathbf P\) is odd, and at \((1,2,2)\) the three slowest eigenvectors are even. The project’s response was a panel of both parities at every size, the slowest member taken as the headline and reported as a lower bound. Its three even members were the flippability structure factor \(F(\mathbf q)\), the total flippable count, and \(|\mathbf P_\perp|^2\). None of those sees the slow mode either, for a second reason beyond parity.

The exact spectrum, and a pseudoscalar character

Lanczos on each parity block, \(k=14\), with the uniform vector deflated in the even block, took 930 and 973 matrix-vector products. Merged, the three slowest modes of the chain are even and the fourth is odd. Per attempt \(\Delta_{\mathrm{even}}=2.5644358\times10^{-3}\) and \(\Delta_{\mathrm{odd}}=4.1225073\times10^{-3}\), a ratio of \(0.622057\), so \(1/\Delta=389.949\) attempts against the \(242.571\) the polarisation can reach. The slowest even eigenvector, written \(v\) below, is simple, separated from the second even mode by \(1.229\times10^{-3}\) in \(\lambda\), so every reading of it is a property of the chain. The slowest odd eigenvalue is fourfold degenerate, which is why the odd readings here are multiplet invariants. Against an independently banked earlier spectrum the maximum difference over the top ten eigenvalues is \(5.44\times10^{-15}\), all ten parities agree, and the lifted eigenvectors satisfy \(\lVert Pv-\lambda v\rVert<10^{-14}\).

The cell’s affine automorphisms are derived from the coordinates rather than assumed: maps \(p\mapsto Mp+t\) modulo the box, with \(M\) a signed axis permutation, are accepted only if they permute the site set, carry every bond to a bond, and carry the derived hexagon set to itself, and where the map exchanges the A and B sublattices the arrow bit complements. At \((1,1,1)\) the construction returns 192 automorphisms, the number of general positions of \(Fd\bar3m\) in one conventional cubic cell, asserted as a test, and at \((1,2,2)\) it returns 256, all acting on the giant component and each verified to commute with \(P\). The character \(\chi(g)=\langle v,gv\rangle\) is \(\pm1\) on every one of them, and

\[\max_g\;\bigl|\chi(g)-\det M_g\bigr| \;=\; 2.66\times10^{-14}\]

over 128 proper and 128 improper operations. So \(v\) transforms in the determinant representation, invariant under every proper operation and sign-reversing under every improper one, with a stabiliser of exactly 128 elements. Under the two transverse translations its character is \(1.000000000000\), so it sits at zero momentum, while the trace characters \(0\), \(0\) and \(-4\) of the fourfold odd multiplet force those four states to the transverse zone-boundary momenta. An even, zero-momentum, pseudoscalar function of a divergence-free field has a short list of candidates, and the first of them is the helicity.

The emergent magnetic helicity

For a divergence-free field the natural bilinear pseudoscalar is \(\mathbf q\cdot(\mathbf P^{*}\times\mathbf P)\). With \(\mathbf q\) along axis \(k\) and \((i,j,k)\) cyclic, the code computes

\[h_k(\mathbf q) \;=\; \operatorname{Im}\bigl(P_i(\mathbf q)^{*}P_j(\mathbf q)\bigr) \;=\; \operatorname{Re}P_i\cdot\operatorname{Im}P_j - \operatorname{Im}P_i\cdot\operatorname{Re}P_j ,\]

three multiplies and a subtraction on polarisation components the sampler already records, with \(P_i\) and \(P_j\) transverse to \(\mathbf q\) by construction. The quantity is even under arrow reversal, being a product of two odd factors, and changes sign under every improper lattice operation because the cross product does. In the continuum it is the magnetic helicity \(\int\mathbf A\cdot\mathbf B\) of the emergent gauge field. Its parity is measured rather than inferred: over all 221,424 states every helicity column satisfies \(h(Rs)=h(s)\) to a relative deviation below \(10^{-12}\), and every non-vanishing polarisation column satisfies \(P(Rs)=-P(s)\). Of 259 named observables whose overlap \(\langle v,O\rangle^2/(\lVert v\rVert^2\lVert O\rVert^2)\) with \(v\) was computed, the eight largest are all helicities.

The span of the eleven independent helicity columns captures \(0.83456\) of \(v\). A least-squares projection onto spin monomials \(\sigma_S=\prod_{b\in S}\sigma_b\), done as a genuine Gram solve because the monomials are not orthogonal once restricted to an ice sector, puts the total degree-two content of \(v\) at \(0.853223\), identical at rank cutoffs \(10^{-8}\), \(10^{-10}\) and \(10^{-12}\). The helicity family therefore accounts for 98 per cent of everything of degree two in the mode, and the best non-helicity overlap is \(0.066707\). Two controls close the identification. The \(y\leftrightarrow z\) swap has determinant \(-1\), so it must send \(v\) to \(-v\), and geometrically it exchanges \(h_y\) with \(h_z\) and flips the helicity’s sign, so \(h_y+h_z\) is odd under it and matches, while \(h_y-h_z\) is even and must be orthogonal, as the measured \(2.3\times10^{-27}\) says it is. And the matrix \(C_{ij}=\langle v,\sigma_i\sigma_j\rangle\) describing the degree-two content has effective rank \(18.35\) with leading eigenvalues in \(\pm\) pairs at \(\pm0.002933\), the signature of an antisymmetric pairing rather than of the square of a single bond-linear form, which would give a rank-one \(C\) of one sign.

The blind spot, in relaxation times

Autocorrelations were computed exactly by 4,000 sparse matrix-vector products, so the numbers below carry no statistical error.

The helicity’s \(\tau_{\exp}\) is the chain’s true relaxation time to the last digit, where the earlier panel tops out at \(263.13\), short of the gap by a factor \(1.61\). Overlaps of the earlier panel’s even members with \(v\) are \(3.4\times10^{-29}\) and below for the four flippability channels, \(7.0\times10^{-30}\) for the flippable count, and \(8.9\times10^{-27}\) and below for the four squared-polarisation channels, with last digits that move between runs. Those are symmetry-enforced zeros rather than small numbers, because the observables are true scalars under improper operations and \(v\) is a pseudoscalar, which is stronger than suppression by powers of \(q_{\min}^2\). The stochastic estimator sees the mode too, which matters because cubic cells can only be sampled: with 512 replicas and 20,000 samples at 39 attempts it returns \(\tau_{\exp}(h_y+h_z)=378.08\pm1.92\) attempts against \(377.55\) for the identical fit applied to the exact \(\rho(t)\).

Cubic geometry

Exact diagonalisation cannot reach cubic cells: cubic \(L=2\) has of the order of \(2\times10^{11}\) states at \(\mathbf W=0\). The question went to the Monte Carlo instead, with the helicity added to the production panel as four even families and left off by default so that the earlier panel reproduces bit for bit. With the same seeds, sampling intervals, burn-ins, replica counts and series lengths, \(\tau_{\exp}(P_\perp)\) agrees with the banked campaign table to between \(8.8\times10^{-9}\) and \(1.8\times10^{-6}\) relative at every size, which is the rounding of the six-figure constants compared against, and ice-rule and winding violations were zero throughout. Write \(R_h(L)=\tau_{\exp}(h,q_{\min})/\tau_{\exp}(P_\perp,q_{\min})\).

\(R_h\) falls monotonically to one half, approaching from above and crossing to \(0.4988\pm0.0025\) at \(L=16\), while the independent control \(|P_\perp|^2/P_\perp\) approaches the same value from below, the two meeting by \(L=8\). All four helicity families pass the resolution gate at every size, with fit \(r^2\) between \(0.9891\) at \(L=2\) and \(1.0000\) for \(L\geq6\), so no helicity number here is a fit to noise. At the Gaussian fixed point the photons are odd under arrow reversal, so the even sector is the sector of even photon number, whose lowest state is two photons at \(\pm q_{\min}\) relaxing at half the odd time. The helicity is a two-photon composite and lands exactly there. It had no obligation to, since it carries a different representation of the cubic point group from \(|P_\perp|^2\), and the two agreeing from opposite sides is better evidence than either alone. In cubic geometry the odd polarisation remains the slowest member of the panel at every size.

Why the thin cell is different

Thinness alone is not the cause. Across the cells the enumerator can reach, \(\Delta_{\mathrm{even}}/\Delta_{\mathrm{odd}}\) sorts by cross-section rather than by volume.

The three quasi-one-dimensional cells sit between \(3.7\) and \(3.9\) and rise slowly with length. \((1,2,2)\) is the only reachable cell with two directions of extent greater than one, and the only one where the even sector is slowest. A helicity needs two independent transverse directions to have a long wavelength, and the measurements say so directly: the helicity about the thin axis has overlap \(0.03464\) with \(v\) against \(0.75859\) for the combination on the two thick axes. Momentum is the other half of the geometry. With \(b=c=2\) the smallest nonzero wavevector along a transverse axis is already the zone boundary, where \(e^{iq_{\min}a}=-1\), and that is exactly where the character analysis puts the fourfold odd multiplet while the helicity sits at zero momentum. A cell two units across has no interior to its Brillouin zone, so the ordering of the two sectors is settled by a two-point comparison a larger cell would not make. Adding a third thick direction restores the Gaussian counting, and the cubic ladder shows that happening.

What remains open

The natural next cell is \((1,2,3)\), which would test whether the helicity stays slowest when one transverse direction is lengthened. Its \(\mathbf W=0\) count is known exactly, 196,465,480 states from a frontier dynamic programme in \(14.9\) seconds. The diagonalisation is what does not fit, at 31 to 243 hours of wall time, the binding constraints being the binary-search index used to build the adjacency and the unaccelerated Lanczos iteration count. That cell would still be one unit thick, so it would sharpen the thin-cell statement rather than test cubic geometry.

The cubic result is a measurement supported by a field-theoretic argument, and a panel bounds only what it measures. An eighth family finding nothing slower than the polarisation at seven sizes narrows the space of hidden even modes without emptying it. Nothing here is certified. The spectra are floating-point computations with residual checks and cross-checks against independently banked values, the symmetry statements are numerical characters agreeing with \(\det M_g\) to \(2.7\times10^{-14}\), and the cubic result is Monte Carlo with jackknife errors, which is a different epistemic status from the project’s SAT-certified and proof-assistant results.

Related work

Magnetic helicity entered fluid mechanics as a topological invariant. Woltjer proved the invariance of \(\int\mathbf A\cdot\mathbf B\) for a perfectly conducting fluid and its role in characterising force-free fields (Woltjer 1958); Moffatt showed that the corresponding integral for a vortex field measures the knottedness and linkage of the field lines (Moffatt 1969); Berger and Field made the topological reading precise for a magnetic field in a bounded volume (Berger and Field 1984). The pseudoscalar character that does the identifying work here is the property that makes helicity a linking number, since a linkage changes sign under reflection.

The emergent electrodynamics of spin ice is why a helicity exists in this model at all. The ice rule is a lattice divergence-free condition on the coarse-grained field, which puts the classical model in a Coulomb phase with dipolar correlations and pinch points in the structure factor (Isakov et al. 2004; Henley 2010), and the excitations above it are deconfined monopoles (Castelnovo, Moessner, and Sondhi 2008, 2012). At the Rokhsar–Kivelson point the quantum model has an emergent photon (Hermele, Fisher, and Balents 2004), and the classical-to-quantum correspondence (Henley 2004) makes the chain’s spectrum that Hamiltonian’s spectrum. I have found no previous use of the helicity as an observable in spin-ice Monte Carlo, classical or quantum: a literature search on the emergent gauge field returns work on the transverse projector, on monopoles and on photon signatures, and nothing on pseudoscalar bilinears. That is a negative result from one search rather than a claim of priority.

The general point that an observable’s autocorrelation time is not the chain’s is old (Sokal 1997), and the practical version is familiar in lattice gauge theory, where the topological charge is a slow mode that couples weakly to most observables and so leaves their error estimates too small (Schaefer, Sommer, and Virotta 2011). The pyrochlore case is sharper than weak coupling, because the overlap is exactly zero and two symmetries in succession enforce it: arrow reversal removes every odd observable, and the improper operations of the point group remove every even scalar. Where that matters, the alternative to guessing a better observable is to compute the subdominant eigenvalue of the stochastic matrix itself, the route Nightingale and Blöte took for the two-dimensional Ising model (Nightingale and Blöte 1996), and what the diagonalisation here does at the one cell where it is affordable.

Appendix: artefacts

All computation was CPU-only on a single GB10 machine, so no HPC service was used and no acknowledgement is owed. icemix/parity.py supplies the parity blocks, the derived automorphism group, the induced action on states and the monomial projection; icemix/observables.py defines \(P_j(\mathbf q)\); icemix/exact_ac.py computes exact autocorrelations and overlaps; icemix/panel.py carries the panel, the helicity families and helicity_ratio. experiments/05_thin_even/results/results.json is the source of every \((1,2,2)\) number here, from a run of 886.9 s at a peak resident set of 4.45 GiB, and experiments/06_helicity_cubic/results/results.json the source of every cubic number, from a run of 3,413.5 s at 8.21 GiB. tests/test_stage4d.py adds 26 tests and tests/test_stage4f.py 17, covering the parity and symmetry machinery from definitions, the panel change and its bit-identity against the pre-change module, and the recorded readings. docs/reports/2026-09-04-stage4d.md and docs/reports/2026-09-04-stage4f.md carry the full working record.

References