Commensurate Cell Scientific Validation

This document records the equations, acceptance criterion, reference fixtures, and performance boundary used by the v_ase commensurate-cell workspace.

Scope

The validated workflow matches two periodic vectors in the global XY plane and rotates the guest about global Z. It supports either:

  • a selected component copied from the host cell for a same-lattice twist; or

  • a separately loaded host and guest lattice.

It does not claim an energy minimum. The result is a geometrically periodic, bounded integer-cell match that should be relaxed with an appropriate physical model before energetic conclusions are drawn.

Integer Search And Rotation

For host and guest row-vector bases A_h and A_g, integer matrices define candidate boundaries

H = M_h A_h
G = M_g A_g.

v_ase enumerates inequivalent 2D Hermite-normal-form cells, Gauss-reduces their bases, and removes the best proper in-plane rotation by an orthogonal Procrustes solve. The remaining deformation that maps the rotated guest onto the host is

D_g = (G Q)^-1 H.

The current source checks all 20 determinant-one 2D matrices with entries in {-1, 0, 1}, including both elementary shear directions. Regression fixtures verify that equivalent unimodular input bases recover the same physical matches in the tested cases.

This is an adaptation of the cited methods: HNF enumeration, Gauss reduction, and a finite orientation set replace the paper’s integer-entry search and alignment construction. The graph retains a ranked representative per 0.01° bucket. The tests establish the listed analytic cases and agreement with unscreened enumeration of this orientation set; they are not a proof of completeness over arbitrary deformations or all lattice representations.

Since 0.3.1, plane alignment requires a normal dot product of at least 1 − 1e-10. The former 0.985 bound admitted visibly tilted periodic cells, which could hide out-of-plane deformation in a projected match.

Two Strain Values

Candidate acceptance uses the conservative, rotation-invariant maximum principal-stretch mismatch

epsilon_max = max_i |sigma_i(D) - 1|,

where sigma_i are singular values. The Maximum strain control applies to this value only.

The optional Paper strain projection uses the small-strain tensor

epsilon = (D + D^T) / 2 - I
epsilon_mean = (|epsilon_11| + |epsilon_22| + |epsilon_12|) / 3.

This is the mean absolute strain plotted against common-cell atom count by Stradi et al. Keeping both values prevents a change of graph view from changing which candidates pass the physical cutoff.

The regression suite also reconstructs all six mean-strain values in Table 3 of Stradi et al. from the published epsilon_11, epsilon_22, and epsilon_12 components. The comparison allows one final-table rounding unit because the printed components are already rounded to two decimal percent.

Numerical References

Equal-lattice hexagonal series

For the standard commensurate twisted-bilayer family,

cos(theta_mn) = (m^2 + 4mn + n^2) / (2(m^2 + mn + n^2))
N = m^2 + mn + n^2.

The test suite checks every (m, m + 1) point through (31, 32), including 21.786789 deg (N=7), 13.173551 deg (N=19), and 1.050121 deg (N=2977), with zero residual boundary strain.

Separate graphene/Cu(111) fixture

The user-facing fixture in periodic host/guest workflow uses separate graphene and ideal Cu(111) primitive cells. Under a one-percent guest-strain cutoff and area limit 16, its smallest match is:

host:  graphene √13
guest: Cu(111) √12
|rotation|: 16.10211375 deg
maximum principal guest strain: 0.1665824397 %
mean absolute guest strain:     0.1110549598 %

expected.json is read directly by the regression suite. The fixture validates lattice matching and visualization; it is not a relaxed interface model.

The default proposal is sorted by the larger host/guest area ratio, then total area, residual strain, and angle magnitude. Increasing the area ceiling to 64 therefore keeps this √13/√12 cell as the initial proposal instead of replacing it with a larger cell closer to zero rotation. Once the user supplies an angle, the viewport deliberately follows the admissible candidate nearest that angle.

The reference values are also reconstructed independently in the tests:

L_h = √13 * 2.46 Angstrom
L_g = √12 * (3.615 / √2) Angstrom
epsilon_max = L_h / L_g - 1
epsilon_mean = 2 * epsilon_max / 3
theta = 30 deg - atan((√3/2) / 3.5)

The expected common cell contains 26 graphene atoms and 12 Cu atoms. The external guest placement is independently validated using

guest minimum z - host maximum z = 3 Angstrom

before the guest rotation and in-plane residual deformation are applied.

Viewport State Validation

The browser regression verifies the complete user workflow rather than only the search result:

  1. enabling the workspace without a selection shows only the host primitive grid and its two vectors;

  2. selecting a same-cell layer adds a separately colored guest grid and two guest vectors without moving the camera;

  3. loading an external guest replaces the selected guest, preserves the 3 Å gap, and keeps separate host/guest integer matrices;

  4. cells-only is the default; optional atom mode adds an opaque core, a one-cell halo, and boundary-crossing bonds;

  5. the graph uses rotation angle, area ratio, and maximum principal strain as distinct dimensions and moves only the current-angle plane during rotation;

  6. materialization requires an explicit Edit-mode action and preserves the camera instead of reframing the result.

Performance Boundary

The following timings are historical release measurements. The current source audit also fixes handedness, oblique vacuum vectors, square coincidence indices and the doubled graphene/MoS₂ result.

The search uses descriptor-space cKDTree screening, vectorized closed-form 2 x 2 rotation/deformation evaluation, and enriches only the best physical candidate in each 0.01-degree bucket. It does not randomly sample candidates.

On the release development machine, the graphene/hBN-like benchmark evaluates:

Maximum area ratio

Compared boundary pairs

Search time

16

1,379

about 0.016 s

32

19,098

about 0.09 s

64

296,075

about 0.82 s

128

4,731,795

about 9.7 s

Hardware changes absolute timing. The explicit maximum of 128 keeps the interactive search bounded; larger-area studies should use a dedicated offline lattice-matching workflow rather than an implicit incomplete sample.

References

  • P. Lazic, “CellMatch: Combining two unit cells into a common supercell with minimal strain,” Computer Physics Communications 197, 324-334 (2015), DOI: 10.1016/j.cpc.2015.08.038.

  • D. Stradi, L. Jelver, S. Smidstrup, and K. Stokbro, “Method for determining optimal supercell representation of interfaces,” Journal of Physics: Condensed Matter 29, 185901 (2017), DOI: 10.1088/1361-648X/aa66f3.

  • J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H. Castro Neto, “Continuum model of the twisted graphene bilayer,” Physical Review B 86, 155449 (2012), DOI: 10.1103/PhysRevB.86.155449. Equations 5-7 give the commensurate angle and primitive integer-cell family used by the analytic same-lattice regression.