DMI validation¶

Physical problem¶

Validation must test the implemented energy, effective field, natural boundary behavior, and planner legality separately. A green constructor test is not a numerical qualification.

Governing equations used by validation¶

The interfacial and bulk energy densities are evaluated with the same sign conventions as their canonical owners:

Symbols and SI units¶

Assumptions and validity¶

Each test must state DMI variant, coefficient sign, normal convention, geometry, mesh/grid, boundary policy, precision, and solver tolerance. A uniform-state test cannot validate boundary twist; a sign test cannot validate the absolute energy scale. Device-capable tests without executed-device identity remain capability evidence, not GPU qualification.

Python API test request¶

The following complete stage-first scenario is the executable authoring fixture for the interfacial-DMI zero-field test. A uniform state makes every spatial derivative vanish; the saved H_dmi field and e_dmi scalar must therefore be zero to the tolerance declared by the validation artifact. This Python example defines the request but does not itself promote a backend lane to qualified status.

# %% Imports and units
import fullmag as fm

nm = 1.0e-9

# %% Deterministic FDM reference study
study = fm.study("interfacial_dmi_uniform_zero_test")
study.engine("fdm")
study.device("cpu", precision="double")
study.mode("strict")
study.objects.mesh.defaults(cell_size=(2 * nm, 2 * nm, 2 * nm))

# %% Geometry, material, uniform state, and isolated DMI term
film = study.geometry(
    fm.Box(size=(40 * nm, 40 * nm, 2 * nm), name="film"),
    name="film",
)
film.Ms = 5.8e5
film.Aex = 15.0e-12
film.Dind = 3.0e-3
film.alpha = 0.02
film.m = fm.init.UniformMagnetization((0.0, 0.0, 1.0))
study.exchange(enabled=False)
study.demag(enabled=False)
study.solver(integrator="rk4", fix_dt=1.0e-14)

# %% Ordered measurement stage
study.stages.add_run(stage_id="measure_zero_field", until=1.0e-13).autosave(
    fm.StageAutosave(
        table=fm.TableAutosave(
            t_sampl=1.0e-14,
            quantities=["t", "mx", "my", "mz", "e_dmi", "e_total"],
        ),
        fields=[fm.FieldAutosave("H_dmi", every=1.0e-14)],
    )
)

The stage-first solver scenarios that execute these terms must use the repository-owned study/stages pattern. The lowering fixture verifies only canonical normalization, not field output.

ProblemIR and provenance¶

The validation record stores the authored term kind, signed coefficient, and optional normal before planning. The resolved record stores solver/device/precision, normalized normal, boundary realization, output quantity, mesh identity, and test artifact identity. The same test name is not evidence when these resolved values differ.

Round-trip and failure semantics¶

Round-trip must preserve interfacial versus bulk DMI and must not erase the sign of \(D\). Validation errors include malformed normals, non-finite coefficients, unsupported FDM orientation, missing matching output terms, and invalid boundary policies. Unsupported combinations are rejected before execution; a fallback to another DMI variant invalidates the test. Requested intent is recorded before planning. Resolved execution is recorded after planning. Unsupported combinations are rejected before execution.

Discrete realization¶

FDM validation compares cell-centered finite differences and boundary stencils. FEM validation compares weak residuals, surface traces, quadrature energy, and recovered fields. CPU/GPU comparisons must use equal precision, equal coefficient, equal normal, equal mesh/grid, and equivalent output location before a tolerance is interpreted.

Implementation mapping¶

Python term classes own input lowering. FDM and FEM variants have separate field paths; the planner owns output legality and normal restrictions. The source map records the stable implementation symbols and tests used by this page.

Validation strategy¶

DMI validation relies on analytic checks, cross-backend comparison, and sign/symmetry tests. Unlike exchange or demagnetization, DMI has no standard problem with a universally accepted reference solution. The validation therefore focuses on:

1. Zero-field test: uniform magnetization produces zero DMI field and zero DMI energy.

2. Sign reversal: \(D \to -D\) reverses the field and energy sign.

3. Linear profile: a linear \(m_z(x,y)\) profile produces known field values that can be verified analytically.

4. Chiral wall reflection: reflecting a chiral domain wall changes the DMI energy sign.

5. Variational consistency: FEM residual matches the energy finite-difference derivative.

6. Cross-backend parity: FDM CPU vs FDM GPU, FEM CPU vs FEM GPU.

7. Tilted normal: FEM accepts arbitrary normalised normals; non-\(+z\) FDM normals are rejected.

Validation status by lane¶

Rotated interfacial DMI: Göbel 2019 bimeron¶

The dedicated Rotated interfacial Dzyaloshinskii–Moriya interaction page records a paper-based thin-film reproduction with \(D=3\,\mathrm{mJ\,m^{-2}}\). The stored strict FDM CUDA FP64 report was produced before the verifier gained its current 19 gates: it historically passed 15/15, preserved \(|Q|>0.999\) through a 100 ps zero-current hold, retained two opposite-sign \(m_z\) cores, and decreased the total energy. Until the run is repeated with the current verifier, this historical evidence is NOT VERIFIED for the current 19-gate contract, including the minimum 20 ps relaxation duration. Its execution receipt identifies an NVIDIA GeForce RTX 4080 SUPER, device operator mask 159/159, and zero fallback; those fields do not replace a fresh verification. The corresponding FDM CPU and FEM bimeron runtimes remain not verified.

Interfacial DMI tests¶

The key analytic checks for interfacial DMI:

1. Uniform \(\mathbf{m}\): for any constant \(\mathbf{m}\), all spatial derivatives vanish, so \(\mathbf{H}_{\mathrm{DMI}}=\mathbf{0}\) and \(E_{\mathrm{DMI}}=0\). Violations indicate stencil boundary errors or quadrature contamination.

2. Sign of \(D\): reversing \(D\) must reverse the effective field direction and the energy sign for any non-uniform state.

3. Linear \(m_z\) gradient: for \(\mathbf{m}=(0,0,1)\) with a small \(m_z(x)\) perturbation, the FDM and FEM fields must agree in sign and magnitude (up to discretization error).

4. FEM residual derivative: the FEM weak residual \(R_{\mathrm{DMI}}(\mathbf{m};\mathbf{v})\) must agree with the finite-difference approximation \([E(\mathbf{m}+\varepsilon\mathbf{v})-E(\mathbf{m}-\varepsilon\mathbf{v})]/(2\varepsilon)\) to within quadrature tolerance.

Bulk DMI tests¶

Bulk DMI tests follow the same structure with the appropriate energy density \(D\,\mathbf{m}\cdot(\nabla\times\mathbf{m})\):

1. Uniform \(\mathbf{m}\): zero field and energy.

2. Helical state: for a helical magnetization \(\mathbf{m}(x)=(\cos kx,\sin kx,0)\), the energy density is \(Dk\) and the field is analytically known.

3. Sign reversal: \(D\to -D\) reverses chirality preference.

Limitations and known gaps¶

• No muMAG-style standard problem exists for DMI validation; the Göbel case is a paper reproduction with an explicit Fullmag acceptance contract.

• Executed-device identity is captured for the rotated-DMI Göbel FP64 case, but not retroactively for every conventional interfacial or bulk DMI test.

• FEM GPU mixed-P1 element qualification for DMI is incomplete.

• Cross-solver (FDM vs FEM) quantitative convergence comparison has not been published.

Scientific bibliography¶

1. S. Rohart and A. Thiaville, “Skyrmion confinement in ultrathin film nanostructures in the presence of Dzyaloshinskii-Moriya interaction,” Physical Review B 88, 184422 (2013). doi:10.1103/PhysRevB.88.184422.

2. FullMag internal notes: docs/physics/0404-interfacial-dmi.md, docs/physics/0405-bulk-dmi.md, docs/physics/0406-rotated-interfacial-dmi.md, docs/physics/0812-fem-dmi-weak-residual-proof-fixture.md.

3. B. Göbel, A. Mook, J. Henk, I. Mertig, and O. A. Tretiakov, “Magnetic bimerons as skyrmion analogues in in-plane magnets,” Physical Review B 99, 060407(R) (2019), doi:10.1103/PhysRevB.99.060407.

Control Room crosswalk¶

Use Model Explorer -> Objects -> <object> -> Physics when PhysicsInteractionPanel exposes the interaction. Status: partial. frontend support is not implemented applies to physical parameters without a matching control. See {doc}/frontend/capability-register; do not infer UI support from backend or Python availability.

Python/API crosswalk¶

The linked Python API page is authoritative for exact functions, arguments, units, and failure semantics. If this page is a foundation or category overview, runnable Python is ot applicable here and must be taken from the terminal API page.

Bibliography and source scope¶

Use the scientific bibliography and source-code index on the linked terminal page. This block adds no new equation or unverified implementation claim.

Source-code index¶