Bulk Dzyaloshinskii–Moriya interaction¶

Last changes: 23:23 29.08.2026Documentation changelog

This is the canonical public reference for isotropic bulk Dzyaloshinskii–Moriya interaction (bulk or Bloch DMI). Common physics is defined once here; FDM/FEM and CPU/GPU differences are realization sections.

The public coefficient is \(D_b\) in \(\mathrm{J\,m^{-2}}\). It multiplies one spatial derivative, so the resulting volume density is in \(\mathrm{J\,m^{-3}}\).

1. Physical problem¶

Let \(\Omega_m\) be the magnetic domain and \(\mathbf M=M_s\mathbf m\) the physical magnetization, with \(\lvert\mathbf m\rvert=1\) on magnetic degrees of freedom:

(1)¶\[w_b(\mathbf m,\nabla\mathbf m)=D_b\,\mathbf m\cdot(\nabla\times\mathbf m).\]

(2)¶\[E_b[\mathbf m]=\int_{\Omega_m}w_b\,\mathrm dV.\]

(3)¶\[\mathbf m\cdot(\nabla\times\mathbf m) =m_x(\partial_y m_z-\partial_zm_y) +m_y(\partial_zm_x-\partial_xm_z) +m_z(\partial_xm_y-\partial_ym_x).\]

This is a volume invariant, distinct from interfacial DMI.

2. Governing equations¶

For an admissible variation \(\mathbf v\),

(4)¶\[\delta E_b(\mathbf m;\mathbf v) =D_b\int_{\Omega_m} \left[\mathbf v\cdot(\nabla\times\mathbf m) +\mathbf m\cdot(\nabla\times\mathbf v)\right]\mathrm dV.\]

Using \(\nabla\cdot(\mathbf v\times\mathbf m) =\mathbf m\cdot(\nabla\times\mathbf v) -\mathbf v\cdot(\nabla\times\mathbf m)\):

(5)¶\[\delta E_b =2D_b\int_{\Omega_m}\mathbf v\cdot(\nabla\times\mathbf m)\,\mathrm dV +D_b\int_{\partial\Omega_m} (\mathbf v\times\mathbf m)\cdot\boldsymbol\nu\,\mathrm dS.\]

With the Fullmag energy-field convention \(\delta E=-\mu_0\int_{\Omega_m}M_s\mathbf H_b\cdot\mathbf v\,\mathrm dV\):

(6)¶\[\mathbf H_b=-\frac{2D_b}{\mu_0M_s}(\nabla\times\mathbf m).\]

The natural boundary term and exchange-coupled free boundary law are

(7)¶\[\delta E_b^{\partial\Omega} =D_b\int_{\partial\Omega_m} [\mathbf m\times\boldsymbol\nu]\cdot\mathbf v\,\mathrm dS.\]

(8)¶\[2A\,\partial_{\boldsymbol\nu}\mathbf m +D_b(\mathbf m\times\boldsymbol\nu)=\mathbf0.\]

The non-periodic FDM planner does not expose this natural closure and fails closed; FEM carries it through the weak residual.

3. Symbols and SI units¶

Symbol	Meaning	SI unit
\(\mathbf M\)	physical magnetization \(M_s\mathbf m\)	\(\mathrm{A\,m^{-1}}\)
\(M_s\)	saturation magnetization	\(\mathrm{A\,m^{-1}}\)
\(\mathbf m\)	reduced magnetization	\(1\)
\(\mathbf v\)	admissible variation or test field	\(1\)
\(D_b\)	bulk DMI coefficient	\(\mathrm{J\,m^{-2}}\)
\(A\)	exchange stiffness in the coupled boundary law	\(\mathrm{J\,m^{-1}}\)
\(\mu_0\)	vacuum permeability	\(\mathrm{N\,A^{-2}}\)
\(\Omega_m\)	magnetic integration domain	\(\mathrm{m^3}\)
\(\partial\Omega_m\)	magnetic boundary	\(\mathrm{m^2}\)
\(\boldsymbol\nu\)	outward unit normal	\(1\)
\(w_b\)	bulk DMI energy density	\(\mathrm{J\,m^{-3}}\)
\(E_b\)	total bulk DMI energy	\(\mathrm{J}\)
\(\mathbf H_b\)	bulk DMI effective field	\(\mathrm{A\,m^{-1}}\)
\(m_\alpha\)	Cartesian magnetization component	\(1\)
\(\partial_\alpha\)	spatial derivative	\(\mathrm{m^{-1}}\)
\(\delta_\alpha\)	centered FDM derivative	\(\mathrm{m^{-1}}\)
\(\nabla_h\times\)	centered discrete curl operator	\(\mathrm{m^{-1}}\)
\(\Delta x\)	FDM x cell dimension	\(\mathrm{m}\)
\(\Delta y\)	FDM y cell dimension	\(\mathrm{m}\)
\(\Delta z\)	FDM z cell dimension	\(\mathrm{m}\)
\(i\)	FDM cell index	\(1\)
\(\mathcal A\)	active FDM cell set	\(1\)
\(V_i\)	FDM cell volume	\(\mathrm{m^3}\)
\(\mathbf m_h\)	FEM finite-element magnetization	\(1\)
\(\mathbf v_h\)	FEM finite-element test field	\(1\)
\(\Omega_e\)	FEM element domain	\(\mathrm{m^3}\)
\(q\)	quadrature point index	\(1\)
\(D_{b,e}\)	element bulk DMI coefficient	\(\mathrm{J\,m^{-2}}\)
\(\mathbf m_q\)	quadrature-point magnetization	\(1\)
\(w_q\)	quadrature weight including element Jacobian	\(\mathrm{m^3}\)
\(g_{b,a}\)	assembled bulk-DMI residual at node a	\(\mathrm{J}\)
\(M_{s,a}\)	nodal saturation magnetization	\(\mathrm{A\,m^{-1}}\)
\(M_a^{\mathrm{lump}}\)	nodal lumped mass	\(\mathrm{m^3}\)
\(H_{b,a}\)	projected nodal bulk-DMI field	\(\mathrm{A\,m^{-1}}\)

4. Assumptions and validity¶

The invariant is isotropic cubic/B20-type bulk DMI with one finite scalar coefficient; either sign is accepted and selects chirality.
FDM uses a centered cell curl. Periodic neighbors wrap; missing/inactive non-periodic neighbors use the center value.
Explicit FDM Bulk DMI requires all three axes periodic. Multilayer FDM rejects it.
FEM uses a nodal vector field, quadrature weak residual, and lumped-mass projection. The element coefficient is the arithmetic mean of nodal Dbulk_field values.
General Lifshitz tensors, lower-symmetry crystals, surface-only DMI, and an independent DMI boundary operator are outside this contract.

5. Python API¶

The current stage-first body API exposes the scalar bulk-DMI route as body.Dbulk. The builder lowers it to a BulkDMI energy term. The FDM planner requires all three axes to be periodic, so the complete helical-state scenario declares that policy explicitly and disables demagnetization to isolate the bulk-DMI field and energy.

# %% Imports and units
import math

import fullmag as fm

nm = 1.0e-9
period = 40 * nm
wave_number = 2.0 * math.pi / period

# %% Fully periodic FDM study required by bulk DMI
study = fm.study("bulk_dmi_helical_state")
study.engine("fdm")
study.device("cpu", precision="double")
study.mode("strict")
study.objects.mesh.defaults(cell_size=(2 * nm, 2 * nm, 2 * nm))
study.pbc(x=True, y=True, z=True)

# %% Geometry, B20 material, transverse helix, and interactions
crystal = study.geometry(
    fm.Box(size=(80 * nm, 16 * nm, 16 * nm), name="crystal"),
    name="crystal",
)
crystal.Ms = 3.84e5
crystal.Aex = 8.78e-12
crystal.Dbulk = 1.58e-3
crystal.alpha = 0.02
crystal.m = fm.texture.helical(
    wavevector=(wave_number, 0.0, 0.0),
    e1=(0.0, 1.0, 0.0),
    e2=(0.0, 0.0, 1.0),
)
study.exchange()
study.demag(enabled=False)
study.solver(integrator="rk4", fix_dt=1.0e-14)

# %% Ordered measurement stage
study.stages.add_run(stage_id="measure_bulk_dmi", until=1.0e-12).autosave(
    fm.StageAutosave(
        table=fm.TableAutosave(
            t_sampl=1.0e-13,
            quantities=["t", "mx", "my", "mz", "e_ex", "e_dmi", "e_total"],
        ),
        fields=[fm.FieldAutosave("H_dmi_bulk", every=2.0e-13)],
    )
)

Python parameter	Type	Default	SI unit	Validation domain	Meaning	Backend support	ProblemIR destination
`BulkDMI.D`	`float`	required	\(\mathrm{J\,m^{-2}}\)	finite; either sign	explicit bulk-DMI coefficient and chirality sign	FDM/FEM CPU/GPU subject to planner and qualification gates	`energy_terms[].D` for `kind=bulk_dmi`
`Material.Dbulk`	`float \| None`	`None`	\(\mathrm{J\,m^{-2}}\)	finite when supplied	material-owned scalar bulk-DMI coefficient	FEM CPU/GPU; FDM planner-dependent	`materials[].bulk_dmi`
`Material.Dbulk_field`	`list[float] \| None`	`None`	\(\mathrm{J\,m^{-2}}\)	finite values; resolved mesh cardinality	spatial nodal bulk-DMI coefficient	FEM CPU/GPU	`materials[].dbulk_field`
`FdmPbc.axes`	`tuple[bool,bool,bool]`	required	\(1\)	exactly three values	periodic/open axes	FDM CPU/GPU	`pbc.axes`
`FdmPbc.demag`	`str`	`"open"`	\(1\)	three declared demag strings	FDM demag policy	FDM	`pbc.demag`
`FdmPbc.image_counts`	`tuple[int,int,int] \| None`	`None`	\(1\)	non-negative; truncated images only	image counts	FDM	`pbc.image_counts`

The other required Material parameters (name, Ms, A, alpha) are documented by the canonical Material API page.

6. ProblemIR¶

Explicit authoring serializes as:

{"kind": "bulk_dmi", "D": 0.003}

Material authoring serializes as:

{"bulk_dmi": 0.003, "dbulk_field": null}

The explicit and material routes are alternative coefficient sources and are not silently summed. The FEM planner can use the material scalar when no explicit scalar supplies the coefficient and promotes a material field into the resolved FEM field. FdmPbc converts booleans to periodic/open strings. Requested Python intent is preserved separately from resolved execution: solver, device, precision, route, boundary policy, projection, and qualification evidence are provenance data.

7. Round-trip and failure semantics¶

Canonical script export preserves explicit BulkDMI(D=…) versus Material(Dbulk=…)/Material(Dbulk_field=…). Normalization never turns bulk DMI into interfacial DMI or invents a non-periodic FDM boundary.

Validation errors include non-finite coefficients, malformed FDM tuples, material-field cardinality mismatch, invalid IR terms, missing FEM context, and missing resident GPU buffers. The FDM planner rejects explicit Bulk DMI unless all three axes are periodic and rejects multilayer FDM.

Unsupported combinations are planner/runtime failures, not CPU fallback, silent term removal, or a claim of backend parity. Requested intent and resolved execution remain separately inspectable.

8. Discrete realization¶

FDM CPU¶

(9)¶\[(\delta_xm_\alpha)_i=\frac{m_{\alpha,i+\hat x}-m_{\alpha,i-\hat x}}{2\Delta x}, \quad (\delta_ym_\alpha)_i=\frac{m_{\alpha,i+\hat y}-m_{\alpha,i-\hat y}}{2\Delta y}, \quad (\delta_zm_\alpha)_i=\frac{m_{\alpha,i+\hat z}-m_{\alpha,i-\hat z}}{2\Delta z}.\]

(10)¶\[\begin{split}(\nabla_h\times\mathbf m)_i= \begin{bmatrix}\delta_ym_z-\delta_zm_y\\ \delta_zm_x-\delta_xm_z\\ \delta_xm_y-\delta_ym_x\end{bmatrix}_i, \qquad \mathbf H_{b,i}^{\mathrm{FDM}}= -\frac{2D_b}{\mu_0M_{s,i}}(\nabla_h\times\mathbf m)_i.\end{split}\]

(11)¶\[E_b^{\mathrm{FDM}}=\sum_{i\in\mathcal A} D_b\left[\mathbf m_i\cdot(\nabla_h\times\mathbf m)_i\right]V_i.\]

FDM GPU¶

FP64 and FP32 CUDA field-combination kernels use the same negative-curl convention and the CUDA reduction sums the bulk density. The planner still requires fully periodic axes; source presence is not device qualification.

FEM CPU¶

(12)¶\[R_{b,e}(\mathbf m_h;\mathbf v_h)=D_{b,e}\int_{\Omega_e} \left[\mathbf v_h\cdot(\nabla\times\mathbf m_h) +\mathbf m_h\cdot(\nabla\times\mathbf v_h)\right]\mathrm dV.\]

(13)¶\[E_{b,e}=D_{b,e}\sum_q \left[\mathbf m_q\cdot(\nabla\times\mathbf m_q)\right]w_q.\]

(14)¶\[H_{b,a}=-\frac{g_{b,a}}{\mu_0M_{s,a}M_a^{\mathrm{lump}}}.\]

The native loop uses the arithmetic mean of nodal Dbulk_field, periodic reduced-node projection, MFEM quadrature, and lumped projection.

FEM GPU¶

The CUDA dmi_element_residual_kernel has a bulk mode for tetrahedral gradients, coefficient, curl, residual, and energy blocks. RK dispatch uses resident h_bulk_dmi, geometry, saturation, lumped-mass, and residual buffers. Final reduction owns a separate Bulk DMI energy slot. Atomic accumulation and reduction order differ from CPU.

9. Implementation mapping¶

Layer	FDM CPU	FDM GPU	FEM CPU	FEM GPU
coefficient	explicit scalar	device D_bulk	scalar or nodal mean	resident material field
field	centered curl	FP64/FP32 centered curl	weak residual + lumped projection	CUDA residual + RK projection
energy	active-cell volume sum	CUDA reduction	quadrature sum	element blocks + final slot
boundary	all-axis periodic gate	same gate	weak residual term	same weak form
failure	non-periodic/multilayer rejection	same gate	invalid context/buffers	missing resources/evidence

10. Validation¶

Check	Expected result	Evidence boundary
constant m	zero curl, field, energy	analytical/unit test
linear Cartesian field	analytic curl components	analytical/unit test
D_b to -D_b	field and energy sign reverse	CPU/backend test
FDM CPU/GPU	same periodic FP64 grid within tolerance	managed device run
FEM weak residual	tetrahedral reference agreement	weak-residual test
FEM CPU/GPU	field and final energy agreement	managed native runtime
planner gate	non-periodic/multilayer fails closed	planner test
Python round-trip	explicit/material routes distinct	Python/IR test

Compilation, source inspection, and skipped GPU tests are not proof of executed-device parity.

11. Limitations¶

Non-periodic FDM natural exchange+DMI boundary conditions are not exposed.
Multilayer FDM Bulk DMI is rejected.
Lower-symmetry/tensorial DMI families are outside this contract.
The material-field element reduction is an arithmetic nodal mean.
Material.Dbulk warning text and the public coefficient contract use \(\mathrm{J\,m^{-2}}\).

12. Scientific bibliography¶

A. N. Bogdanov and D. A. Yablonskii, “Thermodynamically stable vortices in magnetically ordered crystals,” Soviet Physics JETP 68, 101 (1989).
A. N. Bogdanov and U. K. Rößler, “Chiral symmetry breaking in magnetic thin films and multilayers,” Physical Review Letters 87, 037203 (2001), doi:10.1103/PhysRevLett.87.037203.

13. Source-code index¶

Equation or claim	Repository path	Stable symbol	Responsibility	Lane/evidence
explicit constructor and IR	`packages/fullmag-py/src/fullmag/model/energy.py`	`class BulkDMI`	finite scalar and serialization	Python contract
material route	`packages/fullmag-py/src/fullmag/model/structure.py`	`class Material`	scalar/field serialization	Python contract
FDM periodicity	`packages/fullmag-py/src/fullmag/model/problem.py`	`class FdmPbc`	axes/policy validation	Python contract
IR explicit validation	`crates/fullmag-ir/src/validation.rs`	`validate_dmi_energy_terms`	finite explicit value	IR contract
IR material validation	`crates/fullmag-ir/src/validation.rs`	`validate_material_dmi_values`	finite material values	IR contract
FDM planner	`crates/fullmag-plan/src/fdm.rs`	`plan_fdm`	periodic gate and lowering	FDM planning
FEM planner	`crates/fullmag-plan/src/fem.rs`	`plan_fem`	material/explicit resolution	FEM planning
FDM CPU density	`crates/fullmag-engine/src/fdm/cpu/fields.rs`	`dmi_energy_density_from_vectors`	centered curl density	FDM CPU
FDM CPU field	`crates/fullmag-engine/src/fdm/cpu/fields.rs`	`bulk_dmi_field`	negative-curl field	FDM CPU
FDM CPU SoA energy	`crates/fullmag-engine/src/fdm/cpu/fields.rs`	`dmi_energy_from_soa`	volume reduction	FDM CPU
FDM CUDA field	`backends/fdm/gpu/cuda/interactions/demag_fp64.cu`	`combine_effective_field_fp64_kernel`	FP64 bulk field	FDM GPU
FEM CPU realization	`backends/fem/cpu/mfem/interactions/dmi_bulk.cpp`	`compute_bulk_dmi_field`	quadrature/residual/projection/energy	FEM CPU
FEM weak residual	`backends/fem/src/dmi_weak_residual.cpp`	`dmi_accumulate_bulk_residual`	residual algebra	FEM CPU/GPU
FEM projection	`backends/fem/src/dmi_weak_residual.cpp`	`dmi_project_lumped_field`	residual-to-field	FEM CPU/GPU
FEM CUDA kernel	`backends/fem/gpu/cuda/interactions/dmi/dmi_kernels.cu`	`dmi_element_residual_kernel`	residual/energy blocks	FEM GPU
FEM CUDA dispatch	`backends/fem/gpu/cuda/integrators/rk/rk_dmi_fields.cu`	`gpu_rk_compute_dmi_field_contributions`	resident field dispatch	FEM GPU
FEM CUDA reduction	`backends/fem/gpu/cuda/integrators/rk/rk_dmi_energy_reductions.cu`	`gpu_rk_reduce_final_dmi_energy_terms`	final energy slot	FEM GPU

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¶

Public Python and lowering sources are linked by the applicable terminal API page. Runtime realization is in the relevant backends/fdm or backends/fem lane; frontend ownership is apps/control-room/src/modules/inspector/panels/PhysicsInteractionPanel.tsx where a live control exists.