crystod-mag#

Symmetry-adapted spin bases (cluster multipoles / SAMM): every symmetry-distinct magnetic order allowed on the sites of one element, printed with its irrep, its per-atom spin directions, a ready-to-paste VASP or Quantum ESPRESSO magnetization input, and a VESTA file with spin arrows.

I want to …	command
enumerate the magnetic orders at q = 0	`crystod-mag -c POSCAR --element Ni --qpoint 0 0 0`
survey every special k point	`crystod-mag -c POSCAR --element Ni`
get a Quantum ESPRESSO input instead	`... --format qe`
build a magnetic supercell at q ≠ 0	`crystod-mag -c POSCAR --element Ni --qpoint R`

28. Symmetry-adapted spin bases#

Example directory: example/28_spin_basis (testsuite section 28)

Treat the spins on the sites of a magnetic element as axial-vector degrees of freedom and decompose them into space-group irreps at a q point by projection — a complete, symmetry-exhaustive enumeration of the ferromagnetic and antiferromagnetic arrangements, following the cluster-multipole / symmetry-adapted multipole moment (SAMM) framework of M.-T. Suzuki et al. [PRB 95, 094406 (2017); PRB 99, 174407 (2019)]:

crystod-mag -c example/test_POSCARs/221_PPOSCAR_AlNi3 --element Ni --qpoint 0 0 0
crystod-mag -c example/test_POSCARs/221_PPOSCAR_AlNi3 --element Ni --qpoint 0 0 0 --format qe

# survey mode: spin-multipole irreps at ALL special k points
crystod-mag -c example/test_POSCARs/221_PPOSCAR_AlNi3 --element Ni

The output opens with the magnetic sites and the complete irrep enumeration — for the Ni 3c cluster of AlNi3 (the Mn3Ir geometry), 9 spin degrees of freedom decompose into exactly three symmetry-adapted families:

Space group: Pm-3m (#221)
Magnetic sites: Ni x 3
  Ni1: [0.5, 0.5, 0.0]
  Ni2: [0.5, 0.0, 0.5]
  Ni3: [0.0, 0.5, 0.5]

Selected q-point: GAMMA = [0.0, 0.0, 0.0]

Spin (axial-vector) representation on Ni sites: 9 dimensions
Decomposition: 2 x GM4+(3) + GM5+(3)

Symmetry-adapted spin bases:
  GM4+(3): dim 3 [FM, dipole]
  GM4+(3): dim 3 [AFM, octupole]
  GM5+(3): dim 3 [AFM, octupole]

Every basis vector is then printed with its per-atom spin directions, net moment, and the magnetization input. The [111] component of the GM4+ cluster octupole is the experimentally realized 120-degree structure of Mn3Ir — note the exactly vanishing net moment:

=== GM4+(3) [AFM, octupole] ===
  component 111:
    Ni1: S = [ 0.4082,  0.4082,  0.8165]
    Ni2: S = [ 0.4082, -0.8165, -0.4082]
    Ni3: S = [-0.8165,  0.4082, -0.4082]
    net moment: [0.0, 0.0, -0.0]
    spin directions (all atoms, POSCAR order):
      Al1: [0, 0, 0]
      Ni2: [0.4082, 0.4082, 0.8165]
      Ni3: [0.4082, -0.8165, -0.4082]
      Ni4: [-0.8165, 0.4082, -0.4082]
    MAGMOM = 0 0 0   0.4082 0.4082 0.8165   0.4082 -0.8165 -0.4082   -0.8165 0.4082 -0.4082
    written to: POSCAR_AlNi3_spin_GM4+_octupole_111.vesta

Per-atom spin directions and a ready-to-paste noncollinear magnetization input are printed by default for every basis vector. --format selects the input format:

vasp (default) prints a noncollinear MAGMOM line;
qe prints the Quantum ESPRESSO counterpart — noncolin = .true. with per-type starting_magnetization(i) / angle1(i) (polar angle from z) / angle2(i) (azimuth from x) for the &SYSTEM namelist, where the magnetic element is split into one atom type per distinct spin direction (the atom membership of each type is printed as comments for ATOMIC_SPECIES/ATOMIC_POSITIONS).

With --format qe, the same 120-degree octupole becomes (Ni split into three types because the three spins point in three different directions):

    Quantum ESPRESSO noncollinear magnetization (&SYSTEM):
      noncolin = .true.
      ! type 1 (Al): Al1 -- non-magnetic
      ! type 2 (Ni1): Ni2  S = [0.4082, 0.4082, 0.8165]
      starting_magnetization(2) = 1.0000
      angle1(2) = 35.2611
      angle2(2) = 45.0000
      ! type 3 (Ni2): Ni3  S = [0.4082, -0.8165, -0.4082]
      starting_magnetization(3) = 1.0000
      angle1(3) = 114.0928
      angle2(3) = -63.4378
      ! type 4 (Ni3): Ni4  S = [-0.8165, 0.4082, -0.4082]
      starting_magnetization(4) = 1.0000
      angle1(4) = 114.0928
      angle2(4) = 153.4378
      ! angle1 = polar angle from the z axis (deg); angle2 = azimuth from the x axis in the xy plane (deg).
      ! Split Ni into the types above in ATOMIC_SPECIES / ATOMIC_POSITIONS to realize this spin arrangement.

When --qpoint is omitted, the spin (axial-vector) irrep decomposition is listed for every special k point of the space group — the magnetic counterpart of the crystod SALC survey (e.g. for AlNi3: GM: 2.0 [GM4+(3)] + 1.0 [GM5+(3)], R: R2+ + R3+ + R4+ + R5+, …).

For the Mn3Ir-type Ni 3c cluster of AlNi3 this yields 9 dims = 2 x GM4+(3) + GM5+(3): the GM4+ (T1g) cluster dipole (FM), the GM4+ (T1g) cluster octupole (AFM: the experimentally realized 120-degree structure of Mn3Ir, which shares the irrep with the dipole and hence allows the anomalous Hall effect), and the GM5+ (T2g) cluster octupole (AFM). Every AFM basis satisfies sum_i S_i = 0 exactly.

Each basis vector is exported as POSCAR_<formula>_spin_<irrep>_<dipole|octupole|...>_<direction>.vesta (e.g. POSCAR_AlNi3_spin_GM4+_octupole_111.vesta for the 120-degree Mn3Ir-type state), with spin arrows on the magnetic atoms scaled so that the largest spin gets --amplitude Angstroms (default 1.5). For q != 0 (e.g. --qpoint R) the commensurate magnetic supercell is built automatically, with the Bloch phase applied to the spins and the MAGMOM/VESTA output referring to the supercell. --conventional exports the spin structures in the conventional cell instead of the primitive cell (VESTA files get a _conv suffix, and for q != 0 the conventional cell is multiplied until the Bloch phase is commensurate); --tolerance sets the symmetry tolerance (default 1e-5).

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28. Symmetry-adapted spin bases#