Heisenberg Hamiltonian

Hamiltonian

The Spin-Hamiltonian is defined as

Hamiltonian


Zeeman

Zeeman


Anisotropy

The anisotropy term is implemented in terms of three components:

  • uniaxial anisotropy

Uniaxial Anisotropy

  • cubic anisotropy

Cubic Anisotropy

  • biaxial anisotropy

Biaxial Anisotropy

Where for any site \(j\) the vectors \(\hat{K}_j^{(1)}\), \(\hat{K}_j^{(2)}\) and \(\hat{K}_j^{(3)} = \hat{K}_j^{(1)} \times \hat{K}_j^{(2)}\) are pairwise orthonormal.

The uniaxial anisotropy is equivalent to the biaxial anisotropy up to an offset to the total energy when setting

equivalence


Exchange

  • symmetric exchange interaction

Exchange

  • Dzyaloshinskii–Moriya interaction (antisymmetric exchange)

DMI

where it is important to note that <ij> denotes the unique pairs of interacting spins i and j.


Dipole-Dipole Interaction

Dipole-Dipole Interaction


Quadruplet Interaction

Quadruplet Interaction


Gaussian (test-) Hamiltonian

The Hamiltonian is defined as

Gaussian Hamiltonian