magnopy.examples.cubic_ferro_nn

magnopy.examples.cubic_ferro_nn(a: float = 1, J_iso: float = 1, J_21=(0, 0, 0), S: float = 0.5, dimensions: int = 3) → SpinHamiltonian

Prepares ferromagnetic Hamiltonian on the cubic lattice with one atom per unit cell.

Only nearest-neighbor isotropic exchange interactions and on-site quadratic anisotropy are populated. The Hamiltonian has the form

\[\mathcal{H} = \dfrac{1}{2} \sum_{\mu, \nu} J_{iso} \left( \boldsymbol{S}_{\mu} \cdot \boldsymbol{S}_{\mu+\nu} \right) + \sum_{\mu} \boldsymbol{S}_{\mu} \cdot \boldsymbol{J}_{21} \cdot \boldsymbol{S}_{\mu}\]

where

• \(\nu \in {(1,0,0), (-1,0,0)}\) if dimensions == 1.

• \(\nu \in {(1,0,0), (-1,0,0), (0,1,0), (0,-1,0)}\) if dimensions == 2.

• \(\nu \in {(1,0,0), (-1,0,0), (0,1,0), (0,-1,0), (0,0,1), (0,0,-1)}\) if dimensions == 3.

Parameters:

afloat, default 1.0

Lattice parameter of the cubic lattice. The unit cell of the Hamiltonian is defined as

[[a, 0, 0], [0, a, 0], [0, 0, a]]

J_isofloat, default 1.0

Isotropic exchange parameter for the pairs of the nearest-neighbor magnetic sites, given in energy units. Only magnitude is important, sign is ignored.

J_21(3, 3) or (3, ) array-like, default (0, 0, 0)

On-site quadratic anisotropy.

Sfloat, default 0.5

Spin value of the magnetic site.

dimensionsint, default 3

Either 1, 2 or 3. Dimensionality of the spin Hamiltonian.

Returns:

spinhamSpinHamiltonian

Spin Hamiltonian.

Examples

To get an example with the default values use

>>> import magnopy
>>> spinham = magnopy.examples.cubic_ferro_nn()

>>> spinham.cell
array([[1., 0., 0.],
       [0., 1., 0.],
       [0., 0., 1.]])

>>> spinham.atoms.names
['X']
>>> spinham.atoms.spins
[0.5]
>>> spinham.atoms.positions
[[0, 0, 0]]

>>> for _, alphas, parameter in spinham.p21:
...     print(alphas[0], parameter, sep="\n")
0
[[0. 0. 0.]
 [0. 0. 0.]
 [0. 0. 0.]]

>>> for nus, alphas, parameter in spinham.p22:
...     print(alphas[0], alphas[1], nus[0])
...     print(parameter)
0 0 (-1, 0, 0)
[[-1. -0. -0.]
 [-0. -1. -0.]
 [-0. -0. -1.]]
0 0 (0, -1, 0)
[[-1. -0. -0.]
 [-0. -1. -0.]
 [-0. -0. -1.]]
0 0 (0, 0, -1)
[[-1. -0. -0.]
 [-0. -1. -0.]
 [-0. -0. -1.]]
0 0 (0, 0, 1)
[[-1. -0. -0.]
 [-0. -1. -0.]
 [-0. -0. -1.]]
0 0 (0, 1, 0)
[[-1. -0. -0.]
 [-0. -1. -0.]
 [-0. -0. -1.]]
0 0 (1, 0, 0)
[[-1. -0. -0.]
 [-0. -1. -0.]
 [-0. -0. -1.]]

With this function one can customize a few things of the Hamiltonian:

• Lattice parameter

  >>> import magnopy
  >>> spinham = magnopy.examples.cubic_ferro_nn(a=2)
  >>> spinham.cell
  array([[2., 0., 0.],
         [0., 2., 0.],
         [0., 0., 2.]])
  

• Spin values

  >>> import magnopy
  >>> spinham = magnopy.examples.cubic_ferro_nn(S=1.5)
  >>> spinham.atoms.spins
  [1.5]
  

• Value of the isotropic exchange

  >>> import magnopy
  >>> spinham = magnopy.examples.cubic_ferro_nn(J_iso=2)
  >>> for nus, alphas, parameter in spinham.p22:
  ...     print(alphas[0], alphas[1], nus[0])
  ...     print(parameter)
  0 0 (-1, 0, 0)
  [[-2. -0. -0.]
   [-0. -2. -0.]
   [-0. -0. -2.]]
  0 0 (0, -1, 0)
  [[-2. -0. -0.]
   [-0. -2. -0.]
   [-0. -0. -2.]]
  0 0 (0, 0, -1)
  [[-2. -0. -0.]
   [-0. -2. -0.]
   [-0. -0. -2.]]
  0 0 (0, 0, 1)
  [[-2. -0. -0.]
   [-0. -2. -0.]
   [-0. -0. -2.]]
  0 0 (0, 1, 0)
  [[-2. -0. -0.]
   [-0. -2. -0.]
   [-0. -0. -2.]]
  0 0 (1, 0, 0)
  [[-2. -0. -0.]
   [-0. -2. -0.]
   [-0. -0. -2.]]
  

• Diagonal of the on-site quadratic anisotropy

  >>> import magnopy
  >>> spinham = magnopy.examples.cubic_ferro_nn(J_21=(1, 2, -1))
  >>> for _, alphas, parameter in spinham.p21:
  ...     print(alphas[0], parameter, sep="\n")
  0
  [[ 1.  0.  0.]
   [ 0.  2.  0.]
   [ 0.  0. -1.]]
  

• Dimensionality of the nearest neighbors

  1D ferromagnet

  >>> import magnopy
  >>> spinham = magnopy.examples.cubic_ferro_nn(dimensions=1)
  >>> for nus, alphas, parameter in spinham.p22:
  ...     print(alphas[0], alphas[1], nus[0])
  ...     print(parameter)
  0 0 (-1, 0, 0)
  [[-1. -0. -0.]
   [-0. -1. -0.]
   [-0. -0. -1.]]
  0 0 (1, 0, 0)
  [[-1. -0. -0.]
   [-0. -1. -0.]
   [-0. -0. -1.]]
  

  2D ferromagnet

  >>> spinham = magnopy.examples.cubic_ferro_nn(dimensions=2)
  >>> for nus, alphas, parameter in spinham.p22:
  ...     print(alphas[0], alphas[1], nus[0])
  ...     print(parameter)
  0 0 (-1, 0, 0)
  [[-1. -0. -0.]
   [-0. -1. -0.]
   [-0. -0. -1.]]
  0 0 (0, -1, 0)
  [[-1. -0. -0.]
   [-0. -1. -0.]
   [-0. -0. -1.]]
  0 0 (0, 1, 0)
  [[-1. -0. -0.]
   [-0. -1. -0.]
   [-0. -0. -1.]]
  0 0 (1, 0, 0)
  [[-1. -0. -0.]
   [-0. -1. -0.]
   [-0. -0. -1.]]
  

  3D ferromagnet (default)

  >>> spinham = magnopy.examples.cubic_ferro_nn(dimensions=3)
  >>> for nus, alphas, parameter in spinham.p22:
  ...     print(alphas[0], alphas[1], nus[0])
  ...     print(parameter)
  0 0 (-1, 0, 0)
  [[-1. -0. -0.]
   [-0. -1. -0.]
   [-0. -0. -1.]]
  0 0 (0, -1, 0)
  [[-1. -0. -0.]
   [-0. -1. -0.]
   [-0. -0. -1.]]
  0 0 (0, 0, -1)
  [[-1. -0. -0.]
   [-0. -1. -0.]
   [-0. -0. -1.]]
  0 0 (0, 0, 1)
  [[-1. -0. -0.]
   [-0. -1. -0.]
   [-0. -0. -1.]]
  0 0 (0, 1, 0)
  [[-1. -0. -0.]
   [-0. -1. -0.]
   [-0. -0. -1.]]
  0 0 (1, 0, 0)
  [[-1. -0. -0.]
   [-0. -1. -0.]
   [-0. -0. -1.]]