Spin Hamiltonian

Exercises

1. Create a spin Hamiltonian for the 1D chain with competing nearest and next-nearest neighbors (i. e. isotropic parameters of different sign and close absolute values).

2. Change the convention of the spin Hamiltonian. Inspect how the parameters are changing when you do so.

import numpy as np
import magnopy

Exercise 1

# Cubic cell with a = 1
cell = np.eye(3)

# One atom per unit cell
# located in the center of the unit cell
atoms = dict(
    names=["Fe"],
    positions=[[0.5, 0.5, 0.5]],
    spins=[2.5],
    g_factors=[2],
)

# Choose convention
convention = magnopy.Convention(multiple_counting=True, spin_normalized=False, c22=1)

# Create a Hamiltonian
spinham = magnopy.SpinHamiltonian(cell=cell, atoms=atoms, convention=convention)

# Add nearest neighbor parameter
# Choose 1D chain along the first lattice vector
parameter = magnopy.converter22.from_iso(iso=1)
spinham.add_22(alpha=0, beta=0, nu=(1, 0, 0), parameter=parameter)

# Add next-nearest neighbor parameter of opposite sign and slightly smaller absolute value
spinham.add_22(alpha=0, beta=0, nu=(2, 0, 0), parameter=-0.95 * parameter)

_, pe2 = magnopy.experimental.plot_spinham(spinham=spinham, _sphinx_gallery_fix=True)

pe2.show(axes_visible=False, legend_position="left")

Exercise 2

# Create two modified conventions

mod_conv_1 = convention.get_modified(multiple_counting=False)
mod_conv_2 = convention.get_modified(c22=-1)

First, display the parameters in the original convention: they have the same values as the one we passed to magnopy.SpinHamiltonian.add_22() function

for alpha, beta, nu, parameter in spinham.p22:
    print(
        f"{spinham.atoms.names[alpha]} -> {spinham.atoms.names[beta]} at {nu}\n{parameter}"
    )

Fe -> Fe at (1, 0, 0)
[[1. 0. 0.]
 [0. 1. 0.]
 [0. 0. 1.]]
Fe -> Fe at (2, 0, 0)
[[-0.95 -0.   -0.  ]
 [-0.   -0.95 -0.  ]
 [-0.   -0.   -0.95]]
Fe -> Fe at (-1, 0, 0)
[[1. 0. 0.]
 [0. 1. 0.]
 [0. 0. 1.]]
Fe -> Fe at (-2, 0, 0)
[[-0.95 -0.   -0.  ]
 [-0.   -0.95 -0.  ]
 [-0.   -0.   -0.95]]

Next we disable multiple counting. As there are twice as less parameters in the Hamiltonian in the new convention, then values of the parameters shall be two times larger to preserve physical properties of the model (for example classical energy).

spinham.convention = mod_conv_1

for alpha, beta, nu, parameter in spinham.p22:
    print(
        f"{spinham.atoms.names[alpha]} -> {spinham.atoms.names[beta]} at {nu}\n{parameter}"
    )

Fe -> Fe at (1, 0, 0)
[[2. 0. 0.]
 [0. 2. 0.]
 [0. 0. 2.]]
Fe -> Fe at (2, 0, 0)
[[-1.9 -0.  -0. ]
 [-0.  -1.9 -0. ]
 [-0.  -0.  -1.9]]

Finally, we change the sign in front of the two spins & two sites term. The parameter shall change their sign to preserve physical properties of the model.

spinham.convention = mod_conv_2

for alpha, beta, nu, parameter in spinham.p22:
    print(
        f"{spinham.atoms.names[alpha]} -> {spinham.atoms.names[beta]} at {nu}\n{parameter}"
    )

Fe -> Fe at (1, 0, 0)
[[-1. -0. -0.]
 [-0. -1. -0.]
 [-0. -0. -1.]]
Fe -> Fe at (2, 0, 0)
[[0.95 0.   0.  ]
 [0.   0.95 0.  ]
 [0.   0.   0.95]]
Fe -> Fe at (-1, 0, 0)
[[-1. -0. -0.]
 [-0. -1. -0.]
 [-0. -0. -1.]]
Fe -> Fe at (-2, 0, 0)
[[0.95 0.   0.  ]
 [0.   0.95 0.  ]
 [0.   0.   0.95]]