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.

Now we are ready to discuss magnopy.SpinHamiltonian.

It is created on some crystal, which was discussed in the previous section and adds interaction parameters to it.

Read magnopy's dedicated documentation page for the detailed introduction.

Creating a Hamiltonian

Spin Hamiltonian is created from three objects

• cell

• atoms

• convention

Here we create a Hamiltonian for the ferromagnetic nearest-neighbor Heisenberg model with triaxial on-site anisotropy on a simple orthorhombic lattice.

\[\mathcal{H} = \sum_{i} \Bigl( K_x (S_i^x)^2 + K_y (S_i^y)^2 + K_z (S_i^z)^2 \Bigr) + \dfrac{1}{2} J \sum_{i, j} \mathbf{S}_i \cdot \mathbf{S}_j\]

In the notation (not convention, but notation) that resembles magnopy's code the same Hamiltonian can be written as

\[\mathcal{H} = \sum_{\mu,\alpha} \mathbf{S}_{\mu, \alpha} \cdot \mathbf{J}_{\mu, \alpha} \cdot \mathbf{S}_{\mu, \alpha} + \dfrac{1}{2} \sum_{\mu,\nu,\alpha,\beta} \mathbf{S}_{\mu, \alpha} \cdot \mathbf{J}_{\alpha\beta,\nu} \cdot \mathbf{S}_{\mu+\nu, \beta}\]

where

\[\begin{split}\mathbf{J}_{\mu\alpha} = \begin{pmatrix} K_x & 0 & 0 \\ 0 & K_y & 0 \\ 0 & 0 & K_z \end{pmatrix}\end{split}\]

and

\[\begin{split}\mathbf{J}_{\alpha\beta,\nu} = \begin{pmatrix} J & 0 & 0 \\ 0 & J & 0 \\ 0 & 0 & J \end{pmatrix}\end{split}\]

if \(\alpha = \beta\) and \(\nu \in \{(1,0,0), (-1,0,0), (0,1,0), (0,-1,0), (0,0,1), (0,0,-1)\}\) and

\[\begin{split}\mathbf{J}_{\alpha\beta,\nu} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}\end{split}\]

otherwise.

import numpy as np
import magnopy

Creating a Hamiltonian instance

# Orthorhombic cell with a = 1, b=1.5, c=2
cell = np.diag([1, 1.5, 2])

# One atom per unit cell
atoms = dict(
    names=["Fe"],
    positions=[[0.0, 0.0, 0.0]],
    spins=[2.5],
    g_factors=[2],
)

# Convention
convention = magnopy.Convention(
    multiple_counting=True, spin_normalized=False, c21=1, c22=1 / 2
)

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

Adding parameter to the Hamiltonian

Now everything is ready to add some parameters. Magnopy stores the parameters in the form that closely resembles mathematical form of the spin Hamiltonian, which can be found in Spin Hamiltonian page (see "Expanded form").

Magnopy supports terms with up to four spins with full tensors of the interaction parameters

• \(1\times3\) for one-spin parameters

• \(3\times3\) for two-spin parameters

• \(3\times3\times3\) for three-spin parameters

• \(3\times3\times3\times3\) for four-spin parameters.

For the purpose of this tutorial we will focus on the first three terms of the expanded form (i.e. one-spin and two-spin terms).

Two functions are defined for each group of parameter, that add or remove a parameter from the Hamiltonian. For example, to add or remove a two-spin/two-sites parameter one can use magnopy.SpinHamiltonian.add_22() or magnopy.SpinHamiltonian.remove_22().

First, lets add isotropic exchange interaction between nearest neighbors with the value \(J = -1\) meV.

# J < 0 for ferromagnetic coupling (according to the chosen convention: c22 = 1/2)
parameter = magnopy.converter22.from_iso(iso=-1)
print(f"parameter is \n{parameter}")

spinham.add_22(alpha=0, beta=0, nu=(1, 0, 0), parameter=parameter)
spinham.add_22(alpha=0, beta=0, nu=(0, 1, 0), parameter=parameter)
spinham.add_22(alpha=0, beta=0, nu=(0, 0, 1), parameter=parameter)

parameter is
[[-1. -0. -0.]
 [-0. -1. -0.]
 [-0. -0. -1.]]

Note

• alpha and beta are indices of the lists in atoms. In that example they both point to the first atom.

• Due to the translation symmetry of the Hamiltonian it is enough to specify all parameters for some chosen unit cell. This unit cell is commonly labeled as (0, 0, 0). Index alpha specifies the first atom, that is always in the (0, 0, 0) unit cell. Index beta specify the second atom, that is understood to be located in the unit cell specified by nu. In the example above second atom is from (1, 0, 0), (0, 1, 0) or (0, 0, 1) unit cell.

• Any parameter for the term that involves two spins is a 3x3 matrix. An isotropic parameter in the matrix form is a diagonal matrix with all diagonal elements being the same.

One does not need to add parameters for \(\nu \in \{(-1,0,0), (0,-1,0), (0,0,-1)\}\). Since multiple counting is True, magnopy will automatically add those.

for index, (alpha, beta, nu, parameter) in enumerate(spinham.p22):
    atom_000 = spinham.atoms["names"][alpha]
    atom_nu = spinham.atoms["names"][beta]
    print(f"Bond #{index + 1}")
    print(f"{atom_000} in (0, 0, 0) unit cell -> {atom_nu} in {nu} unit cell")
    print(parameter)

Bond #1
Fe in (0, 0, 0) unit cell -> Fe in (0, 0, 1) unit cell
[[-1. -0. -0.]
 [-0. -1. -0.]
 [-0. -0. -1.]]
Bond #2
Fe in (0, 0, 0) unit cell -> Fe in (0, 1, 0) unit cell
[[-1. -0. -0.]
 [-0. -1. -0.]
 [-0. -0. -1.]]
Bond #3
Fe in (0, 0, 0) unit cell -> Fe in (1, 0, 0) unit cell
[[-1. -0. -0.]
 [-0. -1. -0.]
 [-0. -0. -1.]]
Bond #4
Fe in (0, 0, 0) unit cell -> Fe in (0, 0, -1) unit cell
[[-1. -0. -0.]
 [-0. -1. -0.]
 [-0. -0. -1.]]
Bond #5
Fe in (0, 0, 0) unit cell -> Fe in (0, -1, 0) unit cell
[[-1. -0. -0.]
 [-0. -1. -0.]
 [-0. -0. -1.]]
Bond #6
Fe in (0, 0, 0) unit cell -> Fe in (-1, 0, 0) unit cell
[[-1. -0. -0.]
 [-0. -1. -0.]
 [-0. -0. -1.]]

Next, we add on-site anisotropy with \(K_x = -0.1\) meV, \(K_y = -0.2\) and \(K_z = -0.3\) meV.

parameter = np.diag([-0.1, -0.2, -0.3])
print(f"parameter is \n{parameter}")

spinham.add_21(alpha=0, parameter=parameter)

for alpha, parameter in spinham.p21:
    print(
        f"On-site anisotropy for atom {spinham.atoms['names'][alpha]} is\n{parameter}"
    )

parameter is
[[-0.1  0.   0. ]
 [ 0.  -0.2  0. ]
 [ 0.   0.  -0.3]]
On-site anisotropy for atom Fe is
[[-0.1  0.   0. ]
 [ 0.  -0.2  0. ]
 [ 0.   0.  -0.3]]

Changing convention

All parameters that are added to the Hamiltonian are expected to be compliant with the Hamiltonian's convention. The latter can always be checked with

print(spinham.convention)

"custom" convention where
  * Bonds are counted multiple times in the sum;
  * Spin vectors are not normalized;
  * Undefined c1 factor;
  * c21 = 1.0;
  * c22 = 0.5;
  * Undefined c31 factor;
  * Undefined c32 factor;
  * Undefined c33 factor;
  * Undefined c41 factor;
  * Undefined c421 factor;
  * Undefined c422 factor;
  * Undefined c43 factor;
  * Undefined c44 factor.

You can change the convention of the Hamiltonian at any moment. Magnopy will recompute all existing parameters in such a way, that physical properties of the Hamiltonian do not change. All parameters that will be added later shall be compliant with the new convention.

new_convention = magnopy.Convention.get_predefined(name="GROGU")

spinham.convention = new_convention

print(spinham.convention)

"grogu" convention where
  * Bonds are counted multiple times in the sum;
  * Spin vectors are normalized to 1;
  * Undefined c1 factor;
  * c21 = 1.0;
  * c22 = 0.5;
  * Undefined c31 factor;
  * Undefined c32 factor;
  * Undefined c33 factor;
  * Undefined c41 factor;
  * Undefined c421 factor;
  * Undefined c422 factor;
  * Undefined c43 factor;
  * Undefined c44 factor.

Arithmetic operations

Mathematical form of the spin Hamiltonian involves a lot of summation. It might be convenient to be able to sum two Hamiltonians together.

Good news is that magnopy implements addition and subtraction of two Hamiltonians. Moreover, it implements multiplication of the Hamiltonian by any number.

Important

For summation and subtraction the Hamiltonians shall be defined on the same cell and atoms.

Hint

To get an independent instance of spin Hamiltonian with the same cell, atoms and convention, but with no parameters you can use magnopy.SpinHamiltonian.get_empty().

For example, imagine that we want to have isotropic and anisotropic terms of the Hamiltonian separately.

First, let us create an empty Hamiltonian with the same cell, atoms and convention for each term.

aniso_term = spinham.get_empty()
iso_term = spinham.get_empty()

Then we add isotropic exchange to the first one and anisotropy to the second one

Note

Parameters for (0, 0, 0) -> (1, 0, 0) bond and (0, 0, 0)-> (-1, 0, 0) bond are connected via multiple counting. Therefore, when one of them is added, addition of the second one will raise an error. Same problem will appear if we try to add (0, 0, 0) -> (1, 0, 0) bond twice.

This happens because magnopy always stores only one of the equivalent bonds internally, thus by adding the second parameter we essentially are trying to add the parameter to the same bond again. In other words, the same problem appears if we try to add (0, 0, 0) -> (1, 0, 0) bond twice.

When the parameter is added to the same bond again, magnopy raises an error by default. There are a number of strategies on how magnopy can handle such situations without raising an error. They are controlled with when_present argument, that is present in all spinham.add_* methods. Here we use when_present="skip", which means that repeated parameters are safely skipped without raising an error. See magnopy.SpinHamiltonian.add_22() for more options for when_present argument.

for alpha, parameter in spinham.p21:
    aniso_term.add_21(alpha=alpha, parameter=parameter)

for alpha, beta, nu, parameter in spinham.p22:
    iso_term.add_22(
        alpha=alpha, beta=beta, nu=nu, parameter=parameter, when_present="skip"
    )

Now full Hamiltonian can be computed as simple as

full_spinham = aniso_term + iso_term

Moreover, we can define new term - antisymmetric Dzyaloshinskii–Moriya interaction.

dmi_term = iso_term.get_empty()

dmi_term.add_22(
    alpha=0,
    beta=0,
    nu=(0, 1, 0),
    parameter=magnopy.converter22.from_dmi(dmi=(0.5, 0, 0)),
)

full_spinham = iso_term + aniso_term + dmi_term

Visualizing

Note

This functionality of magnopy is still experimental.

One can visualize on-site (involves one site) and two-spin/two-sites parameters using magnopy.experimental.plot_spinham()

pe1, pe2 = magnopy.experimental.plot_spinham(
    full_spinham, distance_digits=3, _sphinx_gallery_fix=True
)

This function returns two instances of magnopy.PlotlyEngine. First (pe1) with the on-site parameters

Hint

Hover over the magnetic cites to see the values of parameters.

Click on the legend to hide some of the elements.

pe1.show(axes_visible=False)

Adn second (pe2) one with all the exchange bonds. DMI vectors are displayed as well.

Hint

Double-click on some element of the legend to hide all other ones.

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