#  Square-lattice antiferromagnet

Source: [python (.py)](square_antiferromagnet.py) | [jupyter (.ipynb)](square_antiferromagnet.ipynb) 


_This tutorial mirrors MATLAB spinW [Tutorial 4](../../../spinw/tutorials/tutorial04)_

 In this tutorial we will show how to calculate the magnon spectrum for a square lattice
 antiferromagnet, and in particular for the case of La$_2$CuO$_4$ (Coldea et al., 
 _Phys. Rev. Lett._ **86** 5377 (2001).

 As previously, we import pyspinw, and this time we also import matplotlib.

```python
from pyspinw import *
import matplotlib.pyplot as plt
```
##  A J1-J2 antiferromagnet on a square lattice
 First we examine the case of a square lattice with a strong antiferromagnetic first neighbour
 and weak _ferromagnetic_ second neighbour Heisenberg interactions with the following parameters:
  - ordering wave vector $\mathbf{k} = (\frac{1}{2} \frac{1}{2} 0)$
  - spins in the $a$-$b$ plane (the spins can have an arbitrary phase so we fix this by defining them
    to be parallel/anti-parallel to $a$).

```python
unit_cell = UnitCell(3, 3, 9)
sites = [Site(0, 0, 0, 1, 0, 0, name="X")]
k = CommensuratePropagationVector(0.5, 0.5, 0)
struc = Structure(sites, unit_cell, supercell=SummationSupercell(propagation_vectors=[k]))
```
 Next we define the exchange interaction with a large positive (AFM) nearest neighbour $J_1$
 and a small negative (FM) next-nearest neighbour $J_2$ and form the Hamiltonian and plot it

```python
exchanges = generate_exchanges(struc, bond=1, j=1, color=[1,0,0]) + \
            generate_exchanges(struc, bond=2, j=-0.1, color=[0,1,0])
hamiltonian = Hamiltonian(struc, exchanges)
hamiltonian.print_summary()
```

```python
view(hamiltonian)
```

![]( j1j2square.png
)

 Now we calculate the magnon spectrum along the main in-plane wavevectors

```python
path = Path([[0,0,0], [0.5,0,0], [0.5,0.5,0], [0,0,0]], n_points_per_segment=201)
fig = hamiltonian.spaghetti_plot(path, show=False)
fig.axes[0].set_ylim(0, 6.5)
```

```python
plt.show()
```

![]( j1j2spectrum.png
)

##  The case of La$_2$CuO$_4$
 We now reproduce the model of La$_2$CuO$_4$, the parent of one of the family of
 high temperature cuprate superconductors, from Coldea et al., _Phys. Rev. Lett._
 **86** 5377 (2001), which has three nearest-neighbour interactions

```python
j1 = generate_exchanges(struc, bond=1, exchange_type=HeisenbergExchange, j=59.65)
j2 = generate_exchanges(struc, bond=2, exchange_type=HeisenbergExchange, j=-3.75)
j3 = generate_exchanges(struc, bond=3, exchange_type=HeisenbergExchange, j=1)
hamiltonian = Hamiltonian(struc, j1 + j2 + j3)
hamiltonian.print_summary()
```

```python
view(hamiltonian)
```

![]( lco_structure.png
)

 This time we use a slightly different path to emphasise the effect of the third neighbour
 interaction in the $X - P, (\frac{1}{2}00)-(\frac{3}{4}\frac{1}{4}0)$ direction.

```python
path = Path([[3/4, 1/4, 0], [1/2, 1/2, 0], [1/2, 0, 0], [3/4, 1/4, 0], [1, 0, 0], [1/2, 0, 0]], n_points_per_segment=201)
fig = hamiltonian.spaghetti_plot(path, show=False)
fig.axes[0].set_ylim(0, 300)
```

```python
plt.show()
```

![]( lco_spectrum.png
)