# Standalone Examples

This page is the figure gallery. For the fastest API overview, equations, and shape conventions, start with [usage.md](usage.md).

The maintained standalone entry points are:
- `examples/standalone_quickstart.py` for a one-file smoke test
- `examples/standalone_gallery.py` for the figure-producing workflows below

The first three examples here are direct excerpts from `examples/standalone_gallery.py` and require `contimod_graphene` plus `matplotlib`.

## 1. ABC Trilayer: Full Model vs 2-Band Reduction

Physics: compare the full six-band rhombohedral trilayer Hamiltonian against the low-energy two-band reduction. The maintained example pins `Delta=0.0` so the plot isolates the outer-layer bias `U`, rather than the preset's small built-in `Delta=-1.15` meV offset.

Near neutrality, the low-energy branch follows the expected ABC scaling $E \propto k^3$.

![ABC trilayer band structure](_static/abc_trilayer_bandstructure.png)

```python
import numpy as np
import jax
import jax.numpy as jnp
import matplotlib.pyplot as plt
import contimod_graphene as cg

params = cg.GrapheneTBParameters.preset("tlg").replace(U=10.0, Delta=0.0)
model = cg.RhombohedralMultilayer(n_layers=3, params=params)

k_lin = 0.28 * jnp.linspace(-0.5, 0.5, 400)
ks = jnp.stack([k_lin, jnp.zeros_like(k_lin)], axis=-1)

Hs_full = model.hamiltonian_batch(ks)
bands_full = jnp.linalg.eigvalsh(Hs_full)  # shape (400, 6)

Hs_low = jax.vmap(
    lambda kx, ky: model.two_band_hamiltonian(kx, ky),
    in_axes=(0, 0),
)(*ks.T)
bands_low = jnp.linalg.eigvalsh(Hs_low)  # shape (400, 2)

mid = len(k_lin) // 2
print(np.round(np.asarray(bands_full[mid]), 3))
print(np.round(np.asarray(bands_low[mid]), 3))
print(np.round(float(bands_low[mid, 1] - bands_low[mid, 0]), 3))

fig, ax = plt.subplots()
for band in bands_full.T:
    ax.plot(k_lin, band, color="black", linewidth=1.5, alpha=0.5)
for band in bands_low.T:
    ax.plot(k_lin, band, color="red", linewidth=1.5, linestyle="--", alpha=0.9)

ax.set_xlabel(r"$k_x\,a$")
ax.set_ylabel("Energy [meV]")
ax.set_ylim(-80, 80)
plt.show()
```

```text
[-382.508 -377.508   -9.014    9.014  377.508  382.508]
[-9.014  9.014]
18.028
```

Interpretation: at `k=0`, the `2 x 2` model reproduces the low-energy pair of the full six-band Hamiltonian, while the other four bands stay hundreds of meV away.

## 2. Bilayer Landau-Level Fan

Physics: diagonalize the dense LL-basis Hamiltonian $H_{\mathrm{LL}}(B)$ as a function of magnetic field and plot the eigenvalues as a fan diagram.

![Bilayer graphene LL fan diagram](_static/blg_landau_level_fan.png)

For `n_layers=2`, the Bernal and rhombohedral kernels describe the same AB bilayer connectivity. The maintained example keeps the public `BernalMultilayer` entry point for consistency with the rest of the docs.

```python
import jax.numpy as jnp
import matplotlib.pyplot as plt
import numpy as np
import contimod_graphene as cg

params = cg.GrapheneTBParameters.preset("blg").replace(U=0.0)
model = cg.BernalMultilayer(n_layers=2, params=params)

B_values = jnp.linspace(0.5, 9.0, 80)

eigvals = []
for B in B_values:
    H = model.landau_level_hamiltonian(float(B), n_cut=40)
    eigvals.append(np.linalg.eigvalsh(np.asarray(H)))  # 158 levels for BLG at n_cut=40

sample = np.linalg.eigvalsh(
    np.asarray(model.landau_level_hamiltonian(10.0, n_cut=8, valley="K"))
)
closest = np.sort(sample[np.argsort(np.abs(sample))[:8]])
print(np.round(closest, 3))

fig, ax = plt.subplots()
for B, e in zip(np.asarray(B_values), eigvals):
    ax.plot(np.full_like(e, B), e, "k.", ms=2)

ax.set_xlabel(r"Magnetic field $B$ [T]")
ax.set_ylabel("Energy [meV]")
ax.set_ylim(-15, 15)
plt.show()
```

```text
[-89.455 -66.373 -39.829   0.195   4.361  50.655  82.021 109.28 ]
```

Interpretation: this shows the near-zero bilayer levels and the first higher LL branches before you commit to the full fan plot.

## 3. ABA Trilayer Zero-Field Bands

Physics: compute the zero-field six-band ABA trilayer dispersion along a one-dimensional $k_x$ scan.

This is intentionally a custom-parameter example. The package does not currently ship a dedicated ABA trilayer preset, so the cleanest maintained path here is `GrapheneTBParameters.from_dict(...)`.

![ABA trilayer band structure](_static/aba_trilayer_bandstructure.png)

```python
import numpy as np
import jax.numpy as jnp
import matplotlib.pyplot as plt
import contimod_graphene as cg

graphene_params_ABA = {
    "gamma0": 3100,
    "gamma1": 370,
    "gamma2": -19,
    "gamma3": 315,
    "gamma4": 140,
    "gamma5": 20,
    "U": 0.0,
    "Delta": 18.5,
    "delta": 3.8,
}

k_lin = 0.28 * jnp.linspace(-0.5, 0.5, 400)
ks = jnp.stack([k_lin, jnp.zeros_like(k_lin)], axis=-1)

params_aba = cg.GrapheneTBParameters.from_dict(graphene_params_ABA)
model_aba = cg.BernalMultilayer(n_layers=3, params=params_aba)

Hs = model_aba.hamiltonian_batch(ks)
bands = jnp.linalg.eigvalsh(Hs)  # shape (400, 6)

print(np.round(np.linalg.eigvalsh(np.asarray(model_aba.hamiltonian(0.0, 0.0))), 3))

fig, ax = plt.subplots()
for band in bands.T:
    ax.plot(k_lin, band, color="black", linewidth=1.5, alpha=0.5)

ax.set_xlabel(r"$k_x\,a$")
ax.set_ylabel("Energy [meV]")
ax.set_ylim(-80, 80)
plt.show()
```

```text
[-514.476  -15.45    -9.25    -0.25    18.75   532.076]
```

Interpretation: the outer pair sits far from neutrality, while the inner four bands show the familiar ABA mix of monolayer-like and bilayer-like low-energy structure.

## 4. Quickstart And Notebooks

`examples/standalone_quickstart.py` is the smallest maintained script and writes a JSON summary of shapes, band extrema, and sample eigenvalues. It is the fastest way to check that the public API works on your machine.

`contimod_example.ipynb` is intentionally the one notebook in `examples/` that assumes the separate downstream `contimod` package is installed.

The older notebooks `bandstructure_plots.ipynb`, `bernal_bands_LL.ipynb`, and `landau_level_fans.ipynb` remain useful exploratory references, but the maintained default workflow should start from `examples/standalone_quickstart.py`, `examples/standalone_gallery.py`, or the short snippets in [usage.md](usage.md).