# Orientation-dependent interactions of anisotropic localized states

Localized states in low-symmetry two-dimensional materials, anisotropic
quantum dots, and Wannier bases need not be rotationally symmetric. Their
interaction can therefore depend on the direction connecting two centers, so a
scalar distance alone is insufficient to characterize the matrix element.

An anisotropic transition field generates this orientation dependence directly.
Consider

$$
\rho(r,\theta)
=
\frac{e^{-r^2}}{\pi}
\left[1+a r^2\cos(2\theta)\right],
\qquad a=0.35.
$$

The exact angular content is $m=0,\pm2$. The corresponding transforms are

$$
F_0(q)=\frac{e^{-q^2/4}}{2\pi},
\qquad
F_{\pm2}(q)=\frac{a q^2}{16\pi}e^{-q^2/4}.
$$

## Decompose and calibrate

```python
dec = make_decomposition()
hcal = HarmonicTransform.converge_parameters(
    dec,
    rtol=1.0e-4,
    atol=1.0e-12,
    q_tail_rtol=1.0e-3,
    method="simpson",
    verbose=False,
)
field = hcal.transform(dec)

hcal.plot_convergence()
field.plot_harmonics()
```

```{figure} ../_static/examples/anisotropic_harmonic_convergence.svg
:class: q2d-figure q2d-figure-standard
:alt: Harmonic-transform self-convergence diagnostics for the anisotropic transition field

Self-convergence diagnostics for the anisotropic harmonic transform.
```


```{figure} ../_static/examples/anisotropic_hankel.svg
:class: q2d-figure q2d-figure-standard
:alt: Harmonic Hankel transforms of the anisotropic transition field

The $m=0$ and $|m|=2$ components are transformed independently. Solid curves
show the analytic Hankel transforms of these components and open markers show
the corresponding QUARTIC2D values.
```

## Resolve the angular dependence

For $m,m'\in\{0,\pm2\}$, the interaction contains only angular differences
$0$, $\pm2$, and $\pm4$. For a real field these combine into

$$
V(\delta,\varphi_\delta)
=
A_0(\delta)
+A_2(\delta)\cos 2\varphi_\delta
+A_4(\delta)\cos 4\varphi_\delta.
$$

The interaction is calibrated on the complete angular scan before the
production values are evaluated:

```python
ical = Interaction.converge_parameters(
    vectors,
    field,
    field,
    yukawa,
    rtol=1.0e-4,
    atol=1.0e-12,
    method="gl4",
    verbose=False,
)
interaction = ical.interaction(vectors, field, field, yukawa)
ical.plot_convergence()
```

```{figure} ../_static/examples/anisotropic_interaction_convergence.svg
:class: q2d-figure q2d-figure-standard
:alt: Interaction self-convergence diagnostics for the anisotropic interaction

Self-convergence diagnostics for the anisotropic interaction calibration.
```


```{figure} ../_static/examples/anisotropic_orientation.svg
:class: q2d-figure q2d-figure-standard
:alt: Orientation dependence of the anisotropic interaction at three separations

The upper plot shows the interaction versus displacement angle at three separations. The lower plot shows the fractional angular modulation relative to each curve's angular mean. The anisotropy is strongest when the localized states substantially overlap and decreases with separation.
```

```{figure} ../_static/examples/anisotropic_pair_contributions.svg
:class: q2d-figure q2d-figure-compact
:alt: Harmonic-pair contributions to the anisotropic interaction

Grouping the returned `V_mm` array by $|m-m'|$ isolates the isotropic, twofold, and fourfold contributions at $\delta=1.5$. The black curve is their sum and reproduces the full interaction to floating-point precision.
```


```{literalinclude} ../_generated/examples/anisotropic_interaction.txt
:language: text
```

The complete calculation is `examples/anisotropic_interaction.py`.