r"""
The Integer Quantum Hall Effect: Quantized Hall Conductance from Chern Numbers
====================================================================================

Von Klitzing, Dorda, and Pepper's 1980 discovery -- a two-dimensional
electron gas in a strong magnetic field develops a Hall conductance
quantized to extraordinary precision, :math:`\sigma_{xy} = \nu\,e^2/h` with
:math:`\nu` an exact integer -- is reproduced here on its lattice (Bloch)
incarnation, the Harper-Hofstadter model
(:func:`~physicskit.condensed.models.harper_hofstadter_hamiltonian`): a
square lattice threaded by a uniform flux :math:`p/q` per plaquette. At
flux :math:`1/3` the spectrum splits into three magnetic sub-bands; feeding
each one, in turn, into
:func:`~physicskit.condensed.topology.compute_chern_number` (the same TKNN
machinery used for the Haldane model) gives the exact integer Chern number
of that band, and summing the Chern numbers of every filled band below a
gap gives that gap's quantized Hall conductance directly -- the numerical
content of the TKNN formula that explains *why* von Klitzing's measurement
came out an integer at all.
"""

# %%
import matplotlib.pyplot as plt
import numpy as np

from physicskit.condensed.models import harper_hofstadter_hamiltonian
from physicskit.condensed.topology import compute_chern_number

# %%
# Three magnetic sub-bands at flux 1/3
# ------------------------------------------
# Odd ``q`` avoids the exact band touchings that occur for some even flux
# denominators, where individual-band Chern numbers become ill-defined.

p, q = 1, 3
k2_grid = np.linspace(0, 2 * np.pi, 200)
bands = np.array([np.linalg.eigvalsh(harper_hofstadter_hamiltonian(0.0, k2, p, q)) for k2 in k2_grid])

chern_numbers = compute_chern_number(lambda k1, k2: harper_hofstadter_hamiltonian(k1, k2, p, q), grid_size=30)
sigma_xy = np.cumsum(chern_numbers)
print(f"flux = {p}/{q}: band Chern numbers = {chern_numbers} (sum = {sum(chern_numbers)})")
print(f"quantized Hall conductance below each gap (units of e^2/h): {list(sigma_xy[:-1])}")

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(11, 4.3))
for n in range(q):
    ax1.plot(k2_grid, bands[:, n], color="steelblue")
ax1.set_xlabel(r"$k_2$")
ax1.set_ylabel("E")
ax1.set_title(f"Hofstadter sub-bands at flux {p}/{q} (k1 = 0)")

ax2.bar(range(1, q + 1), sigma_xy, color="firebrick")
ax2.axhline(0, color="black", lw=0.5)
ax2.set_xticks(range(1, q + 1))
ax2.set_xlabel("bands filled")
ax2.set_ylabel(r"$\sigma_{xy}$  ($e^2/h$)")
ax2.set_title("Quantized Hall conductance below each gap")
fig.tight_layout()

# %%
# Every gap's conductance is an exact integer, and the three bands' Chern
# numbers sum to exactly zero -- the full three-band Hilbert space, taken
# together, is topologically trivial, as it must be for any complete set of
# bands of a lattice Hamiltonian; the nontrivial physics is entirely in how
# that zero splits across the individual gaps.

plt.show()