r"""
The Anomalous Hall Effect: from the Anomalous Velocity to the Berry Phase
============================================================================

In 1954 Karplus and Luttinger found that the electrons of a ferromagnet
acquire an *anomalous velocity* transverse to an applied electric field,
a Hall current with no magnetic field. Half a century later it was
recognized as a Berry-phase effect (Jungwirth, Niu and MacDonald 2002;
Haldane 2004): a band's velocity picks up :math:`-\dot{\mathbf{k}}\times
\boldsymbol\Omega_n(\mathbf{k})`, and the intrinsic Hall conductivity is the
Berry curvature summed over the occupied states,

.. math::

    \sigma_{xy}(E_F) = \frac{e^2}{h}\,\frac{1}{2\pi}\int_{BZ} d^2k\,
    \sum_n f(E_n)\,\Omega_n(\mathbf{k})\, .

In a gap, every band is either full or empty, and the integral is the Chern
number: the *quantum* anomalous Hall effect. Inside a band, it is whatever
fraction of the curvature lies below the Fermi level -- not quantized, and
no longer zero even in a topologically trivial phase.

:meth:`~tbkit.kspace.KSpace.hall_conductivity` evaluates the Kubo formula
behind it at any Fermi level, for the Haldane model below (the sign is that
of :meth:`~tbkit.kspace.KSpace.chern_number`, see its docstring).
"""
import numpy as np
import matplotlib.pyplot as plt

import tbkit.lattices as lattices
from tbkit.bridges import finite_system
from tbkit.kspace import KSpace
from tbkit.plot import Plot


def haldane(M, t2=0.2):
    '''Haldane model: nearest-neighbour t = 1, i t2 on the second neighbours, mass +-M.'''
    hal = KSpace(lattices.honeycomb())
    hal.set_hopping([{'i': 0, 'j': 1, 'R': R, 't': 1.} for R in [(0, 0), (-1, 0), (0, -1)]])
    for R in [(0, 1), (-1, 0), (1, -1)]:  # a2, -a1, a1 - a2: 120 degrees apart
        hal.set_hopping([{'i': 0, 'j': 0, 'R': R, 't': 1j*t2}, {'i': 1, 'j': 1, 'R': R, 't': -1j*t2}])
    hal.set_onsite({'a': M, 'b': -M})
    return hal


# %%
# The lattice
# -----------
# The Haldane model on a patch of :math:`5\times4` cells
# (:func:`~tbkit.bridges.finite_system`), each bond drawn with a width
# proportional to its amplitude: the honeycomb lattice's two sublattices
# ('a' and 'b', in two colours), the nearest-neighbour hopping :math:`t`
# (thick) and the complex second-neighbour hoppings :math:`\pm it_2`
# within each sublattice (thin), which break time-reversal symmetry.

fig_lat = Plot(finite_system(haldane(0.3), (5, 4))).lattice(plt_hop=True, ms=12, figsize=(5.5, 4.5))


# %%
# The Hall conductivity across the spectrum
# ---------------------------------------------
# One diagonalization of a 150 x 150 mesh gives every Fermi level at once.
# The topological phase (:math:`M = 0.3`, below Haldane's
# :math:`M_c = 3\sqrt3\,t_2 \approx 1.04`) and the trivial one
# (:math:`M = 1.5`):

e_f = np.linspace(-3.6, 3.6, 721)
phases = {'topological, M = 0.3': haldane(0.3), 'trivial, M = 1.5': haldane(1.5)}
sigma = {name: hal.hall_conductivity(e_f, nk=150) for name, hal in phases.items()}

for name, hal in phases.items():
    en = hal.mesh_bands(150)
    gap = (en[:, 0].max(), en[:, 1].min())
    chern = hal.chern_number([0], 60)
    in_gap = (e_f > gap[0] + 0.02) & (e_f < gap[1] - 0.02)
    plateau = sigma[name][in_gap]
    print('{}: gap ({:.3f}, {:.3f}), C = {:.4f}, sigma_xy in the gap from {:.6f} to {:.6f} e^2/h'.format(
        name, *gap, chern, plateau.min(), plateau.max()))
    # the plateau is the Chern number
    assert np.allclose(plateau, chern, atol=1e-6)
    # an empty and a full model carry no Hall current
    assert sigma[name][0] == 0. and abs(sigma[name][-1]) < 1e-10

# Inside the bands the value is not quantized: at E_F = -1.2, in the
# upper part of the lower band, it is far from any integer in both phases
# -- nonzero even in the trivial phase.
e_band = -1.2
mid = np.argmin(np.abs(e_f - e_band))
for name in phases:
    print('{}: sigma_xy(E_F = {:.2f}) = {:.4f} e^2/h'.format(name, e_f[mid], sigma[name][mid]))
    assert 0.1 < sigma[name][mid] < 0.9

# sphinx_gallery_thumbnail_number = 2
fig, ax = plt.subplots(figsize=(7, 4.5))
for (name, s), color in zip(sigma.items(), ('C0', 'C3')):
    ax.plot(e_f, s, color=color, label=name)
ax.axhline(1., color='k', lw=0.5, ls=':')
ax.axhline(0., color='k', lw=0.5)
ax.set_xlabel('$E_F$')
ax.set_ylabel(r'$\sigma_{xy}$ ($e^2/h$)')
ax.set_title('Haldane model: anomalous Hall conductivity')
ax.legend()

# %%
# Where the curvature sits
# ---------------------------
# Filling the lower band adds the curvature of its states, from the bottom
# up. In the topological phase the curvature has one sign over the whole
# band, peaked near K and K' at its top, so :math:`\sigma_{xy}` rises
# monotonically to :math:`C = 1`. In the trivial phase the curvature at K
# and at K' have opposite signs (the mass term dominates the Haldane term
# at one valley and not at the other), so :math:`\sigma_{xy}` rises and
# then falls back to exactly 0 as the band fills: a Hall current carried
# by a metal whose filled bands are topologically trivial.

lower = e_f < -0.74
topo, triv = sigma['topological, M = 0.3'][lower], sigma['trivial, M = 1.5'][lower]
assert np.all(np.diff(topo) >= -1e-9)
peak = np.argmax(triv)
print('trivial phase: sigma_xy peaks at {:.3f} e^2/h for E_F = {:.2f}'.format(triv[peak], e_f[lower][peak]))
assert triv[peak] > 0.15 and np.all(np.diff(triv[peak:][e_f[lower][peak:] > -0.46]) <= 1e-9)

# %%
# Temperature and convergence
# ------------------------------
# A finite temperature smooths the band edges (the Fermi-Dirac tails
# empty the top of the lower band), and speeds up the convergence in the
# mesh: at :math:`T = 0.05`, a 60 x 60 mesh already agrees with 240 x 240
# to :math:`10^{-4}` in the bands.

hal = phases['topological, M = 0.3']
e_band = np.array([-2.2, -1.4, 1.2])
coarse = hal.hall_conductivity(e_band, temperature=0.05, nk=60)
fine = hal.hall_conductivity(e_band, temperature=0.05, nk=240)
print('T = 0.05, nk = 60 vs 240: max difference {:.2e}'.format(np.max(np.abs(coarse - fine))))
assert np.max(np.abs(coarse - fine)) < 1e-4

warm = hal.hall_conductivity(e_f, temperature=0.1, nk=150)
fig2, ax2 = plt.subplots(figsize=(7, 4.5))
ax2.plot(e_f, sigma['topological, M = 0.3'], label='$T = 0$')
ax2.plot(e_f, warm, label='$T = 0.1$')
ax2.set_xlabel('$E_F$')
ax2.set_ylabel(r'$\sigma_{xy}$ ($e^2/h$)')
ax2.set_title('Topological phase: the plateau at $C = 1$, smoothed by temperature')
ax2.legend()