The TKNN Invariant: Exactly Quantized Chern Numbers

Thouless, Kohmoto, Nightingale, and den Nijs showed that the quantized Hall conductance is a topological invariant: the integral of the Berry curvature of the occupied Bloch bands over the Brillouin zone, now called the (first) Chern number \(C\). Because \(C\) can only change when a bulk gap closes, it is exact and disorder-independent – the moment band theory became topological band theory. compute_berry_curvature() and compute_chern_number() implement the Fukui-Hatsugai-Suzuki lattice discretization of this integral, returning exactly quantized integers for any gapped Bloch Hamiltonian.

import numpy as np

from physicskit.condensed.models import haldane_model
from physicskit.condensed.topology import compute_berry_curvature, compute_chern_number
from physicskit.condensed.visualizers import plot_berry_curvature

Any gapped two-band Bloch Hamiltonian – here, Haldane’s model

The FHS algorithm needs nothing but a function H(k1, k2); it makes no assumption about the model beyond a spectral gap.

H = lambda k1, k2: haldane_model(k1, k2, t=1.0, t2=0.2, phi=np.pi / 2, M=0.0)

Berry curvature concentrated near the gapped Dirac points

The per-plaquette Berry curvature of the lower band is sharply peaked near the honeycomb lattice’s two (former) Dirac points, where the time-reversal-breaking mass gap is smallest.

F = compute_berry_curvature(H, grid_size=30, band_index=0)
fig, ax = plot_berry_curvature(F)
ax.set_title("Berry curvature of the lower Haldane band")

Text(0.5, 1.0, 'Berry curvature of the lower Haldane band')

Integrating the curvature gives an exact integer

Summing the curvature over the whole Brillouin zone and dividing by \(2\pi\) returns an exact integer for every band – not approximately quantized, but exactly, by construction of the FHS link variables.

C = compute_chern_number(H, grid_size=30)
print("Chern numbers (per band):", C)
print(f"raw curvature sum / 2*pi for the lower band: {F.sum() / (2 * np.pi):.10f}")

Chern numbers (per band): [1, -1]
raw curvature sum / 2*pi for the lower band: 1.0000000000