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Sandbox Physics

Topological Matter 03 · competing Dirac masses

Haldane Honeycomb · Hall Phase at Zero Net Flux

Watch one transition in both the honeycomb lattice and its Brillouin zone. A sublattice potential gives the two valleys equal-sign masses, while complex next-nearest-neighbor hopping gives opposite signs. When one Dirac mass crosses zero, Berry curvature rearranges and the Chern number changes.

Interactive modelHaldane Honeycomb · Hall Phase at Zero Net Flux
Chern number CCC=+1C=+1
Valley mass mKm_K−1.25 t-1.25\,t
Valley mass mK′m_{K'}+1.25 t+1.25\,t
Direct gap Δ\Delta2.50 t2.50\,t

Physics tutorial

The Haldane model: Hall topology at zero net flux

BackgroundA honeycomb lattice has two inequivalent Dirac valleys. Sublattice potential and complex next-nearest-neighbor hopping gap them in different ways.

Why it mattersHaldane showed that quantum Hall topology does not require a uniform external field, establishing the minimal lattice picture of a Chern insulator.

Start with the essentials

Focus question
How can the Chern number be nonzero when net magnetic flux through every unit cell is zero?
One-sentence intuition
A local flux pattern can break time reversal while averaging to zero over a cell. The relative signs of the two valley masses determine whether their Chern contributions add or cancel.

Core mathematical model

Effective valley masses

mK=M−33 t2sin⁡ϕ,mK′=M+33 t2sin⁡ϕm_K=M-3\sqrt3\,t_2\sin\phi,\qquad m_{K'}=M+3\sqrt3\,t_2\sin\phi

MM contributes with the same sign at both valleys, while complex hopping contributes with opposite signs.

Chern-number criterion

C=12[sgn⁡(mK′)−sgn⁡(mK)]C=\frac12\left[\operatorname{sgn}(m_{K'})-\operatorname{sgn}(m_K)\right]

Opposite mass signs add to a nontrivial phase; equal signs cancel.

Phase boundary

∣M∣=33 ∣t2sin⁡ϕ∣|M|=3\sqrt3\,|t_2\sin\phi|

Exactly one Dirac mass vanishes on the boundary, closing the gap at one valley.

Common difficulties

Zero net flux is not zero flux everywhere

Typical misconceptionIf unit-cell flux sums to zero, magnetic structure cannot affect the bands.

Better mental modelSpatially patterned flux, or equivalent complex hopping, can still break time reversal; only its cell integral cancels.

The valleys are not independent integers

Typical misconceptionEach isolated Dirac valley contributes one full Chern number.

Better mental modelA low-energy valley carries a half-integer signed contribution; only both valleys in the full lattice produce a physical integer.

Run the experiment

  1. 01

    Use only sublattice mass

    Set t2t_2 to zero and vary MM.

    What to observe: Both valley masses keep the same sign, their Berry-curvature peaks cancel, and the Chern number stays zero.
  2. 02

    Turn on complex hopping

    Set M=0M=0 and move ϕ\phi to positive ninety degrees.

    What to observe: The valley masses acquire opposite signs and their curvature contributions add to a nonzero Chern number.
  3. 03

    Close only one valley

    Choose the gap-closing preset and nudge the sublattice mass.

    What to observe: One curvature peak sharpens and flips while the other valley remains gapped.