Q062 · Complex hopping

Haldane Lattice · Turn the Phase

Tune the phase of next-neighbor hopping in a honeycomb lattice. Compare a topological gap with an ordinary sublattice gap, select the injection state and collect positions after propagation.

Clean bulk Chern number $\text{—}$

Sampled direct gap $\text{—}$

Model: outer two rows $\text{—}$

Recorded displacement $\text{—}$

01 / CHOOSE A STATE. FOLLOW ITS MOTION.

Put something in its path.

Play the packet. Turn on the defect and drag the coral ring. Then collect positions to check the motion.

Prepare fromSublattice mass

m/J

0

Next-neighbor phase

\phi/\pi

0.5

Next-neighbor coupling

t_2/J

0.2

Injection momentum

k_1/\pi

0.85

Defect potential

V/J

0

Packet, finite size & sampling

Packet width

\sigma_1

2.5

Ribbon width

N_2

10

Fixed onsite disorder

W/J

0

Trials per acquisition

M

400

Sampling seed

s

2468

The disorder pattern stays fixed. The seed changes recorded outcomes only. Save a sample before changing settings.

Compare the square-lattice model →

Spinless, noninteracting Haldane model with no net flux per cell. Complex hopping is an effective coupling, not a simulated magnetic field. Finite packets, corners and defects can mix energies and boundaries.

FINITE OPEN SAMPLE

t=0

Circle area: model cell probabilityRing: defect handle

Observation time

tJ/\hbar

No recorded trials.

02 / BUILD THE EVIDENCE

Did the position distribution move?

Initial recordsFinal recordsDashed: direct model

M=0

\text{—}

Each trial uses two independent preparations: one initial position and one position after evolution. These are not particle paths. The displacement and its standard error use recorded pairs only.

03 / CLEAN PERIODIC BULK

Add up the Berry flux.

\text{—}

Reciprocal coordinates follow the two marked primitive vectors. The square is a parameter torus, not a real-space detector or a Cartesian picture of the honeycomb Brillouin zone.

04 / CLEAN RIBBON

Choose a state in the gap.

$k_1/\pi$ · $E/J$

Lower edgeUpper edgeGray: bulk-like

Selected eigenstate

\text{—}

Periodic along the first primitive vector, open across the second. Color shows occupation in the outer two rows. A Gaussian envelope localizes this clean eigenvector; the resulting finite packet has a momentum and energy spread.

Save a comparison

Trial	Initial cell	Final cell

Latest twelve trials shown; CSV retains all records and settings. Up to six saved samples can be restored.

What is calculated, and what is not

Single particle, static Hamiltonian, two orbitals per cell, open finite boundaries. The defect acts equally on both orbitals in selected cells. The clean bulk and clean ribbon references exclude defects and disorder. A direct band gap is not necessarily an indirect insulating gap. No leads, scattering conductance, filled Fermi sea, interactions, loss or experimental detector response are modeled.

Chern numbers use gauge-invariant plaquette overlaps on two meshes. At a known gap closing or an unresolved mesh, the integer label is withheld. This check is a numerical diagnostic, not a rigorous convergence bound.

\text{—}

Asbóth et al. · Chapters 2 & 6 Fukui, Hatsugai & Suzuki · 2005 Haldane · 1988

Start with the essentials

Focus question: Reverse a hopping phase. Can the same edge carry the wave the other way?
One-sentence intuition: A gapped bulk constrains the net edge chirality. It does not force an arbitrary finite wave packet to stay on an edge forever.

Core mathematical model

Bulk Hamiltonian

H(\mathbf k)=d_0\mathbb I+\mathbf d\cdot\boldsymbol\sigma,\quad d_z=m-2t_2\sin\phi[\sin k_1-\sin k_2+\sin(k_2-k_1)]

Dimensionless reciprocal cycles follow the marked primitive vectors. Energies use the nearest-neighbor reference coupling. The real-space matrix is built from the same hopping convention.

Gauge-invariant flux

C=\frac{1}{2\pi}\sum_{\square}\arg(U_1U_2U_1^{-1}U_2^{-1})

Links are normalized lower-band spinor overlaps around oriented plaquettes. Two mesh sizes are compared. Known gap closings invalidate a separated-band invariant.

What the records estimate

\widehat{\Delta n_1}=\frac1M\sum_{r=1}^{M}(n_{1,r}^{\mathrm{final}}-n_{1,r}^{\mathrm{initial}})

Initial and final sites are sampled from independent preparations. The uncertainty is one standard error of the paired differences. It is not a Chern-number measurement.

Common difficulties

Three different systems

Typical misconceptionThe bulk number includes the displayed defect.

Better mental modelThe torus and ribbon references are clean. Only the finite sample includes the local potential and fixed disorder.

Real-space direction

Typical misconceptionThe sign of a Chern number alone says left or right.

Better mental modelDirection also requires a boundary orientation and a consistent reciprocal-coordinate convention. Opposite edges carry opposite motion.

Finite samples

Typical misconceptionAny decrease of outer-row probability disproves topology.

Better mental modelCorners, the opposite boundary, packet energy width and ordinary bulk components all matter. The displayed outer-row statistic excludes the vertical end walls.

Run the experiment

01
Predict the direction
Use the edge-channel preset. Inspect the colored ribbon state, then play the packet.
What to observe: The packet follows the finite Hamiltonian, not a prescribed edge path.
02
Measure a contrast
Acquire positions and save them. Reverse the chirality, acquire another sample at the same time, and compare the displacement.
What to observe: Histogram bars and estimates use records; dashed curves are direct model probabilities.
03
Try to break the channel
Turn on the defect and drag it into the packet path. Switch to the ordinary-gap preset, choose a bulk-like state, or narrow the strip.
What to observe: Compare edge weight, energy, finite-size mixing and side-wall warnings. A bright edge alone is insufficient evidence.