Q062 · Complex hopping

Haldane Lattice · Turn the Phase

If you reverse the hopping phase in a Haldane lattice, does the edge carry waves the other way? Yes. In the Haldane model, reversing the phase of the next-nearest-neighbor hopping flips the sign of the Chern number of the topological gap, so the edge channel now carries waves the opposite way. A gap opened only by an energy difference between the two sublattices is ordinary and has no protected edge channel.

Turn the phase of the next-neighbor hopping on a honeycomb lattice and compare a topological gap with an ordinary gap from the sublattice offset. Choose which state to inject, let it propagate, then sample where it ends up.

Clean bulk Chern number $\text{—}$

Sampled direct gap $\text{—}$

Model: outer two rows $\text{—}$

Recorded displacement $\text{—}$

01 / Choose a state and 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 only the recorded outcomes. Save a sample before you change settings.

Compare the square-lattice model →

This is the spinless Haldane model without interactions, and the net flux through each cell is zero. The complex hopping is an effective coupling; no magnetic field is simulated. Finite packets, corners and defects can mix different energies and different edges.

Finite sample with open edges

t=0

Circle area: model cell probabilityRing: defect handle

Observation time

tJ/\hbar

No recorded trials.

02 / Collect position data

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 / Periodic bulk without defects

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 / Ribbon without defects

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 come from gauge-invariant plaquette overlaps on two meshes. At a known gap closing, or when the mesh cannot resolve the result, no integer is shown. 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: If you reverse the hopping phase in a Haldane lattice, does the edge carry waves the other way?
One-sentence intuition: A gapped bulk constrains the net chirality of the edge, that is, which way the edge states circulate overall. It does not keep an arbitrary finite wave packet on the 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)]

The momentum coordinates are dimensionless, and each runs through one full period along a marked primitive vector. Energies are in units of the nearest-neighbor coupling. The real-space lattice uses the same hopping convention.

Chern number as summed Berry flux

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

Each link is the normalized overlap of lower-band spinors at neighboring grid points, taken around an oriented plaquette. The sum is computed on two grid sizes and compared. Where the gap is known to close, the bands are no longer separate and this invariant is not defined.

Average displacement from sampled positions

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

The starting and final sites are sampled from independent preparations of the packet. The uncertainty is one standard error of the paired differences. This estimates how far the packet moved; it does not measure the Chern number.

Common difficulties

Three different systems

Typical misconceptionThe Chern number of the bulk already accounts for the defect I placed.

Better mental modelThe torus used for the bulk number and the ribbon used for the spectrum are both clean. Only the finite sample contains the local defect potential and the fixed disorder.

Direction in real space

Typical misconceptionThe sign of the Chern number by itself tells you whether the wave goes left or right.

Better mental modelA direction also needs the orientation of the edge and a consistent convention for the momentum coordinates. Opposite edges carry the wave in opposite directions.

Finite samples

Typical misconceptionIf the probability on the outer rows drops at all, the topological picture must be wrong.

Better mental modelCorners, the opposite edge, the packet's spread in energy and any ordinary bulk components all affect it. The outer-row fraction counts only the top and bottom rows and leaves out the vertical end walls.

Run the experiment

01
Predict the direction
Choose the edge-channel preset. Look at the colored state in the ribbon spectrum and guess which way it will go, then play the packet.
What to observe: The packet evolves under the Hamiltonian of the finite lattice. Any motion along the edge comes out of that evolution; no path is set in advance.
02
Compare the two directions
Acquire positions and save them. Reverse the chirality, acquire a second sample at the same time point, and compare the two displacements.
What to observe: The histogram bars and the estimates come from the sampled positions. The dashed curves are probabilities taken directly from the model.
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: Check the edge weight, the energy, the finite-size mixing and any side-wall warnings. A bright edge by itself does not show that the wave travels along it.