Chern number calculator for 2D materials.

In

• Mathematica (v11.3)
• python (python3.7.2)

The Chern calculator is generated in chern.nb notebook. After then, a chern.m is generated for the next time use. One can either

• Evaluate chern.nb each time in current (working) kernal, or
• import chern from chern.m in the user's working notebook

to get the chern calculator in the working memory. Then use in two ways:

• chern[hk]
• chern[hk, discretized]

To be noticed:

• It accepts one required arg + one optional arg in the calculator, i.e. chern[hk, discretized]

  • required: hk, the Hamiltonian
  • optional: discretized, the discretization to use in calculation, default to 16. (e.g., discretized=16 stands for 16-by-16 discretization of the Brillioun zone).

• The Hamiltonian should be two-dimensional, in the sense that it should be either

  • Function that can accept two arguments (..#1..#2..&), or
  • Expression in terms of {kx, ky} (not a function, but when {kx, ky} replaced by numbers it should reduces to matrix with all entries numeric).

  These two form of Hamiltonian can both be accepted by our calculator.

• The Hamiltonian is not limited to two-band models (example: graphene, haldane etc.), but it should be Hermittian and periodic in ($2\pi\times2\pi$) brillioun zone.

• The Hamiltonian should not contain other unevaluated Symbols except {kx, ky}, which means that after we assign values to {kx, ky}, the Hamiltonian should be numeric in all its entries.

• It returns the list of Chern number for all bands, which should be EXACTLY INTEGERs. If it's not, the model possibly be gapless --- which is, by the theory, not allowed to calculate the Chern for gapless bands in the first place.

• See tutorial for the Mathematica version tutorial:mathematica
• See example of the Haldane model in Mathematica example:haldane.m

git clone https://github.com/atom-sun/chern.git
cd python/chern
pip install .
python examples/haldane.py

See tutorial:python for the script.

pip uninstall chern

• Haldane honeycomb 2band model

python example/haldane.py

or

python examples/haldane.py --m 0.5 --phi 0

to specify a set of $${m, \phi}$$ values of haldane model. Also in example:haldane.ipynb.

The algorithm is given by a Japanese group in chern(2005) this paper. Credits to them.

• Ning Sun
  • email: [email protected]