Beyond nearest interaction in continuous Hamiltonian.
Dear Kwant core team, Dirac or Majorana Fermions might be used in kwant with Discretizing continuous Hamiltonian on a square lattice. However the way that kwant descritizes the Dirac Hamiltonian gives rise to what is called doubling Fermions (even the desicritization preserves the hermicity ’symmetric derivative') . In this context, several approaches have been used to get rid of the extra Dirac cones in the Brillouin zone. Without going in deeper details, each of those proposed methods has some issue related to preserving either chiral or symplectic symmetry.To avoid this issue recent methods goes beyond nearest neigbour interaction and then my question come to raise. *How can we include second or third nearest interaction in kwant.continuum.discretize *. Best regards Adel
Adel Belayadi wrote:
Dirac or Majorana Fermions might be used in kwant with Discretizing continuous Hamiltonian on a square lattice. However the way that kwant descritizes the Dirac Hamiltonian gives rise to what is called doubling Fermions (even the desicritization preserves the hermicity ’symmetric derivative') . In this context, several approaches have been used to get rid of the extra Dirac cones in the Brillouin zone.
Hi Adel, what approaches do you have in mind specifically? I guess that you are aware of the “nogo” theorem that states that getting rid of Fermion doubling has a price. Recently, I noticed one practical way of getting rid of the “doublers” at the cost of going to the third dimension [1]. It seems to me that implementing something like this is beyond the scope of Kwant’s discretizer module, but it would be perhaps an interesting project to integrate it into Kwant in a generic way. The code of the above paper is available [2], but as far as I can see from having a quick look, it’s not a generic toolbox (like discretizer) but rather an implementation of the concrete model using Kwant.
Without going in deeper details, each of those proposed methods has some issue related to preserving either chiral or symplectic symmetry.To avoid this issue recent methods goes beyond nearest neigbour interaction and then my question come to raise. How can we include second or third nearest interaction in kwant.continuum.discretize .
Kwant’s discretizer in its current form is actually quite simpleminded: it works on a square lattice using the most basic finite difference discretization scheme (firstorder, central) [3]. It’s still very useful because it helps with tedious bookkeeping. One can see this by looking at the source: the inner loop that discretizes individual terms of the Hamiltonian consists of the function _discretize_summand() which in terms uses _differentiate() (both are in the file continuum/discretizer.py). If you see ways to extend the discretizer in general and useful ways, and are willing to work on it, I’m sure that people here (including me) would be happy to give advice. Cheers Christoph [1] https://arxiv.org/abs/2302.07024 [2] https://zenodo.org/record/7625331 [3] https://en.wikipedia.org/wiki/Finite_difference
participants (2)

Adel Belayadi

Christoph Groth