# How ITensor simulate Kitaev honeycomb model?

+1 vote

Dear ITensor team,
Thans for your excellent software. I am beginner of ITensor and my questions maybe silly, sorry about that. I think ITensor can realized small size (3X4 unit cell) Kitaev honeycomb model. But I don't know how to modify example code. Please give me some advices or sample code if possible, Thank you very much !

Hi, thanks for the question. Are you interested in doing this for the C++ or Julia version? I’ll assume C++ and for Julia it’s actually quite similar.

We have an example code of doing a 2D system you can use here:
http://itensor.org/docs.cgi?vers=cppv3&page=formulas/2d_dmrg

The only difference from a 1D code in that example is that the AutoMPO part of the code runs over site numberings which are obtained from an array of “LatticeBond” objects that are returned from the triangularLattice function. You can see the code for this function here:
https://github.com/ITensor/ITensor/blob/v3/itensor/mps/lattice/triangular.h

Despite the apparent complexity of that function (triangularLattice) it is actually doing something really simple: it is making an array of pairs of integers, such as (n,n+1) and (n,n+Ny-1), etc. which define the sites on which the Hamiltonian terms should act, using a 1D ordering of the sites. This is because an MPS always has a 1D ordering of its sites, so to implement a 2D Hamiltonian you have to define some mapping of the 2D bonds (pairs of sites) to pairs of sites in the 1D, MPS ordering. The AutoMPO system takes care of the rest. Usually the 2D to 1D mapping used is just a “zig zag” pattern that goes up each column of the 2D system.

To implement the honeycomb lattice, then, all you have to do is come up with such a 2D to 1D mapping. I would recommend reshaping the honeycomb lattice first to a square lattice but with some bonds missing and perhaps some bonds added if necessary. Then on paper enumerate the sites from 1,2,...,N where N=Nx*Ny. Now make a function which generates the integer pairs of all the 2D Hamiltonian bonds following this enumeration and returns an array of structs (data structures) of these integer pairs.

A helpful practice step is to modify the 2D DMRG code linked above to just print out the values bnd.s1 and bnd.s2. Compare these values to the numberings of sites of the triangular lattice following the zig zag convention I mentioned above.

Best regards,
Miles

commented by (320 points)
Thanks, miles! I can convert 2D honeycomb lattice into 1D chain and I can use DMRG to calculate ground state and GS energy, right? I can convert square lattice to 1D chain and get correct result.
commented by (52.6k points)
Hi yes, you can. This is always how calculations of 2D systems with MPS and DMRG are done i.e. by converting to a 1D site ordering. So that's good if you already know how to do it for a square lattice and then doing it for a honeycomb lattice is very similar in spirit. Of course the interactions that result in the 1D system are different but the basic idea is the same.

Best,
Miles