Hi KevinH,
This may not answer all of your questions, but first of all I can tell you a bit more about what the data in a LatticeBond object means so you can be sure to write the third neighbor terms correctly.
A key thing is getting the s1 and s2 arguments (n,n+Ny and things like that) correct when making a LatticeBond. The meaning of these integers are the sites along a one-dimension “zig zag” path going through the lattice. This is because the primary use of LatticeBond is for making MPO tensor networks which are one-dimensional.
First, I’m providing a drawing of what I mean by a zig-zag path through an (Nx=4,Ny=4) square lattice. Note how the site numbers go up the first column (1,2,3,4), then jump to the bottom again (5,6,7,8) and so on.
I’ve drawn all of the first and second neighbor bonds, for the case of yperiodic=false and I drew two of the third-neighbor bonds that you want to make.
Now let’s take the first-neighbor bonds coming out of site 6 as an example. You can see that the one going up connects to a site number that is one higher, so then we would make the LatticeBond for this bond as
LatticeBond(n,n+1,x,y,x,y+1)
Then let’s take the bond going to the right out of site 6. This one connects to site 10 which is 6+4 or 6+Ny where Ny=4 in this example. So for that kind of bond, we make a LatticeBond like
LatticeBond(n,n+Ny,x,y,x+1,y)
So for third-neighbor bonds, the pattern you will need is something like n,n+2 for the vertical bonds going up and n,n+2*Ny for the horizontal bonds going to the right. But then in the code you can see additional care is needed to make these bonds only when there is actually a valid site to connect to. So for the example of first-neighbor bonds going up, we only make them when y < Ny otherwise we are at the top of the lattice and there are no more sites above to connect to.
I hope that helps explain most of the non-periodic bonds and how to make them.
For the y-periodic bonds, what you do is imagine another copy of the system above, with the same site numberings and everything. So in the drawing above (though I don’t show any periodic bonds explicitly there), if yperiodic=true then there would be a first-neighbor bond coming out of the top of site 4 connecting to the site “above” which is actually site 1 again. So we need a bond connecting site 1 to site 4 and that is what is made in the code for the case of if yperiodic && y == 1. You would do something similar for second- and third-neighbor bonds, drawing a second copy of the system above and working out the connections of the bonds that come out of the top of the lattice.’’
Here is a drawing of how to see the case of yperiodic=true and the first-neighbor bonds sticking out of the top:
Hope that helps!
Finally, I would suggest putting your function through a lot of tests. Count the number of bonds it returns: is it the right number for a bunch of different lattices? Maybe you can also run DMRG on an MPO made from your modified lattice function for a solvable case of your Hamiltonian (like say if it is a fermion system and there is a non-interacting limit) so you can compare your DMRG result to a numerically exact result.