Does any lattice or lattice shape has a periodic boundary condition?

[qijiongli]

If not, then what are the conditions for us to construct a periodic boundary condition(PBC)?
If so, then please help me construct a PBC for the lattice shape in the attachments.

I want to ask that what lattice site m's left neighbor is and what lattice site i's down neighbor is.From the picture, both m's left and i's down neighbor is the BLUE site, but in the PBC, the neighbors are GREEN and RED correspondingly (right?). However, the GREEN site and the RED site are impossible to be the same site in the PBC(right?). So I'm confused with it.

thanks a lot

 

[Mentz114]

Are you joining adjacent edges or opposite edges ? Try rotating the lattice through 45 deg, working out the PBC and then rotating back.

 

[qijiongli]

Thank you very much for your help.

Do you mean that the direction of the PBC is along the direction of the BOUNDARY EDGE( the dotted GREEN line showed in the attachment)? Then both m's left and i's down neighbor is the BLUE site (K) on the opposite edge.If so, it seems to say that all PBCs are along the direction of the BOUNDARY EDGES?

 

[Mentz114]

I don't think I understand the problem. Have a look at the picture, I have shown a point in solid yellow and its neighbours in outlines yellow. The corners will have widely 'separated' neighbours.

 

[qijiongli]

thanks a lot!

I don't think I understand the problem. Have a look at the picture, I have shown a point in solid yellow and its neighbours in outlines yellow. The corners will have widely 'separated' neighbours.

For the above lattice shape, it can be constructed from the square lattice by connecting the next-nearest-neighbor sites.And each site's neighbors are easy to find by the original square lattice.

But I find that it is just a spatial case. And now I want to simulate a Ising model on the following lattice, whose boundary edges are the lines connecting one of the next-next-nearest-neighbors on the square lattice(dotted lines shows). The crossing points of the solid lines will be placed spins. Under the PBC, what the RED site's right and down neighbor will be?