Recent content by emily1986

It seems like you have found the crux of the issue, but I don't quite understand yet. Would you mind explaining your previous post in more detail? For instance, what exactly is being represented by these sets of numbers? The (-1,0) represents the x and y components of a reciprocal lattice...

Yeah the graph is skewed for some reason. Here is a better version.

That's right. X corresponds to point (.5,0) and Y corresponds to (.5,.5)

Sure. Here is an example 9x9 matrix. The potential V can be any value. Units are atomic.

I don't understand what you are getting at. My potential in real space is just a product of two delta functions summed over all lattice vectors. The Fourier transform of a delta function will be a constant regardless of the dimension of my matrix.

How would changing my potential back to real space fix this? Wouldn't it just make things more complicated? My matrix would still have N dimensions so I wouldn't be removing any approximations.

By approximation do you mean limiting the size of my matrix?

The delta function in 2D is V=(∑iδ(x−ia))(∑jδ(y−ja)) The sums must be multiplied together in order to form a square lattice. Also, the potential does has an influence on the spectrum, but only at certain points. You can see from the picture I posted that there are bands gaps, just not at every...

Band structure is plotted as energy vs. reciprocal lattice vectors k. To find the eigenvalues as a function of k, the hamiltonian needs to be expressed in reciprocal space. This means that we take need to take the Fourier transform of the delta functions which results in some chosen constant V...

DrDu, I diagonalize the hamiltonian in reciprocal space so the off-diagonal terms are just constants.

I'm studying a two dimensional lattice with delta functions at the lattice points. When I diagonalized the hamiltonian I noticed that if I kept the dimension of my matrix less than 5 I would get band gaps at all the BZ boundaries. However, if I increased my matrix to a dimension 5 or higher...

After thinking about it, my question didn't really make sense. Thanks for your help, everything is much clearer now.

Thanks. This example really clicked with me. I believe I have some idea of what's going on now. You could take a flattened maps of the world, like one of these, and then drew a series of parallel vectors across some path. When the map is flattened, the vectors are all parallel. When you fold...

Just to make sure I understand, this means that we will never get a difference in phase as long as we travel along a geodesic? If this is the correct way of thinking about this, then the change in phase of our vector is merely an accumulation of error? We ignore higher order terms, but over...

Wouldn't you have to normalize your vector at some point? If you keep projecting a vector on some surface that isn't in the plane of the vector, the projection length will be shorter than the original vector, correct?