- #1
Frigorifico
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- TL;DR Summary
- I'm close to the eigenvalues of this hamiltonian, but I can't quite get them right
I have this hamiltonian whose eigenvalues I wanna find. If you are curious it's equation 12 in this paper
I've attached a pdf showing what I did, really all that matters is the very end, the last matrix and equations 19 and 20, but in case a summary would be useful:
1.- Get the fourier transform
2.- Expand on sigma, group things that multiply the same pairs of operators
3.- Expand on k, use anticommutation to get the right order of the operators
(I know the spinors I'm using aren't quite the standard, but my advisor insisted I use those)
4.- Then I used some basic assumptions about square lattices and nearest neighbors to simplify a few terms and put them in the matrix
5.- Put the matrix in Wolfram|Alpha to get the eigenvalues
The problem is in the nested square root. We get a term like \Delta^2 sin^2(k_y) and there's nothing similar for k_x, but my advisor says that the eigenvalues MUST be symmetric on k because the matrix is
I mean, the matrix doesn't seem symmetric in k to me, since the imaginary term is always related to k_x and not k_y, but maybe that's wrong, I don't know
When my advisor pointed out other problems it wasn't always easy to fix, but it was relatively easy to see I had done something wrong. This time however I simply can't find any mistakes. I read what I did over and over, trying to find any mistakes, and sometimes I'll find a wrong sing or a missing index, and I fix it, but it never changes the final result
I'm all out of ideas, so I decided to ask here
I've attached a pdf showing what I did, really all that matters is the very end, the last matrix and equations 19 and 20, but in case a summary would be useful:
1.- Get the fourier transform
2.- Expand on sigma, group things that multiply the same pairs of operators
3.- Expand on k, use anticommutation to get the right order of the operators
(I know the spinors I'm using aren't quite the standard, but my advisor insisted I use those)
4.- Then I used some basic assumptions about square lattices and nearest neighbors to simplify a few terms and put them in the matrix
5.- Put the matrix in Wolfram|Alpha to get the eigenvalues
The problem is in the nested square root. We get a term like \Delta^2 sin^2(k_y) and there's nothing similar for k_x, but my advisor says that the eigenvalues MUST be symmetric on k because the matrix is
I mean, the matrix doesn't seem symmetric in k to me, since the imaginary term is always related to k_x and not k_y, but maybe that's wrong, I don't know
When my advisor pointed out other problems it wasn't always easy to fix, but it was relatively easy to see I had done something wrong. This time however I simply can't find any mistakes. I read what I did over and over, trying to find any mistakes, and sometimes I'll find a wrong sing or a missing index, and I fix it, but it never changes the final result
I'm all out of ideas, so I decided to ask here
