Graduate Can't find eigenvalues for s-wave superconductor

[Frigorifico]

TL;DR

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

 

[Vlad Tepesh]

Comparison: Your Result vs. The Correct Expression

The correct expression for the eigenvalues λ is:

λ = ± [ |l|² + Δ² + ε'² ± 2 √( ε'² |l|² + Δ² (|l|² - l_x²) ) ]^(1/2)

Your derived eigenvalues (Equation 19) were:

λ = ± [ 4α²(sin²k_x + sin²k_y) + Δ² + ε'² + (J S_z)²
± 2 ( ε'² (4α²(sin²k_x + sin²k_y) + (J S_z)²)
+ Δ² (4α² sin²k_y + (J S_z)²) )^(1/2) ]^(1/2)

The second part of the inner product is: Δ² ( 4α² sin²k_y + (J S_z)² ).

The term in bold is Δ² l_y². The correct term should be Δ² (l_y² + l_z²) or Δ² (|l|² - l_x²). Your advisor is correct that the absence of k_x dependence in that Δ² term is the symmetry-breaking error.