# Wave function ansatz for disclinated graphene with spin

I am currently investigating spin dynamics in disclinated graphene. More information about my approach can be found in my other post. I would like to know if my approach is somewhat correct to find the properties of the wave function.

For a given K-point, I have the following set of equations (obtained by acting the Hamiltonian on the wave function)

$m \tau\sin(\alpha)\psi_{A\uparrow} + m\tau\cos(\alpha)\psi_{A\downarrow} - \frac{i}{r}\left[ \left( \partial_r + \frac{1}{2}\right) \tau + \nu_{\tau}(n) \right] \psi_{B\uparrow} = E \psi_{A\uparrow}$

$m \tau\cos(\alpha)\psi_{A\uparrow} - m\tau\sin(\alpha)\psi_{A\downarrow} - \frac{i}{r}\left[ \left( \partial_r + \frac{1}{2}\right) \tau + \nu_{\tau}(n) \right] \psi_{B\downarrow} = E \psi_{A\downarrow}$

$-m \tau\sin(\alpha)\psi_{B\uparrow} - m\tau\cos(\alpha)\psi_{B\downarrow} - \frac{i}{r}\left[ \left( \partial_r + \frac{1}{2}\right) \tau - \nu_{\tau}(n) \right] \psi_{A\uparrow} = E \psi_{B\uparrow}$

$-m \tau\cos(\alpha)\psi_{B\uparrow} + m\tau\sin(\alpha)\psi_{B\downarrow} - \frac{i}{r}\left[ \left( \partial_r + \frac{1}{2}\right) \tau - \nu_{\tau}(n) \right] \psi_{A\downarrow} = E \psi_{B\downarrow}$

Where $\psi_{A\uparrow}$ is the wave function of a spin-up particle on the A sublattice, etc. $\nu_{\tau}(n)$ is just some function of $n$ which is any integer from $1$ to $6$ (depending on the topology of the system), and $\tau = \pm 1$ depending on which K-point we're looking at. $\alpha$ is the opening angle of the cone (which remains constant).

I use the following ansatz for the 4-component spinor:

$\left(\begin{array}{c}{\psi_{A\uparrow} \\ \psi_{A\downarrow} \\ \psi_{B\uparrow} \\ \psi_{B\downarrow}} \end{array} \right) = \frac{1}{r^a}\left(\begin{array}{c}{C_1 \\ C_2 \\ C_3 \\ C_4} \end{array} \right)e^{ikr}$, where $C_n$ are just coefficients (that I will try to find later).

My first goal is to find $a$, which should tell me how the function decays with $r$. So I substitute the ansatz back into the set of equations, and the first one gives me (after rearranging and acting the $\partial_r$ operator):

$(m\tau\sin(\alpha) - E)C_1 + m\tau\cos(\alpha)C_2 - \frac{i \tau}{r} \left(\frac{1}{2} + \nu_{\tau}(n) - a - rk \right) C_3 = 0$

Then, I group all the terms in $r^{-1}$ and say that if the entire expression is $0$, then the contribution of these terms must also be $0$.

$\left(\frac{-i \tau}{2r} - \frac{i \tau \nu_{\tau}(n)}{r} + \frac{i \tau a}{r} \right)C_3 = 0$

$a = \frac{1}{2} + \nu_{\tau}(n)$
$a = \frac{1}{2} - \nu_{\tau}(n)$
So it seems that the only way I can get a consistent $a$ is if $\nu = 0$... but this is not the case. What am I doing wrong? Or is my entire approach too clumsy for this kind of problem?