clarify construction of U in w2_majorana/braiding.ipynb

[basnijholt]

In the discussion on how $U$ looks, the following is suddenly concluded:

And because it has to preserve fermion parity, it can only depend on their product, that is on the parity operator

I see that: parity must be conserved, [U, P_tot]=0, it should depend on the two operators $\gamma_n$ and $\gamma_m$, but fail to see why that it leads to $U$ depending on the product of $\gamma_n$ and $\gamma_m$.

A clarification would be in order.

[basnijholt]

I tried to derive the U_nm matrices myself, and I think you've made some minus mistake somewhere.

Derivation of exchange operator for four Majoranas

The fermionic modes expressed in Majorana modes (and visa versa) are:

c_{1}=\dfrac{\gamma_{1}+i\gamma_{2}}{2},\,c_{1}^{\dagger}=\dfrac{\gamma_{1}-i\gamma_{2}}{2},


\gamma_{1}=c_{1}+c_{1}^{\dagger},\,\gamma_{2}=\dfrac{c_{1}-c_{1}^{\dagger}}{i}=i\left(c_{1}^{\dagger}-c_{1}\right).


Which we can rewrite as




The operator U_{12}
  (Eq. [eq:U_nm]) has the product \gamma_{1}\gamma_{2}
  which in fermionic modes is

\gamma_{1}\gamma_{2}=\left(c_{1}+c_{1}^{\dagger}\right)i\left(c_{1}^{\dagger}-c_{1}\right)=i\left(c_{1}c_{1}^{\dagger}-c_{1}c_{1}+c_{1}^{\dagger}c_{1}^{\dagger}-c_{1}^{\dagger}c_{1}\right)=i-2ic_{1}^{\dagger}c_{1},


and gives for U_{12}


U_{12}=\tfrac{1}{\sqrt{2}}\left(1\pm\gamma_{1}\gamma_{2}\right)=\tfrac{1}{\sqrt{2}}\left(1\pm i\left(1-2c_{1}^{\dagger}c_{1}\right)\right).


If we let this operator act on the basis \left|00\right\rangle ,\left|01\right\rangle ,\left|10\right\rangle ,\left|11\right\rangle 
  as defined in Eq. [eq:basis] we get

U_{12}=\frac{1}{\sqrt{2}}\left(\begin{array}{cccc}
1\pm i & 0 & 0 & 0\\
0 & 1\pm i & 0 & 0\\
0 & 0 & 1\mp i & 0\\
0 & 0 & 0 & 1\mp i
\end{array}\right)
 which only has diagonal terms.

Finding U_{23}
 is slightly more work, because here we exchange two Majoranas out of two different pairs. We again define

c_{2}=\dfrac{\gamma_{3}+i\gamma_{4}}{2},\,c_{2}^{\dagger}=\dfrac{\gamma_{3}-i\gamma_{4}}{2},


\gamma_{3}=c_{2}+c_{2}^{\dagger},\,\gamma_{4}=i\left(c_{2}^{\dagger}-c_{2}\right)


and use Eq. [eq:majoranamodes] to write out Eq. [eq:U_nm]

U_{23}=\tfrac{1}{\sqrt{2}}\left(1\pm\gamma_{2}\gamma_{3}\right)=\tfrac{1}{\sqrt{2}}\left(1\pm i\left(c_{1}^{\dagger}-c_{1}\right)\left(c_{2}+c_{2}^{\dagger}\right)\right)=\tfrac{1}{\sqrt{2}}\left(1\pm i\left(c_{1}^{\dagger}c_{2}+c_{1}^{\dagger}c_{2}^{\dagger}-c_{1}c_{2}-c_{1}c_{2}^{\dagger}\right)\right)


Similar to the previous case, this operator acts on the basis \left|00\right\rangle ,\left|01\right\rangle ,\left|10\right\rangle ,\left|11\right\rangle 
  and in matrix form looks likeU_{23}=\frac{1}{\sqrt{2}}\left(\begin{array}{cccc}
1 & 0 & 0 & \mp i\\
0 & 1 & \mp i & 0\\
0 & \pm i & 1 & 0\\
\pm i & 0 & 0 & 1
\end{array}\right).