For
$$
K=\begin{pmatrix}0&2\\2&0\end{pmatrix},
\qquad
K^{-1}=\begin{pmatrix}0&1/2\\1/2&0\end{pmatrix},
$$
the vectors $q_e=(1,0)^T$ and $q_m=(0,1)^T$ have
$$
q_e^TK^{-1}q_e=q_m^TK^{-1}q_m=0,
\qquad
q_m^TK^{-1}q_e=\frac12.
$$
The <K-matrix> formula therefore gives bosonic self-exchange for $e,m$ and mutual full-braiding phase
$$
\exp(2\pi i q_m^TK^{-1}q_e)=e^{i\pi}=-1.
$$
Fusion adds vectors. Because
$$
2q_e=K(0,1)^T,
\qquad
2q_m=K(1,0)^T,
$$
both double fusions lie in $K\mathbb Z^2$ and braid trivially with every vector; hence they represent the vacuum in the <anyon lattice of an Abelian Chern--Simons theory>. Finally $q_f=q_e+q_m$ has $q_f^TK^{-1}q_f=1$, so its exchange phase is $e^{i\pi}=-1$. These are exactly the fusion and braiding data of the <surface-code anyon model>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 342 / 1 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-342/1/d/solution