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

Solution

For

K = (0220), K^{- 1} = (0 1/2 1/2 0),

q_{e} = (1, 0)^{T}

and

q_{m} = (0, 1)^{T}

have

q_{e}^{T} K^{- 1} q_{e} = q_{m}^{T} K^{- 1} q_{m} = 0, q_{m}^{T} K^{- 1} q_{e} = \frac{1}{2} .

The K-matrix formula therefore gives bosonic self-exchange for

e, m

and mutual full-braiding phase

exp (2 πi q_{m}^{T} K^{- 1} q_{e}) = e^{iπ} = - 1.

Fusion adds vectors. Because

2 q_{e} = K (0, 1)^{T}, 2 q_{m} = K (1, 0)^{T},

both double fusions lie in

K 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}^{T} K^{- 1} q_{f} = 1

, so its exchange phase is

e^{iπ} = - 1

. These are exactly the fusion and braiding data of the surface-code anyon model.

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