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 Incoming links: Anyon lattice of an Abelian Chern--Simons theory

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.

K = (0 M M 0), K^{- 1} = (0 M^{- 1} M^{- 1} 0) .

If both

q \in L

and

q^{'}

condense, choose their boundary excursions so that the two string operators cross once. Their commutator is their mutual full-braiding phase:

W_{q} W_{q^{'}} = e x p (2 πi q^{T} K^{- 1} q^{'}) W_{q^{'}} W_{q} .

Both operators act as the identity on every ground state, so consistency requires

q^{T} K^{- 1} q^{'} \in Z for every q = (q_{1}, q_{2}, 0, 0)^{T} \in L .

q^{'} = (q_{top}^{'}, q_{bot}^{'})

, this says

M^{- 1} q_{bot}^{'} \in Z^{2} .

l_{top} = M^{- 1} q_{bot}^{'}

and choose

l_{bot} = 0

. Then

K l = (0, q_{bot}^{'})^{T}

, and

u = q^{'} - K l = (q_{top}^{'}, 0)^{T} \in L .

Therefore every additionally condensable anyon has the form

q^{'} = u + K l, u \in L, l \in Z^{4} .

The

K l

term is a local particle in the anyon lattice of an Abelian Chern--Simons theory, so the condensate is maximal modulo local excitations, as required for a Lagrangian subgroup of Abelian anyons.