ABSTRACT - DRUM - University of Maryland

to non-unitary evolution induced by system-environment coupling. Our assumption
on the locality of the interactions in the system implies that
[Ĥt, ˆΣ ab ] = 0, [Ŝ, ˆΣ ab ] = 0. (5.4)
The time evolution of the expectation values of σ(t) is given by
d〈σ〉
dt
= d ∂σ dˆρ
Tr σ ˆρ = Tr ˆρ(t) + Tr σ
dt ∂t dt . (5.5)
With (5.4), it is straightforward to check that
Tr σ[Ĥ, ˆρ] = 0, Tr σ( Ŝ ˆρŜ† − 1 2 {Ŝ† Ŝ, ˆρ} ) = 0. (5.6)
Therefore we have
d〈σ(t)〉
dt
= Tr ∂σ(t) ˆρ(t) + Tr σ(t) ∂ ˆρ(t)
∂t
∂t
= ∂ t Tr [σ(t)ˆρ(t)]. (5.7)
As we have defined, ∂ t means that all changes come from the change in the basis
{ˆγ ai (t)}. Since after the braiding the system returns to its initial configuration, the
operators ˆγ ia undergo unitary transformations. So if the braiding starts at t = t i and
ends at t = t f , we have the simple result 〈σ(t i )〉 = 〈σ(t f )〉. However, the operators
ˆΓ(t f ) are different from ˆΓ(t i ). One can easily verify that the operators ˆΓ a satisfy
Ivanov’s rule [35, 111] under braiding of vortices a and b:
ˆΓ a → ˆΓ b , ˆΓ b → −ˆΓ a . (5.8)
And the transformation of 〈ˆσ〉 is identical to the case without any midgap states. In
conclusion, in terms of physically measurable quantities, the non-Abelian statistics
is well-defined in the presence of excited midgap states localized in the vortex core.
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