ABSTRACT - DRUM - University of Maryland

considerations, we should have β 1λ = β 2λ ≡ β λ , up to exponentially small corrections.
Similar argument also applies to vortices in 2D p x + ip y superconductors.
As mentioned above, we will make the approximation that each excited state
can be treated independently. So we consider the effect of one of the excited states
first and omit the label λ temporarily. Again we write ˆd = 1 √
2
(ˆξ + iˆη). Without
loss of generality we also assume that β is real. We find from the solution of timedependent
BdG equation that
)
ˆγ 1 → 4β2 Et
sin2
E2 2 ˆγ 1 +
(1 − 4β2 Et
sin2 ˆγ
E2 2 + 2√ √
2βε sin 2 Et
2 2β sin Et
ˆξ + ˆη
2
E 2
E
)
ˆγ 2 → −
(1 − 4β2 Et
sin2 ˆγ
E2 1 − 4β2 Et
sin2
2 E2 2 ˆγ 2 + 2√ √
2βε sin 2 Et
2 2β sin Et
ˆξ + ˆη
E 2
E
ˆξ → 2√ (
) √ .
2βε sin 2 Et
2
(−ˆγ
E 2 1 + ˆγ 2 ) + cos Et + 8β2 sin 2 Et
2 2ε sin Et
ˆξ − ˆη
E 2 E
ˆη → 2√ 2β sin 2 Et
2
(−ˆγ 1 + ˆγ 2 ) +
E
Here again E = √ ε 2 + 4β 2 .
ε sin Et
E
ˆξ + cos Et ˆη
(6.45)
At first glance the physics here is very similar to what has been discussed
for local bound states: non-adiabatic transitions cause changes of fermion parity in
the ground state subspace. The crucial difference between local bound states and
a continuum of extended states is that, in the former case, local fermion parity is
still conserved as long as we count fermion occupation in the excited states while in
the latter, it is impossible to keep track of the number of fermions leaking into the
continuum so the notion of local fermion parity breaks down. These non-adiabatic
effects may pose additional constraints on manufacturing of topological qubits. Let’s
consider to what extent the braiding statistics is affected. A useful quantity to look
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