ABSTRACT - DRUM - University of Maryland

Berry connection of this state then reads [34]
〈n 1 , . . . , n M |∂|n 1 , . . . , n M 〉 = 〈g|∂|g〉 +
⎛ ⎞
M∑
u
(u ∗ n i
, vn ∗ ni
i
)∂ ⎜ ⎟
⎝ ⎠ . (6.13)
v ni
i=1
So the difference between the Berry phase of a state with quasiparticles and the
ground state is simply the sum of “Berry phase” of the corresponding BdG wavefunctions.
Since the Berry phase of ground state |g〉 can be eliminated by a global
U(1) transformation, only the difference has physical meaning.
According to (6.13), the relevant term to be evaluated is
⎛ ⎞
u 01 + iu 02
(u ∗ 01−iu ∗ 02, v01−iv ∗ 02)∂
∗ ⎜ ⎟
⎝ ⎠ = 2Re (u∗ 1∂u 1 + u ∗ 2∂u 2 )+2iRe(u ∗ 1∂u 2 −u ∗ 2∂u 1 ),
v 01 + iv 02
(6.14)
where we have made use of the Majorana condition v = u ∗ .
The first term in
(6.14) vanishes because ∫ u ∗ ∂u must be purely imaginary. The second term has a
non-vanishing contribution to the total Berry phase. However, due to the localized
nature of zero-energy state, the overlap between u 1 and u 2 is exponentially small:
∫ T
0
dt Re(u ∗ 1∂ t u 2 − u ∗ 2∂ t u 1 ) ∼ e −|R 1−R 2 |/ξ . (6.15)
Therefore the Berry phase can be neglected in the limit of large separation R. This
completes our discussion of non-Abelian statistics. The above calculation can be
easily generalized to the case of many anyons.
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