ABSTRACT - DRUM - University of Maryland

minor modifications in the details, should apply to the fractional quantum Hall
non-Abelian qubits [136], real p-wave superconducting systems based on solids [46]
and quantum gases [137], topological insulator-superconductor heterostructures [47],
and semiconductor-superconductor sandwich structures [49] and nanowires [59, 125].
Our results are quite general and are independent, in principle, of the detailed methods
for the anyonic braiding which could vary from system to system in details.
However, it is worthy to point out that the susceptibility of the systems to the nonadiabatic
effects is sensitive to the microscopic details, such as the size of the bulk
gap, the overlap between the various eigenstates which will become further clarified
later.
6.1 Quantum Statistics of Majorana Fermions
6.1.1 Quantum Statistics and Adiabatic Evolution
We first briefly review how quantum statistics is formulated mathematically in
terms of the adiabatic evolution of many-body wavefunctions, following a recent exposition
in Ref. [138]. Consider the general many-body Hamiltonian Ĥ[R 1(t), . . . , R n (t)]
where parameters {R i } represent positions of quasiparticles. We assume the existence
of well-defined, localized excitations which we call quasiparticles. At each moment
t, there exists a subspace of instantaneous eigenstates of Ĥ[R 1 (t), . . . , R n (t)]
with degenerate energy eigenvalues. Instantaneous eigenstates in the subspace are
labeled as |α(t)〉 ≡ |α({R i (t)})〉. We constraint our discussion in the ground state
subspace with zero energy eigenvalue.
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